Secondary Steelmaking: Principles and Applications

SECONDARY STEELMAKING Principles and Applications Ahindra Ghosh, Sc.D. AICTE Emeritus Fellow Professor (Retired) Indian...

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SECONDARY STEELMAKING Principles and Applications

Ahindra Ghosh, Sc.D. AICTE Emeritus Fellow Professor (Retired) Indian Institute of Technology, Kanpur Department of Materials and Metallurgical Engineering

CRC Press Boca Raton London New York Washington, D.C.

©2001 CRC Press LLC

0264 Disclaimer Page 1 Thursday, November 2, 2000 11:07 AM

Library of Congress Cataloging-in-Publication Data Ghosh, Ahindra Secondary Steelmaking : Principles and Applications p. cm. Includes bibliographical references and index. ISBN 0-8493-0264-1 1. Steel. I. Title. TN730 .G48 2000 672—dc21

00-060865

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

© 2001 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-0264-1 Library of Congress Card Number 00-060865 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper


Dedication to Dr. G. P. Ghosh (Late) Prof. T. B. King (Late) Prof. A. K. Seal


Preface With the passage of time, customers who buy steel are becoming more and more quality conscious. In view of this, steelmakers are attempting to improve steel quality as a continuing endeavor. The product of steelmaking is liquid steel, which is then cast primarily via the continuous casting route. Liquid steel of superior quality should have a minimum of harmful impurities and nonmetallic inclusions, the desired alloying element content and casting temperature, and good homogeneity. The primary steelmaking furnaces, such as the basic oxygen furnace and electric arc furnace, are not capable of meeting quality demands. This has led to the growth of what is known as secondary steelmaking, which is concerned with further refining and processing of liquid steel after it is tapped into the ladle from the primary steelmaking furnace. Secondary steelmaking is a major thrust area in modern steelmaking technology and has witnessed significant advances in the last 30 years. Its scope is wide and includes deoxidation, degassing, desulfurization, homogenization, temperature control, removal, and modifications of inclusions, etc. This text consists of 11 chapters. The first chapter provides a brief overview of secondary steelmaking. Chapters 2 through 4 briefly review relevant scientific fundamentals, viz., thermodynamics, fluid flow, mixing, mass transfer, and kinetics relevant to secondary steelmaking. Chapters 5 through 10 deal with reactions, phenomena, and processes that are of concern in secondary steelmaking. Since some topics do not justify a full chapter for each, a chapter on miscellaneous topics (Chapter 8) provides coverage of these issues. The technology to manufacture what is known as clean steel calls for a variety of measures at different processing stages. An attempt has been made to present an integrated picture of this in Chapter 10. Mathematical modeling is an important component of process research nowadays. The basics as relevant to secondary steelmaking, along with application examples, are presented in Chapter 11. Although the present text deals primarily with principles and applications for the secondary steelmaking processes, it contains brief information on the processes and modern technological advances as well. Synthesis of science with technology is one of the objectives. The textbook style of writing has been adopted. Some examples and their solutions also have been included. References have been included at the end of each chapter. Hence, the author hopes that this text will be found useful not only by students and teachers, but also by steelmakers and research and development engineers interested in the field. Ahindra Ghosh


Acknowledgments The author gratefully acknowledges the contribution of his colleague Dr. D. Mazumdar, who wrote Chapter 11 and provided help in other aspects, and assistance provided by Dr. S. K. Choudhary, Dr. T. K. Roy, Mr. K. Deo, Mr. A. Sharma, and Ms. S. Ghosh at certain stages of preparation of the manuscript. Thanks are due to Mr. B. D. Biswas and Mr. J. L. Kuril for careful typing of the manuscript, Mr. A. K. Ganguly for tracing figures, and Dr. M.N. Mungole for helping with photographs. Financial assistance from the Centre for Development of Technical Education, Indian Institute of Technology, Kanpur, is gratefully acknowledged. Lastly, the work would not have been possible without the patience and cooperation of author’s wife Radha and other members of his family.


About the Author Professor Ahindra Ghosh was born at Howrah, West Bengal, India, in 1937. He studied for his B.E. degree in Metallurgical Engineering at Bengal Engineering College and received the degree from Calcutta University in 1958. Subsequently, he received his Sc.D. degree from the Massachusetts Institute of Technology in 1963, specializing in extractive metallurgy. He served as Research Associate at Ohio State University, U.S.A., from 1963–64. Since 1964, he has been with the Department of Materials and Metallurgical Engineering, Indian Institute of Technology, Kanpur, where he retired as Professor in June, 2000, and is currently an Emeritus Fellow of All India Council of Technical Education. During this period, he also has spent short periods at the Imperial College, London, as well as the Massachusetts Institute of Technology as a visiting scientist; and at Metallurgical and Engineering Consultants, Ranchi, and Tata Research Development and Design Centre, Pune, as an advisor. Professor Ghosh has guided many research students and scholars. He has to his credit 2 books and about 75 original research publications in reviewed journals. He also has delivered invited lectures at many conferences and has published several review papers in conference proceedings, etc. For the last three decades, his principal interest has been in the theory of metallurgical processes in ironmaking and steelmaking, with specific emphasis on sponge ironmaking, secondary steelmaking, ingot casting, and continuous casting. In these endeavors, Professor Ghosh also had significant interaction with industry in addition to his work with metallurgical fundamentals. He is also involved in basic research in solidification of metals and high-temperature oxidation of alloys. Professor Ghosh has served as an editor of the Transactions of the Indian Institute of Metals and as a member or advisor for many professional activities. In recognition, he has been elected a Fellow of the Indian National Academy of Engineering for his distinguished contribution to engineering.


Contents

Preface About the Author List of Symbols with Units Chapter 1

Introduction

1.1 History of Secondary Steelmaking 1.2 Trends in Steel Quality Demands 1.3 Scientific Fundamentals 1.4 Process Control References Chapter 2

Thermodynamic Fundamentals

2.1 Introduction 2.2 First and Second Laws of Thermodynamics 2.3 Chemical Equilibrium 2.4 ∆G0 for Oxide Systems 2.5 Activity–Composition Relationships: Concentrated Solutions 2.6 Activity–Composition Relationships: Dilute Solutions 2.7 Chemical Potential and Equilibrium 2.8 Slag Basicity and Capacities References Appendix Appendix Appendix Appendix

2.1 2.2 2.3 2.4

Chapter 3

Flow Fundamentals

3.1 Basics of Fluid Flow 3.2 Fluid Flow in Steel Melts in Gas-Stirred Ladles References Appendix 3.1 Chapter 4 4.1 4.2 4.3

Mixing, Mass Transfer, and Kinetics

Introduction Mixing in Steel Melts in Gas-Stirred Ladles Kinetics of Reactions among Phases


4.4 Mass Transfer in a Gas-Stirred Ladle 4.5 Mixing vs. Mass Transfer Control References Appendix 4.1 Chapter 5

Deoxidation of Liquid Steel

5.1 Thermodynamics of Deoxidation of Molten Steel 5.2 Kinetics of the Deoxidation of Molten Steel 5.3 Deoxidation in Industry References Appendix 5.1 Chapter 6

Degassing and Decarburization of Liquid Steel

6.1 Introduction 6.2 Thermodynamics of Reactions in Vacuum Degassing 6.3 Fluid Flow and Mixing in Vacuum Degassing 6.4 Rates of Vacuum Degassing and Decarburization 6.5 Decarburization for Ultra-Low Carbon (ULC) and Stainless Steel References Chapter 7

Desulfurization in Secondary Steelmaking

7.1 Introduction 7.2 Thermodynamic Aspects 7.3 Desulfurization with Only Top Slag 7.4 Injection Metallurgy for Desulfurization References Chapter 8

Miscellaneous Topics

8.1 Introduction 8.2 Gas Absorption during Tapping and Teeming from Surrounding Atmosphere 8.3 Temperature Changes of Molten Steel during Secondary Steelmaking 8.4 Phosphorus Control in Secondary Steelmaking 8.5 Nitrogen Control in Steelmaking 8.6 Application of Magnetohydrodynamics References Chapter 9

Inclusions and Inclusion Modification

9.1 Introduction 9.2 Influence of Inclusions on the Mechanical Properties of Steel 9.3 Inclusion Identification and Cleanliness Assessment 9.4 Origin of Nonmetallic Inclusions 9.5 Formation of Inclusions during Solidification 9.6 Inclusion Modification References Chapter 10

Clean Steel Technology

10.1 Introduction ©2001 CRC Press LLC

10.2 Summary of Earlier Chapters 10.3 Refractories for Secondary Steelmaking 10.4 Tundish Metallurgy for Clean Steel References Chapter 11 Modeling of Secondary Steelmaking Processes Dipak Mazumdar, Ph.D. 11.1 Introduction 11.2 Modeling Techniques 11.3 Modeling Turbulent Fluid Flow Phenomena 11.4 Modeling of Material and Thermal Mixing Phenomena 11.5 Modeling of Heat and Mass Transfer between Solid Additions and Liquid Steel 11.6 Numerical Considerations 11.7 Concluding Remarks References


List of Symbols with Units* a

specific surface area

m–1

a

acceleration vector

ms–2

A

area

m2

ai

activity of component i in a solution

—

Bi

Biot number

C

specific heat

CD

drag coefficient

Ci

concentration of component i in solution

— –1

–1

–1

Jmol K , Jkg K–1 — kg m–3

slag capacity for component i d

—

diameter

m

Di

molecular diffusivity of species i

m s–1

Dt

turbulent diffusivity

m2 s–1

E

internal energy, activation energy

Jmol–1

energy input in a gas-stirred bath

J

e

j i

Eu F F, F FD fi Fr Frm

2

first-order interaction coefficient describing influence of solute j on fi

—

Euler number

—

view factor

—

force, force vector

N

drag force

N

activity coefficient of solute i in a solution in 1 wt.pct. standard state

—

Froude number

—

modified Froude number

— ms–2

g

acceleration due to gravity

G

Gibbs free energy

Jmol–1, J

Gibbs free energy at standard state

Jmol–1, J

finite change in G, GO

Jmol–1, J

GO ∆G, ∆GO Gi m

partial molar Gibbs free energy of component i in solution

Gi

partial molar Gibbs free energy of mixing of component i in solution

Gr

Grasshof number

H hi

enthalpy

Jmol–1 J mol–1 — J mol–1, J

height of liquid bath

m

activity of solute i in a solution in 1 wt.% standard state

—

* — indicates a dimensionless quantity, mol means gram · mole.


I

intensity of turbulence

—

i, j

tensor arrays

—

Ji,x

flux of species i along x-coordinate

mol · m–2 s–1 Jkg–1

k

turbulent kinetic energy per unit mass of fluid

kc

specific chemical rate constant

ki

empirical rate constant for first-order process

ms–1

mass transfer coefficient for species i

ms–1

km,i K KM l Li m, M m˙ , M˙

ms–1, etc.

equilibrium constant

—

deoxidation constant for deoxidizer M

—

equilibrium constant involving metal M

—

a length parameter

m

partition coefficient of species i between two phases

—

mass

kg

rate of change of mass

mi

mass fraction of component i

Mi

molecular/atomic mass of species i

kg · s–1 — g · mol–1

Mo

Morton number

—

Nu

Nusselt number

—

P

pressure

pi

partial pressure of component i in a gas mixture

atm, Nm–2 atm

Pe

Peclet number

—

Pr

Prandtl number

—

q

quantity of heat

Q

r

volumetric gas flow rate

m s

—

heat flow rate

W

radial coordinate

m

universal gas constant vessel radius circulation rate of metal in vacuum degassing degree of desulfurization

S

–1

activity quotient

reaction rate R

J 3

entropy

mol · s–1 Jmol–1 K–1, m3 atm mol–1 K–1 m kg s–1 — Jmol–1 K–1, JK–1

Danckwerts surface renewal factor

s–1

source term in differential equation

as applicable

Sc

Schmidt number

—

Sh

Sherwood number

—

t

time


s

tc

circulation time

s

te

exposure time

s

mixing time

s

tr

residence time

s

T

temperature

tmix

K

velocity, velocity vector

ms–1

ux

velocity along x-coordinate

ms–1

V

volume

w

quantity of work done

u, u

m3 J

We

Weber number

—

Wi

weight percent of component i in a solution

—

rectangular coordinates

m

Xi

mole/atom fraction of component i in solution

—

Y

degree of mixing

—

x, y, z

slag rate in desulfurization

kgt–1

Greek Symbols α

volume fraction of gas in gas-liquid mixture

—

αi

a-function for component i in a solution

—

Pauling electronegativity

—

γi

activity coefficient of component i in a solution

—

o i

Henry’s law constant for solute i in binary solution

γ

Γ

general symbol for diffusivity of heat, mass, momentum

δ

partial differential

∆

finite change of a quantity

— 2 –1

ms

as applicable

δc,eff

effective concentration boundary layer thickness

m

δu,eff

effective velocity boundary layer thickness

m

rate of dissipation of energy

W

emissivity of surface

—

ε εm

rate of dissipation of energy per unit mass

Wkg–1

θ

angle

λ

geometrical scale factor

λt

turbulent thermal conductivity

Λ

optical basicity

µ

viscosity

N sm –2

µi

chemical potential of component i in a solution

J mol–1

ν

kinematic viscosity

ρ

density


degree, radian — W m–1 s–1 —

m2s–1 kg m–3

σ

surface/interfacial tension Stefan–Boltzmann constant

τ

general symbol for dependent variable in differential equation

Other Symbols []

metal phase

()

slag/oxide phase

∇

gradient of a scalar quantity

Some Physical Constants acceleration due to gravity (g) = 9.81 ms–2 atmospheric pressure, 1 atm

= 760 mm Hg = 1.013 × 105 Nm–2 = 1.013 bar

gas constant (R)

= 8.314 × J · mol–1 k–1 = 82.06 × 10–6 m3 · atm · mol–1 k–1


W m–2 K–4 N m–2

shear stress dimensionless residence time

φ

N m–1

— as applicable

1

Introduction

1.1 HISTORY OF SECONDARY STEELMAKING Prior to 1950 or so, after steel was made in furnaces such as open hearths, converters, and electric furnaces, its treatment in a ladle was limited in scope and consisted of deoxidation, carburization by addition of coke or ferrocoke as required, and some minor alloying. However, more stringent demands on steel quality and consistency in its properties require controls that are beyond the capability of the steelmaking furnaces. This is especially true for superior-quality steel products in sophisticated applications. This requirement has led to the development of various kinds of treatments of liquid steel in ladles, besides deoxidation. These have witnessed massive growth and, as a result, have come to be variously known as secondary steelmaking, ladle metallurgy, secondary processing of liquid steel, or secondary refining of liquid steel. However, the name secondary steelmaking has more or less received widest acceptance and hence has been adopted here. Secondary steelmaking has become an integral feature of modern steel plants. The advent of the continuous casting process, which requires more stringent quality control, is an added reason for the growth of secondary steelmaking. Steelmaking in furnaces, also redesignated now as primary steelmaking, is therefore increasingly employed only for speedy scrap melting and gross refining, leaving further refining and control to secondary steelmaking. There are processes, such as vacuum arc refining (VAR) and electroslag remelting (ESR), that also perform some secondary refining. However, they start with solidified steel and remelt it. Hence, by convention, these are not included in secondary steelmaking. Harmful impurities in steel are sulfur, phosphorus, oxygen, hydrogen, and nitrogen. They occupy interstitial sites in an iron lattice and hence are known as interstitials. The principal effects of these impurities in steel are loss of ductility, impact strength, and corrosion resistance. When it comes to detailed consideration, each element has its own characteristic influence on steel properties. These will be briefly mentioned in subsequent chapters associated with them. Oxygen and sulfur are also constituents of nonmetallic particles in steel, known as inclusions. These particles are also harmful to properties of steel and should be removed as much as possible. Carbon is also present as interstitial in iron lattice. However, unlike the other interstitials, it is generally not considered to be harmful impurity and should be present in steel as per specification. But, today, there are grades of steel in which carbon also should be as low as possible. Historically, the Perrin process, invented in 1933, is the forerunner of modern secondary steelmaking. Treatment of molten steel with synthetic slag was the approach. Vacuum degassing (VD) processes came in the decade of 1950–1960. The initial objective was to lower the hydrogen content of liquid steel to prevent cracks in large forging-quality ingots. Later on, its objective also included lowering of nitrogen and oxygen contents. Purging with inert gas (Ar) in a ladle using porous bricks or tuyeres (IGP) came later. Its primary objective was stirring, with consequent homogenization of temperature and composition of melt. It offered the additional advantage of faster floating out of nonmetallic particles. It was also found possible to lower carbon to a very low value in stainless steel by treatment of the melt with oxygen under vacuum or along with an


argon stream. This led to development of vacuum-oxygen decarburization (VOD) and argon-oxygen decarburization (AOD). Synthetic slag treatment and powder injection processes of molten steel in a ladle were started in late 1960s and early 1970s with the objective of lowering the sulfur content of steel to the very low level demanded by many applications. This led to the development of what is known as injection metallurgy (IM). Injection of powders of calcium bearing reagents, typically calcium silicide, was also found to prevent nozzle clogging by Al2O3 and lead to inclusion modification, which are of crucial importance in continuous casting as well as for improved properties. The growth of secondary steelmaking is intimately associated with that of continuous casting of steel. Up to the decade of the 1960s, ingot casting was dominant. Now, most of world’s steel is cast via the continuous casting route. The tolerance levels of interstitial impurities and inclusions are lower in continuous casting than in ingot casting, and this has made secondary refining more important. For good quality finished steel, proper macrostructure of the casting is also important, in addition to the impurity level. This requires close control of the temperature of molten steel prior to teeming into the continuous casting mold. In traditional pitside practice, without ladle metallurgical operations, the temperature drop of molten steel from furnace to mold is around 20–40°C. An additional temperature drop of about 30–50°C occurs during secondary steelmaking. Continuous casting uses pouring through a tundish, causing some further drop of 10–15°C. Therefore, provisions for heating and temperature adjustment during secondary steelmaking are very desirable. This has led to the development of special furnaces such as the vacuum arc degasser (VAD), ladle furnace (LF), and ASEA-SKF ladle furnace. These are very versatile units, capable of performing various operations. There have been further developments in this direction recently. Efforts are being made to install one unit only and even then achieve a flexible manufacturing program. Table 1.1 summarizes the features of various processes.1 It shows the capabilities of each. However, it is to be borne in mind that some versatile units of today are really combinations of several processes. For example, some modern vacuum degassers have provisions for oxygen blowing and powder injection. Hence, good desulfurization and decarburization also can be attained in them. It ought to be noted here that a significant fraction of sulfur in blast furnace hot metal is removed by pretreatment in a ladle during transfer to steelmaking shop. Similarly, phosphorus is removed primarily in a basic oxygen furnace and to some extent during pretreatment of hot metal. Shima2 has reviewed the development of steelmaking technology in Japan, dealing broadly with these.

1.2 TRENDS IN STEEL QUALITY DEMANDS The world steel market was somewhat stagnant and did not witness significant growth during the decade of the 1980s. Scholey3 has discussed this with special emphasis on Europe. Table 1.2 presents world consumption of steel products in 1990 and predictions of the same for A.D. 2000 as per statistics prepared by the International Iron and Steel Institute (IISI).4 Table 1.2 shows that predicted growth of consumption is large in Asia but either insignificant or negative in other countries. However, according to IISI, lack of tonnage growth does not indicate stagnancy. With continuous improvement in quality, less and less quantity of steel is being consumed for the same applications. If this point is taken into consideration, then there has been remarkable progress in steel technology on the quality front, and also improved yield. Figure 1.1 shows the change in product mix in the U.S.A. from 1925 to 1990, as compiled by Stubble.5 It demonstrates a massive shift in favor of sheet and strip; so much so that, in 1990, more than 50% of the product was in this shape, as compared to about 20% in 1925. This is the worldwide trend also. It is to be recognized that this shift was technologically possible to a large extent due to improvement in steel quality through secondary steelmaking. Near net shape casting, which is commercially expected in the near future, will require even more stringent control of impurities and inclusions. ©2001 CRC Press LLC

FIGURE 1.1 Product mix in the United States of America: 1925 vs. 19805 (reprinted by permission of Iron & Steel Society, Warrendale, PA, U.S.A.).

TABLE 1.1 Various Secondary Steelmaking Processes and Their Capabilities Processes Item Desulfurization Deoxidation

VD

VOD

IGP

IM

VAD

LF

ASEA-SKF

minor

minor

minor

yes

yes

yes

yes

yes

yes

yes

yes

yes

yes

yes

Decarburization

minor

yes

no

no

no

no

yes

Heating

no/yes

yes*

no

no

yes

yes

yes

Alloying

minor

yes

minor

minor

yes

yes

yes

Degassing

yes

yes

no

no

yes

no

yes

Homogenization

yes

yes

yes

yes

yes

yes

yes

Achieving more cleanliness (i.e., less inclusions)

yes

yes

yes

yes

yes

yes

yes

Inclusion modification

no

no

minor

yes

yes

yes

yes

*chemical heating only Source: data primarily from Ref. 1.

With the passage of time, customers are demanding better and better quality steels, which means 1. 2. 3. 4. 5.

fewer impurities more cleanliness (i.e., lower inclusion content) more stringent quality control, i.e., less variation from cast to cast microalloying to impart better properties (for plain carbon and low alloy steels) better surface quality and homogeneity

The above demands, combined with other requirements such as (a) the need for cost reduction in view of competition from polymers etc., (b) environmental pollution control, and (c) a relatively stagnant world steel market, pose enormous challenge to the steelmaking community. Before any ©2001 CRC Press LLC

TABLE 1.2 World Steel Consumption Millions of Metric Tonnes of Steel Products 1990 (actual) North America

2000 (predicted)

96.5

99

115.5

117

Japan

92.6

85

Latin America

22.3

35

P. R. China

54.9

80

Other Asian countries

78.6

120

European Community

Africa Former USSR and Eastern Europe Others Total

8.6

9

148.5

100

36.4

41

653.9

686

Source: data from Ref. 4.

secondary steelmaking treatment, the lowest levels of impurities attainable with present-day practices, including a metal pretreatment facility, would be approximately as follows: sulfur: phosphorus: nitrogen: hydrogen: carbon: oxygen:

100 ppm 20 ppm 40 ppm 5 ppm 400 ppm variable

Upon traditional deoxidation in a ladle, oxygen can be brought down to lower than 30 ppm. Thus, the minimum total of S + O + P + N would be about 200 ppm, and including carbon about 600 ppm. Changing demands on quality may be illustrated with the example of line pipe steel for North Sea gas.6 Maximum ppm in Steel, by Element Year 1983 1990 Long-term

C

S

P

H

N

O

400–600

20

150

–

100

–

10

15

–

35

20

S + O + P + N = 45 ppm

Ramaswamy6 has reviewed the subject and has outlined some of the quality requirements of line pipe steel for sour gas applications, steels for offshore platforms, bearing steels, steel for the rod and wire industry, and for power plant rotors. Figure 1.2 shows the trends in residuals attained by Japanese Steel Works.7 A special mention may be made of a recent spurt in demand for ultralow carbon steel (C < 30 ppm or so) for the manufacture of thin sheets by cold rolling with continuous annealing for automobiles. These steels not only have ultra-low C but have other residuals also at ultra-low levels, e.g., N < 15 ppm, S < 10 ppm, P < 15 ppm, H < 2 ppm. In addition, inclusion contents are also drastically lower as compared to regular steels. An expansion ©2001 CRC Press LLC

FIGURE 1.2 Minimum residual levels in steel in Japan.7

of the market for fine steels by approximately a factor of three has been predicted from 1985 to year 2000 by Japan’s fine steel study group.7 As shown in Table 1.1, all kinds of secondary steelmaking operations are capable of yielding steels with more cleanliness. Inclusions are generally harmful to the mechanical properties and corrosion resistance of steels. The choice of deoxidation practice combined with proper stirring is one of the measures to remove inclusions. However, a more serious source of harmful inclusions (i.e., larger sizes) is erosion of refractory lining. In addition, reaction of lining with the melt is a source of impurity at such low impurity levels. Therefore, the success of secondary steelmaking processes is intimately linked with the development or use of newer refractory materials such as those high in alumina, zircon, magnesia, dolomite, etc. The cleanliness consciousness has increased to such an extent that trials are going on for filtering molten steel through ceramic filters to remove nonmetallic inclusions. The technique is still in the experimental stages. Inclusion modification is one of the techniques to render inclusions less harmful to the properties of steel. Injection of calcium into the melt is done for this purpose. Sometimes, rare earths are also employed.

1.3 SCIENTIFIC FUNDAMENTALS The application of scientific fundamentals is an important contributing factor to the progress of secondary steelmaking technology. This has been possible due to growth of applied sciences, including metallurgical sciences, and their application. The laws of thermodynamics had been well laid out by the turn of the 19th century. However, their application to high-temperature systems had to wait because of a lack of thermochemical data. Collection of such data had started on a modest scale by the beginning of this century. The pace accelerated as years went by, and it began on a really massive scale after the 1940s. By about 1970, fairly reliable data were available on most of the systems and reactions of interest in pyrometallurgy. Equilibrium process calculations call for experimental data on activity vs. composition relationships in liquids that may be broadly grouped into metallic solutions, SiO2-based slag solutions, etc. Most of these solutions are multicomponent ones. The development of metallurgical thermodynamics called for new techniques to handle them. The participation of renowned physical chemists other than metallurgists made these possible. ©2001 CRC Press LLC

Kinetics is a late comer as compared to the thermodynamics of pyrometallurgical reactions. Scientific investigations were started after 1950. However, they picked up quickly and, for the last three decades, the field has been pursued vigorously. As a result, kinetics of pyrometallurgical reactions and processes is a subject of engineering science in its own right. Knowledge already available in chemical engineering has been instrumental in its development. It had been recognized several decades back that lack of proper mixing in the liquid bath adversely affects the efficiency of steelmaking processes. Many investigations have been carried out on mixing, especially in the last two decades. Again, mixing, mass transfer, and phase dispersions depend on fluid flow in the bath. Such a flow is turbulent in steelmaking processes. Turbulence is a very complex phenomenon. Scientists and engineers in a variety of disciplines are concerned with the solution of problems involving turbulent flow. Experimental investigations on fluid flow and mixing at steelmaking temperatures are difficult. In this connection, water modeling (i.e., cold modeling or physical modeling) has contributed significantly to our understanding of these aspects. Here, water typically simulates liquid metal. Transparent perspex or glass vessels allow flow visualization. Similarity criteria have been employed to various extents. A quantitative approach in the area of fluid flow, mixing, and mass transfer is based on fluid mechanics—especially as related to turbulent flow. Such computations involve computer-oriented numerical methods. Considerable advances have been made in this direction—so much so that these are being employed for interpretation of results, design, and process prediction.

1.4 PROCESS CONTROL A variety of process control measures must be adopted if desirable benefits are to be obtained from secondary steelmaking. It is neither possible nor necessary to list all these. Only some will be briefly mentioned below in view of their special significance.

1.4.1

IMMERSION OXYGEN SENSOR

Dissolved oxygen in molten steel is a key scientific as well as quality parameter in secondary steelmaking. Its measurement has been possible due to development of immersion oxygen sensor over the last two decades or so. It is actually an oxygen concentration cell with a solid electrolyte (typically ZrO2 + MgO or ZrO2 + CaO variety). The EMF of the cell allows estimation of dissolved oxygen content through thermodynamic relations. Since the signal is electrical and obtained within 15 s of immersion, it is widely used to measure and control oxygen in molten steel. Through thermodynamic relations, it allows us also to know the soluble aluminum content of steel, which again is another valuable piece of information that steelmakers desire. An immersion oxygen sensor has also been widely employed in a variety of scientific and technological investigations related to deoxidation reactions and behavior of oxygen at different stages of steelmaking. The pioneering contribution of Kiukkola and Wagner (1956), who first set up such a cell for thermodynamic measurements in laboratory, is to be recognized. Iwase and McLean8 have reviewed sensors for iron and steelmaking. It may be noted that immersion electrochemical sensors for other elements, such as silicon and phosphorus, are being developed but are essentially based on oxygen sensors.

1.4.2

SOME OTHER PROCESS CONTROL MEASURES

Gases such as oxygen and nitrogen are picked up from surrounding air during teeming and pouring. This can significantly increase gas contents in liquid steel. Unless this is prevented, most secondary steelmaking operations will not provide any benefit. For continuous casting, the use of either a submerged nozzle or shrouding of the nozzle by inert gas is the solution. For ingot casting, this is difficult to practice; in this case, management of teeming stream is the strategy. ©2001 CRC Press LLC

For efficient deoxidation, synthetic slag treatment, and injection processes, it is essential to prevent too much slag from primary steelmaking furnaces from being carried over into ladles. All steelmakers know the associated difficulties if we wish to avoid lots of metal being left out untapped. Therefore, through considerable efforts, significant advances have been made in techniques of tapping with a very low quantity of slag. It is then modified by suitable additions for further processing. In traditional ladles, refractory lined stoppers were employed for flow control during teeming through the nozzle. A major development has been slide gate, which is superior as a flow control device. The traditional method of addition of aluminum to liquid steel as ingots or shots makes the efficiency of aluminum deoxidation poor as well as irreproducible, leading to serious control problems. The technology of mechanized feeding of aluminum wire is a significant improvement in this connection. Today, many plants have facilities for feeding wires consisting of Ca or CaSi powders clad in steel as well. This is an alternative to the injection of these powders into the melt by injection metallurgy techniques. Fruehan9 has reviewed some of these topics in a concise fashion. Of course, advances in instrumentation as well as the use of computers have contributed significantly, as in all other fields. The modern installations employ extensive computer control. Increasing efforts are being made to employ software based on mathematical models, as well as expert control systems by application of artificial intelligence techniques. The review by Bozenhardt and Shafer provides some information.10 Good process control is not possible without fast and reliable chemical analysis techniques. There have been considerable advances in this direction. Emphasis is also being given to in-situ analysis without the need of transferring samples to a separate analytical unit. These advances are being utilized not only in secondary steelmaking but in other areas as well. Stirring is an integral part of secondary steelmaking. It is done primarily by gas purging. However, electromagnetic stirring is an alternative. Electromagnetic (EM) stirring during induction furnace melting of steel has been known from the beginning of 20th century. A major application of EM stirring from the 1970s was in continuous casting. EM devices are also being employed increasingly in recent years in the secondary steelmaking area not only for stirring, but also for flow control, slag control, etc. This offers many advantages, including flexibility in the nature and intensity of fluid motion.

REFERENCES 1. Srinivasan, C.R., in Proc. of National Seminar on Secondary Steelmaking, Tata Steel and Ind. Inst. Metals, Jamshedpur, 1989, p. 15. 2. Shima, T., in Proc. of the 6th Iron and Steel Cong., the Iron and Steel Institute Japan, Nagoya, 1990, Vol. 3, p. 1. 3. Scholey, R., in Proc. 69th Steelmaking Conference, ISS-AIME, Washington, D.C. 1986, p. V. 4. McAloon, T.P., Iron and Steelmaker, Dec. 1992. 5. Stubbles, J.R., in Steelmaking Conference Proceedings, ISS–AIME, vol. 75, 1992, p. 132. 6. Ramaswamy, V., in Srinivasan, p. 71. 7. Y. Adachi, in Shima, Vol. 5, p. 248. 8. Iwase, M., and, McLean, A., in Shima, Vol. 1, p. 521. 9. Fruehan, R.J., Ladle Metallurgy, ISS-AIME, Warrendale, PA, U.S.A., 1985. 10. Bozenhardt, H.F., and Shafer, J.D., Iron and Steel Engineer, June 1993, p. 41.


2

Thermodynamic Fundamentals

2.1 INTRODUCTION Metallurgical thermodynamics belongs to the field of chemical thermodynamics, which is employed to predict whether a chemical reaction is feasible. It also allows quantitative calculation of the state of equilibrium of a system in terms of composition, pressure, and temperature, as well as determination of heat effects of reactions and processes. Laws of thermodynamics are exact. Therefore, calculations based on them are, in principle, sound and reliable. There are standard books dealing with the basics of chemical-cum-metallurgical thermodynamics.1,2 The following is a very brief review only, with special emphasis on topics of relevance to secondary steel making. All reactions and processes tend towards the thermodynamic equilibrium. If sufficient time is allowed, then attainment of equilibrium is possible. Steelmaking reactions and processes are very fast due to their high temperatures. As a result, some of these have been found to approach equilibrium closely within the short processing time. Examples of this in secondary steelmaking are provided in subsequent chapters. Therefore, a full knowledge of thermodynamics is required for the understanding, control, design, and development of metallurgical processes. A discussion of thermodynamics requires precise definitions of some terms. For example, a system is defined as any portion of the universe selected for consideration. The rest of the universe outside the system is known as surroundings. An open system exchanges both matter and energy, a closed system exchanges only energy, and an isolated system exchanges nothing with the ambient. The state of a system is defined at any instant by specifying all state variables and properties such as temperature, pressure, volume, surface tension, viscosity, etc. A complete listing of all the properties of a state is superfluous, because many of them are often mutually interdependent. Pressure (P), volume (V), and temperature (T) are the most common state variables. When a state is described by such variables, assumptions are made implicitly or explicitly. For example, if there is no mention of magnetic field intensity, then it implies that the magnetic effect is insignificant. Similarly, if surface tension forces are ignored, then there is the underlying assumption that surface energy is negligible. Again P, V and T are interrelated. For example, for an ideal gas, PV = nRT

(2.1)

where n is the number of moles occupying volume V. In general, for a thermodynamic substance, if V/n = v, where v is molar volume, then v = f ( P, T )

(2.2)

where the R.H.S. of Eq. (2.2) denotes some appropriate function of P and T. Therefore, the state of a thermodynamic substance can be defined by any two of the above three variables, provided that the only work done is against pressure.


It should be noted that, among these variables, V is a property that depends on the amount of substance under question. On the other hand, P and T are not dependent on mass. A variable such as volume, which depends on the amount of substance in the system, is known as an extensive variable. Variables such as temperature, pressure, etc., which do not depend on mass, are known as intensive variables. It goes without saying that, if an equation contains a variable denoting an extensive property, then there must also be a term denoting mass or mol as in Eq. (2.1). If the latter is missing, there is an implicit assumption that the extensive property is per mass/mol, such as v, in Eq. (2.2), which becomes an intensive property. Thermodynamic relations among intensive properties are of more general validity. A state can be characterized by state variables only when the system has come to equilibrium with respect to those variables. Then and only then can the state be correctly defined in terms of these variables. This also implies that the magnitudes of related intensive properties throughout the system are the same. Thermodynamic equilibrium necessarily requires the attainment of mechanical, thermal, and chemical equilibria. Mechanical interaction of a system with the surroundings is most commonly in the form of pressure. Therefore, in the absence of a field of force, mechanical equilibrium generally means pressure equilibrium, i.e., uniform pressure throughout the system. Similarly, thermal equilibrium implies uniformity of temperature, and chemical equilibrium, in a broad sense, means uniformity of chemical potential for all species in the system. At chemical equilibrium, there is no tendency for further reaction. It is possible that the system is at equilibrium with respect to some variables but not some others. This is known as partial equilibrium, and thermodynamics is capable of handling this as well. However, a precondition for handling any chemical equilibrium is the establishment of mechanical and thermal equilibria.

2.2 FIRST AND SECOND LAWS OF THERMODYNAMICS 2.2.1

STATEMENT

OF THE

FIRST LAW

dE = δq – δw, for an infinitesimal change where

(2.3)

dE = an infinitesimal change in the internal energy (E) of the system δq = an infinitesimal quantity of heat absorbed by the system δw = an infinitesimal quantity of work done by the system

For a finite change, ∆E = q – w

(2.4)

The first law of thermodynamics is nothing but a statement of the law of conservation of energy. Careful experiments have revealed that q is not equal to w for many processes, apparently violating the law of conservation of energy. To make these findings conform to the law of conservation of energy, the concept of internal energy (E) was proposed. Internal energy is the energy stored in the system. In chemical thermodynamics, E is taken as the energy of atoms and molecules. Experiments have proved that E is a state property. Ignoring other fields of forces, and for a closed system, E = f ( P, T ) ©2001 CRC Press LLC

(2.5)

Again, we do not know or cannot measure the absolute value of E. All we can measure are changes in E (∆E for a finite change, dE for an infinitesimal change).

2.2.2

ENTHALPY

AND

SPECIFIC HEAT

Every substance, in a given state at a certain temperature, has a characteristic value of heat content or enthalpy (H). By definition, H = E + PV

(2.6)

and hence is a state property. Differentiating, dH = ( dE + P dV ) + V dP = δq + Vdp

(2.7)

at constant P,

dP = 0

and therefore,

dH = δq

(2.8)

or,

∆H = q

(2.9)

Therefore, at constant P, q is related to the change of state property (H) and hence can be calculated from the initial and final states only. We do not have to consider the path. This is a great simplification. Most of the processes are carried out approximately at constant pressure. Even though the pressure fluctuates, it does not introduce any significant error if q is taken as ∆H. Molar ∆H (i.e., ∆H per mole) values for a variety of processes have been determined experimentally and are available in thermodynamic data books. Using them, heat requirements of processes can be calculated, and process heat balances can be worked out. The molar specific heat of a substance is the heat required to raise temperature of one mole of a substance by 1 Kelvin. Specific heat at a constant pressure is given by ∂q ∂H C p =  ------- =  -------  ∂T  p  ∂T  p

(2.10)

Experimental Cp values are expressed as functions of temperature as c C p = a + bT + -----2 T

(2.11)

where T is temperature in Kelvins, and a, b, and c are empirical constants. Values of Cp may be found in standard thermodynamic data books. Table 2.1 presents values of Cp and enthalpies of transformations for iron.

2.2.3

STATEMENT

OF THE

SECOND LAW

The second law of thermodynamics is based on universal experience. It may be stated in a variety of ways. For the purpose of the ensuing discussions, the following statement would be useful: “Spontaneous processes, i.e., processes taking place without any outside intervention, such as diffusion, free expansion of a gas, heat flow, etc., are not thermodynamically reversible.” ©2001 CRC Press LLC

TABLE 2.1 Specific Heats and Enthalpies of Transformation for Iron Transformation Reaction

Temperature (K)

Specific heat (Cp) (Jmol–1 k–1)

Enthalpy change (∆Η) (J mol–1)

Feα →Feβ

1033

Feα = 17.49 + 24.769 × 10–3 T

+ 5105

Feβ →Fe

1187

Feβ = 37.66

+

670

Feγ →Feδ

1665

Feγ = 7.70 + 19.5 × 10–3 T

+

837

Feδ →Feliq

1809

Feδ = 28.284 +7.53 × 10–3 T

+13807

Feliq = 35.4 + 3.74 × 10–3 T Source: F.R. DeBoer, R. Boom, W.C.M. Mattens, A.R. Miedema, and A.K. Niessen, Cohesion in Metals—Transition Metal Alloys, Cohesion and Structure Series, North-Holland, Amsterdam (l988).

2.2.4

REVERSIBLE PROCESSES

Heat and work are not properties of state. They are energy in transition, and thus the magnitude of q and w would depend on the path that the process takes in going from an initial to the final state. That is why δq and δw rather than dq and dw have been employed in Eq. (2.3). This is a great mathematical limitation. Hence, considerable effort has been made by thermodynamicists to examine under what conditions δw and δq can be related to state properties. Obviously, the path has to be defined. This is where the concept of reversible processes has assumed importance. In a reversible process, the system is displaced from equilibrium infinitesimally and then allowed to attain a new equilibrium, then again displaced infinitesimally and so on. Thus, it may be defined as “the hypothetical passage of a system through a series of equilibrium states.” A reversible process is very slow and impractical. No practical process is reversible in strict sense. However, the concept is very useful and a key one in thermodynamics. The term reversible has been coined because such a process can be reversed along the same path without leaving any permanent change in the system or its surrounding.

2.2.5

ENTROPY (S)

A system may go from an initial to the final state by any of the innumerable paths available to it. These paths would be mostly irreversible. Some of them, however, would be or can be treated as reversible. It can be proved on the basis of Carnot’s Cycle that the quantity δqrev/T is dependent only on the initial and final states, where δqrev refers to δq along a reversible path. The following relationship has been thereby proposed. δq

rev ----------∑ T A→B

= S B – S A = ∆S

(2.12)

or, in differential form, δq rev ----------- = dS T

(2.13)

where A and B designate the initial and final states respectively, and S is a state function (i.e. state property) known as entropy. ©2001 CRC Press LLC

According to the third law of thermodynamics, the entropy of a substance at zero Kelvins (i.e., absolute zero), and at complete internal equilibrium, is zero if there is perfect order in that state, e.g., in perfectly crystalline solids, but not in metastable vitreous phases. This allows evaluation of absolute values of entropy, which are also tabulated in the thermodynamic data books. Example 2.1 Calculate (a) entropy (So) of 1 mole of liquid iron at 2000 K, and (b) enthalpy change ( ∆H ° 2000 – ∆H ° 298 ) in heating 1 mole of iron from 298 to 2000 K. Note that ( S° 298 ( α – Fe ) = 27.15J mol K ) –1

–1

Solution (a) Entropy of liquid iron at 2000 K, i.e. 1033

S

o 2000

(l) = S

o 298

( α – Fe ) +

∫

C p(α) ∆H a → B --------------- dT + -----------------+ T 1033

298 1665

∫

∫

C p(β) ∆H B → γ -+ --------------- dT + ----------------1187 T

1033

C p(γ ) ∆H γ → δ -+ ------------- dT + ----------------1665 T

1187

1187

1809

∫

C p(δ) ∆H -------------- dT + -----------m- + 1809 T

1665

2000

∫

C p(l) ------------- dT T

(E1.1)

1809

Substituting the values of Cp and ∆H for various transformations from Table 2.1. 1033

S

o 2000 ( l )

= 27.15 +

17.49 5105 –3 ------------- + 24.769 × 10 dT + -----------T 1033

∫ 298

1187

+

∫

37.66 670 ------------- dT + ------------ + T 1187

1033

1665

∫

7.7 837 –3 ------- + 19.5 × 10 dT + -----------T 1665

1187

1809

+

∫

2000

28.284 13807 –3 ---------------- + 7.531 × 10 dT + --------------- + T 1809

1665

∫

35.4 –3 ---------- + 3.745 × 10 dT T

1809

–1

= 105.5 J mol K

–1

(Ans.)

(b) Enthalpy change in heating 1 mole of iron from 298 to 2000 K 1033 o

o

H 2000 ( l ) – H 298 ( s ) =

∫

1187

C p ( α ) dT + ∆H α → b +

298

1665

+

∫ 1187

∫

C p ( β ) ( dT + ∆H β → γ )

1033

1809

C p ( γ ) dT + ∆H γ → δ +

∫

2000

C p ( δ ) dT + ∆H m +

1665

Substituting the values of Cp and ∆H from Table 2.1, ©2001 CRC Press LLC

∫ 1809

C p ( l ) dT

(E1.2)

1033

H

o 2000 ( l )

–H

o 298 ( s )

∫

=

1187

[ 17.49 + 24.769 × 10 T ] dT + 5105 + –3

298

∫

37.66 dT + 670

1033

1665

+

∫

[ 7.7 + 19.5 × 10 T ] dT + 837 –3

1187

1809

+

∫

[ 28.284 + 7.531 × 10 T ] dT + 13807 –3

1665

2000

+

∫

[ 35.4 + 3.745 × 10 T ] dT –3

1809

= 24971 + 5105 + 5800 + 670 + 16972 + 837 + 5957 + 13807 + 7489 = 821788 J mol

–1

(Ans.)

2.2.6

COMBINED EXPRESSIONS

OF

FIRST

AND

SECOND LAWS

For a reversible process and a closed system, if the only work done is against pressure, then combining the Eqs. (2.3) and (2.13) we obtain Eq. (2.14), i.e., dE = T dS – P dV

(2.14)

dH = dE + P dV + V dP

(2.15)

dH = T dS + V dP

(2.16)

again,

Combining Eqs. (2.8) and (2.14),

2.3 CHEMICAL EQUILIBRIUM 2.3.1

FREE ENERGY

AND

CRITERION

OF

EQUILIBRIUM

In Eq. (2.14), internal energy E is expressed as a function of entropy S and volume V, both of which are independent state variables. Experimental control of temperature and pressure is easier. Gibbs, therefore, defined a new function G, where G = E + PV – TS = H – TS

(2.17)

G is known as Gibbs free energy, which is a state property from the definition of G. Differentiating Eq. (2.17), dG = dE + P dV + V dP – T dS – S dT ©2001 CRC Press LLC

(2.18)

For a closed system, and for a reversible process (or at equilibrium), if the only work done is against pressure, then combining Eqs. (2.14) and (2.18), dG = V dP – S dT

(2.19)

at equilibrium, under constant temperature and pressure, (dG)P,T = 0, i.e. (∆G)P,T = 0, for a finite process

(2.20)

For an irreversible (spontaneous) process, it can be shown that dG < V dP – SdT

(2.21)

Therefore, at constant temperature and pressure, a spontaneous, (i.e., natural or irreversible) process would occur if (dG)P,T < 0, i.e. (∆G)P,T < 0, for a finite process

(2.22)

Thus, the Gibbs free energy provides us with a criterion to predict equilibrium or possibility of occurrence of a spontaneous process at constant T and P.

2.3.2

ACTIVITY, EQUILIBRIUM CONSTANT

Consider the following isothermal reaction, which occurs at a temperature T. aA + bB = lL + mM

(2.23)

Here A, B, L, and M are general symbols of chemical species and a, b, l, and m denote the number of moles of each. The word isothermal implies that the initial temperature at the beginning of the reaction and the final temperature (when equilibrium is reached) are the same. It is not necessary that the temperature remain unchanged throughout the progress of the reaction. The free energy change for reaction represented by Eq. (2.23) may be expressed as ∆G = ( 1G L + mG M ) – ( aG A + bG B )

(2.24)

where G i is the partial molar free energy of the species i. The standard state is the stablest state of the pure substance at the same temperature (T) and at a pressure of 1 atmosphere. The standard state could thus be a pure solid or liquid or ideal gas at 1 atmosphere of pressure. The magnitude of a variable for any standard state is indicated by a superscript o. It can be shown that o

G i – G i = RT ln a i o

where, G i = free energy of species i at its standard state f a i = -----oi = activity of species i at partial molar free energy G i fi fi = the fugacity of i at the state under consideration o = the fugacity at its standard state fi ©2001 CRC Press LLC

(2.25)

For ideal gases, fugacity equals partial pressure, expressed in atm (i.e., standard atmosphere = 760 mm Hg). By definition, activity ai is 1 when species i is at its standard state. If all reactants and products are at their standard states, then for the reaction of Eq. (2.23), ∆G = ( lG L + mG M ) – ( aG A + bG B ) o

o

o

o

o

(2.26)

where ∆G is the standard free energy change of reaction represented by Eq. (2.23) at temperature T. Combining Eqs. (2.24) through (2.26), o

aL ⋅ aM -------------a b a A ⋅ aB 1

∆G = ∆G + RT ln o

m

(2.27)

or, ∆G = ∆G + RT ln Q

(2.28)

aL ⋅ aM Q = --------------a b a A ⋅ aB

(2.29)

o

where 1

m

Q is called the activity quotient. Equation (2.27) has been derived assuming an isothermal condition, i.e., the same temperature for reactants and products. If it is further assumed that the reaction is isobaric, i.e., the initial and final pressures are the same, and also that thermodynamic equilibrium prevails, then ∆(G)P,T = 0 from Eq. (2.20). Combining this with Eq. (2.28), ∆Go = –RT ln[Q]e = –RT ln K

(2.30)

where K is the value of the activity quotient at equilibrium. K is known as the equilibrium constant. Equation (2.27) is the basis for prediction of the feasibility of reactions. A reaction is spontaneous or feasible if ∆(G)p,T is negative. It is impossible when ∆(G)p,T is positive. Equation (2.30) is used to calculate the equilibrium condition of a reaction. Thermodynamic predictions and calculations can be made if the following conditions are satisfied: 1. The process should take place isothermally (i.e., the initial and final temperature should be the same) and the temperature should be known. 2. The standard free energy change of reaction (∆Go) should be available. 3. Activity versus composition relations for all species involved should be known. Since changes in pressure as encountered in metallurgy do not affect thermodynamic properties significantly, the condition that P should be constant is of no importance in situations we normally encounter. Hence, P = constant restriction shall be omitted from here on. ©2001 CRC Press LLC

2.4 ∆G0 FOR OXIDE SYSTEMS In secondary steelmaking, we primarily encounter formation or decomposition of inorganic oxides. Therefore, a brief write-up is presented on free energies of oxide systems. The standard free energies of formation reactions, representing formation of compounds from the elements, are now known for all inorganic compounds of interest in secondary steelmaking. o These are called standard free energies of formation ( ∆G f ) . A number of books carry compilations o of such data.3–6 Some values of ∆G f for compounds of interest in secondary steelmaking are presented in Appendix 2.1. Consider formation of an oxide from the elements represented by the following general reaction: 2X 2 ------- M + O 2 ( g ) = ---M X O Y Y Y

(2.31)

where M denotes a metal. X and Y are general symbols for oxide stoichiometry. Traditionally, free energy data shown in diagrams would be for a reaction such as Eq. (2.31), where the formation reaction involves only one mole of oxygen. This would make it convenient to compare the data for different oxides. If the metal, oxygen and oxide are in their standard states, then the free energy change is related to temperature as ∆G f = ∆H f – T∆S f o

o

o

(2.32)

where ∆H f and ∆S f are standard heat and entropy of formation, respectively. o

o

According to Kirchoff’s law, in the absence of any phase transformation between T and T1, T

∆H f = ∆H f ( at T 1 ) + o

o

∫ ∆C p dT o

(2.33)

T1 T

∆S f = ∆S f ( at T 1 ) + o

o

∆C p - dT ∫ --------T o

(2.34)

T1

where ∆H f and ∆S f are standard heat and standard entropy of formation at temperature T, and o ∆C p is the difference of specific heats of products and reactants at standard states. The values o o o of ∆C p are generally very small and, therefore, one may assume that ∆H f and ∆S f are o essentially independent of temperature. This allows us to express dependence of ∆G f on temperature as: o

o

∆G f = A + BT o

(2.35)

where A and B are constants. o Equation (2.35) is an approximate one. A more precise representation of ∆G f as a function of T is ∆G f = A + BT + CT ln T o

(2.36)

However, data at steelmaking temperatures in standard compilations are available in the form of Eq. (2.35), for the limited temperature range of steelmaking. ©2001 CRC Press LLC

Appendix 2.1 provides values of A and B for oxides as well as some other compounds of o importance in secondary steelmaking. Figure 2.1 presents a diagram for oxides. ∆G f values are per gm mol of O2. This normalization allows us to compare stabilities of oxides directly from such figures. For example, Al2O3 is stabler than SiO2, since the free energy of formation of the former is more negative as compared to that of the latter. Quantitatively speaking, we are interested in the following reaction: 4 2 --- Al + SiO 2 = --- A1 2 O 3 + Si 3 3

(2.37)

∆Go [for the reaction of Eq. (2.37)] = ∆G f (2/3Al2O3) – (SiO2) is a negative quantity, and hence the reaction is feasible if all reactants and products are at their respective standard states (i.e., pure substances) in accordance with free energy criteria [Eq. (2.22)]. However, if they are not pure (e.g., present as solution in molten iron or slag), then ∆Go does not provide a correct guideline, and we have to find out ∆G by using Eq. (2.27). These will require knowledge of activity as a function of composition. o

FIGURE 2.1 Standard free energy of formation for some oxides.


2.5 ACTIVITY–COMPOSITION RELATIONSHIPS: CONCENTRATED SOLUTIONS Crudely speaking, activity is a measure of “free” concentration in a solution, i.e., concentration that is available for chemical reaction. Also, by definition, activity is dimensionless. In metallurgical processing, the gases behave as ideal, and molecules are free. Hence, activity of a component i in a gas mixture is equal to its concentration. Numerically, by convention, ai = pi, where pi is partial pressure of i in atmosphere. The composition of a solution can be altered significantly during processing only if mixing and mass transfer are rapid. Solid state diffusion is very slow. Hence, during the short processing time, its composition does not change. For example, a particle of CaO will remain CaO as long as it does not dissolve in slag. It may get coated by another solid such as Ca2SiO4 or CaS during steel processing. Here, solid CaO remains pure and its activity, by definition, is 1, since this is its standard state. However, liquid steel contains variable concentrations of impurities and alloying elements. Molten slag is also a solution of oxides with a variety of compositions. Hence, activity versus composition relationships are required here for equilibrium calculations. As already stated, a pure element or compound constitutes its conventional standard state. For example, pure Fe is the conventional standard state for liquid steel, and aFe = 1 for pure iron. Similarly, pure SiO2 is the standard state for a slag containing silica. In the conventional standard state, an ideal solution obeys Raoult’s law, which states, ai = Xi

(2.38)

where Xi is mole fraction of solute i in the solution. For example, let liquid steel contain chromium and nickel. Then, XCr is to be calculated from weight percent composition as follows.

X Cr

W Cr --------M Cr = -----------------------------------------W Cr W Ni W Fe --------+ --------- + --------M Cr M Ni M Fe

(2.39)

where Wi denotes weight percent and Mi molecular mass of species i. Most real solutions do not obey Raoult’s law. They either exhibit positive or negative departures from it. For a binary solution (i.e., containing two species such as Fe + Ni or CaO + SiO2), this is illustrated in Figure 2.2. For example, molten Fe-Mn, Fe-Ni, FeO-MnO solutions are ideal. Molten Fe-Si, CaO-SiO2, FeO-SiO2, MnO-SiO2, etc. show negative departures. Liquid Fe-Cu exhibits positive departure. Departures from Raoult’s law are quantified using a parameter, known as the activity coefficient (γ), which is defined as: γ i = a i /x i

(2.40)

Activities in slag systems use conventional standard states as reference. However, industrial slags are multicomponent systems. Hence, presentation of activity versus composition diagrams is more complex and different from that of a binary solution. Figure 2.3 shows values of activity of SiO2 in CaO-SiO2-Al2O3 ternary system at 1550°C (1823 4 K). These are in the form of isoactivity lines for SiO2. Similarly, there would be diagrams presenting isoactivity lines for CaO and Al2O3. The liquid field is bounded by liquidus lines. In this diagram, Al2O3 has been written as AlO1.5. This is because molecular mass of CaO, SiO2, and AlO1.5 are ©2001 CRC Press LLC

FIGURE 2.2 Raoult’s law and real systems showing positive and negative deviations.

FIGURE 2.3 Activity of SiO2 in CaO-SiO2 – Al2O3 ternary system at 1823 K; the liquid at various locations on liquidus is saturated with compounds as shown.4


close, being equal to 56, 60, and 51, respectively. Therefore, the mole fraction scale is approximately the same as the weight fraction scale. Slag activity data are available from several sources, but the most comprehensive is the Slag Atlas.7 However, this is quite unsatisfactory, since 1. slags are multicomponent and not ternary, and 2. thermodynamic calculations can be performed properly if the activity vs. composition relationship can be expressed by equations. This allows easier interpolations and extrapolations of laboratory experimental data in a composition regime. Example 2.2 Solid iron is in contact with a liquid FeO-CaO-SiO2 slag and gas containing CO and CO2 at 1300°C. The activity of FeO in slag is 0.45, and the p CO / p CO2 ratio in gas is 20/1. Predict whether it is possible to oxidize iron. Also, calculate equilibrium value of p CO / p CO2 ratio in gas. Solution We are to consider the following reaction: Fe(s) + CO2 (g) = (FeO) + CO(g)

(E2.1)

For the reaction of Eq. (E2.1), ∆G = ∆G f [ CO ( g ) ] + ∆G f [ FeO ( s ) ] – ∆G f [ CO 2 ( g ) ] o

o

o

o

(E2.2)

The standard state for FeO is solid pure FeO, since its melting point is 1368°C. With the help of Appendix 2.1, ∆Go at 1300°C (1573 K) = –249.8 – 161.3 + 395.7 = –15.38 kJ mol–1 = –15.38 × 103 J mol–1 (a) From Eq. (2.27), p CO × ( a FeO ) 0 ∆G = ∆G + R × T ln ---------------------------[ a Fe ] × p CO2

(E2.3)

As discussed earlier, solid iron would remain essentially pure in a limited time period. So, aFe may be taken as 1. Going through the calculations, ∆G = + 13.36 kJ mol–1 Since ∆G is positive, oxidation of Fe is not possible. (b) At equilibrium, ∆Go = –RT ln K

(2.30)

where p CO × ( a Feo ) K = ln ----------------------------[ a Fe ] × p CO2 ©2001 CRC Press LLC

at equilibrium

Using the value of ∆G , the p CO ⁄ P CO2 ratio at equilibrium with Fe and the slag turns out to be 7.20. (Note that R = 8.314 J mol–1 K–1.) o

2.5.1

A NOTE ON SOLUTION MODELS

FOR

MOLTEN SLAGS

Whitley8 made the earliest effort in this direction. He assumed the slag to consist of 2CaO · SiO2, 3CaO · P2O5, etc. to estimate “free CaO” in slag as an index of aCaO. However, slags are really ionic liquids, and compounds like CaO, SiO2, etc. do not exist as such. In contrast to these models, the other group of models has been termed as ionic models, where some kind of ionic structure is assumed. The first ionic model of salt melts is that of Temkin (1945), who assumed ideal mixing (i.e., ideal solution) among cations and ideal mixing among anions but no interaction between cations and anions. The last assumption is too simplistic and has not been accepted. However, the first assumption, namely, ideal mixing among cations and among anions separately, constituted the basis for some later models. Flood et al.9 utilized it for reaction of sulfur between liquid steel and slag and obtained the analytical relation for the equilibrium constant as follows: log K h, S =

∑i X′i log K h,S i

(2.41)

where Kh,S denotes the equilibrium constant for sulfur reaction between metal and slag containing several cations. i denotes a cation. X′ i is an electrically equivalent fraction of i among all cations. i K h,S is the equilibrium constant if i is the only cation in slag. This is a useful equation. It allows i calculation of Kh,S in slag from knowledge of K h,S of various cations. Hence, this approach was later extended to the reaction of phosphorus as well. Slag modeling for thermodynamic calculations is of considerable interest to steelmaking. Some recent studies10 indicate efforts to apply the approach of Flood et al. with refinements. Of course, thermodynamic predictions are independent of structural considerations. This provides another approach. Analytical relations based on a regular solution model have proved to be the most popular among structure-independent predictions. For a binary solution, the regular solution model predicts RT lnγ 1 = αX 2 2

(2.42)

where X2 is the mole fraction of component 2 in the binary 1–2, and α is a constant. For a multicomponent solution, the general form of the equation for the regular solution model is10 RT lnγ i =

∑ jα ijX j + ∑ j∑ k( αij + αik – α jk ) X j X k 2

(2.43)

where α values are constants, known as interaction energies between subscripted solutes. Ban-ya11 has recently summarized mathematical expression of slag-metal reactions in steelmaking processes by quadratic formalism based on regular solution model. If the melt is not a strictly regular solution, then for a real solution, RT lnγ i =

∑ jα ijX j + ∑ j∑ k( αij + αik – α jk ) X j X k + I 2

(2.44)

where I has been termed the conversion factor of the activity coefficient from the hypothetical regular solution to the real solution. ©2001 CRC Press LLC

Experimental data of various slag-metal and slag-gas equilibria for many slag compositions were statistically fitted with Eq. (2.44). These have yielded some values of αij and I.11 Example 2.3 Calculate γi in a multicomponent solution of slag at 1873 K. Composition of slag in weight percent is as follows: MnO = 4, CaO = 50, Al2O3 = 35, SiO2 = 8, FeO = 3 Take MnO as species i. Solution From Eq (2.39): Mole fractions of various species in slag are X MnO = 0.0384, X CaO = 0.6058, X Al2 O3 = 0.2339,X SiO2 = 0.091, X FeO = 0.0284 Interaction energies between various cations are11 αMn-Ca = –92050, αMn-Al = –83680, αMn-Si = –75310, αMn-Fe = 7110, αCa-Al = –154810, αCa-Si = –133890, αCa-Fe = –31380, αAl-Si = –127610, αAl-Fe = –41000, αSi-Fe = –41840 For reaction MnO(s) = MnO (regular solution), I = –32470 + 26.14T, J Performing calculations on the basis of Eq. (2.44) and using the above data, γMnO = 0.163 (Ans.).

2.6 ACTIVITY–COMPOSITION RELATIONSHIPS: DILUTE SOLUTIONS 2.6.1

ACTIVITIES

WITH

ONE WEIGHT PERCENT STANDARD STATE

Liquid steel comes primarily in the category of dilute solution, where concentration of solutes (carbon, oxygen etc.) are mostly below 1 wt.% or so except for high alloy steels. Solutes in dilute binary solutions obey Henry’s law, which is stated as follows: ai = γ i Xi o

(2.45)

where γ i is a constant. Deviation from Henry’s law occurs when the solute concentration increases. Therefore, activities of dissolved elements in liquid steel are expressed with reference to Henry’s law and not Raoult’s law. Since we are interested in finding values directly in weight percent, the composition scale is weight percent, not mole fraction. With these modifications, in dilute solution of species i in liquid iron, o


1. If Henry’s law is obeyed by species i, then hi = Wi

(2.46)

2. If Henry’s law is not obeyed by species i, then hi = fi Wi

(2.47)

hi is activity and fi is the activity coefficient in the so-called one weight percent standard state. This is because, at 1 wt.%, hi = 1, if Henry’s law is obeyed. Again, it can be shown that fi is related to Raoultian activity coefficient γi as: γ f i = -----oi γi

(2.48)

It is to be noted that the standard free energy change for reaction is not going to be the same if the standard state is changed. For example, at 1600°C, Si ( 1 ) + 0 2 ( g ) = SiO 2 ( s ); ∆G 49 = – 571.5 kJ mol o

–1

(2.49)

with pure liquid silicon as standard state. However, for the 1 wt.% standard state of Si dissolved in liquid iron, [ Si ] wt.pct. + 0 2 ( g ) = SiO 2 ( s ); ∆G 50 = –406.4 kJ mol * o

–1

(2.50)

∆G 49 and ∆G 50 are related to each other as o

o

∆G 50 = ∆G 49 – [ G Si + – G Si ] at 1 wt. pct. std. state for Si in liquid iron o

o

o

m

= ∆G 49 – [ G Si ] at 1 wt. pct. std. state for Si in liquid iron o

m

(2.51) (2.52)

o

where G i = G i – G i is known as the partial molar free energy of mixing of solute i into a solution. Again, from Eq. (2.25), m

G Si = RT ln [ a Si ] at 1 wt. pct. std. state Si in liquid iron = RT ln γ Si [ X Si ] at 1 wt. pct. o

(2.53) (2.54)

With reference to Eq. (2.39), in Fe-Si binary,

X Si

W Si -------M Si = -----------------------W Si W Fe -------- + --------M Si M Fe

* Note: Si dissolved in liquid metal is denoted either as [Si] or Si, SiO2 dissolved in slag is indicated by (SiO2).


(2.55)

On the basis of Eq. (2.55), XSi = 0.02 at WSi = 1. Noting that γ Si = 1.25 × 10–3, the value of ∆G 50 in Eq. (2.50) was obtained. m Appendix 2.2 presents values of G i for some solute in liquid iron. o

2.6.2

SOLUTE–SOLUTE INTERACTIONS

IN

o

DILUTE MULTICOMPONENT SOLUTIONS

It has been found that solutes in a multicomponent solution interact with one another and thus influence activities of other solutes. Figure 2.4 illustrates this for activity of carbon and oxygen in liquid iron at 1833 K. In the Henry’s law region of Fe-C binary (i.e., without any other added element), fC = 1, i.e., log fC = 0. In the presence of a third element in liquid iron solution, fC keeps changing systematically. It has been derived that if, in a dilute multicomponent solution, A is solvent (Fe in case of liquid steel), and B, C,..., i, j, etc. are solutes, then log f i = e i ⋅ W B + e i W C + … + e i W i + e i W j + …… B

C

i

j

(2.56)

j

where e values are constants. e i is called the interaction coefficient, describing the influence of solute j on fi, which is defined as ∂ ( log f i ) j e i = -------------------∂W j

0.10

C

(2.57)

Wj→O

Si

P Co

0.05

Sb Cu

Te Al

Ni

LOG f

j

N

As

Se S (< 4 % ) Sn (< 4 % )

0

W Cr

Ti

-0.05

Mn

Mo

V

0

1

2

3

4

5

ALLOYING ELEMENT, mass% j

FIGURE 2.4 Influence of alloying elements on the activity coefficient of nitrogen dissolved in molten iron at 1823 K.


i

e i is known as the self interaction coefficient and has a non-zero value only if Fe-i binary deviates from Henry’s law. Again, ∂ ( log f ) i e j = --------------------j ∂W i

Wi → O

j M –2 M i – M j = e i ⋅ -------j + 0.434 × 10 -----------------Mi Mi

(2.58)

Appendix 2.3 presents values of interaction coefficient for some common elements dissolved in liquid iron. Equation (2.56) contains only first-order interaction coefficients. It is, in general, all right for dilute solutions of liquid iron. However, sometimes, even here, second-order interaction coefficients j ( r i ) are to be employed. On the other hand, if solute–solute interactions are not significant, log fi vs. weight percent of the added element exhibit good linear behavior over a long range. This is demonstrated by Figure 2.4 for nitrogen dissolved in liquid iron. The figure is based on several data sources and taken from the review by Iguchi.12 In such cases, Eq. (2.56) may be fairly all right up to reasonably high concentrations of solutes. Iguchi12 has recently reviewed the subject, especially the work of Ban-ya and his coworkers, who had been active in this field for about two decades. The following two approaches have been seriously explored. The first approach is application of quadratic formalism, originally proposed by Darken and applied to several binary systems by Turkdogan and Darken.10 Ban-ya examined its use in Fe-C-j ternary melts. The second approach is application of the interstitial solution model originally proposed by Chipman.13 Elements P, C, S, N, etc. may be treated as interstitial atoms, and this model has been applied to ternary iron alloys containing these elements to high concentrations. It predicts linearity between log ψ with Yj, where ψi is a modified activity coefficient of i, and Yj is atom ratio of j in a ternary containing i and j. Figure 2.5 shows its application to the effect of iron on the activity coefficient of nitrogen in Cr-Fe-N ternary melts. Good linear relation up to a high concentration of iron may be noted. Example 2.4 Liquid steel is being degassed by argon purging in a ladle at 1873 K (1600°C). The gas bubbles coming out of the bath have 10 percent CO, 5 percent N2, 5 percent H2, and the rest Ar. Assuming these to be at equilibrium with molten steel, calculate the hydrogen, nitrogen, and oxygen concentrations in steel in parts per million (ppm). The steel contains 1 percent carbon, 2 percent manganese, and 0.5 percent silicon. The total gas pressure may be taken as 1 atm. Solution (a) For hydrogen, the reaction may be written as 1 [ H ] wt.pct. = --- H 2 ( g ) 2

(E4.1)

1905 log K H = ------------ + 1.591 T

(E4.2)

for which

Again, at equilibrium, 1⁄2

KH


p H2 = ---------[ hH ]

(E4.3)

FIGURE 2.5 Effect of iron on the activity coefficient of nitrogen in liquid chromium.13

Now, p H 2 = 0.05 atm, and KH = 405.58 at 1873 K So, hH at equilibrium = 5.513 × 10–4 = fH · WH Again, log f H = e H ⋅ W C + e H ⋅ W Mn + e H ⋅ W Si C

Mn

Si

(E4.4)

Assume interactions of dissolved H, N, and O on fH as negligible. This is justified in view of their j very small concentrations. Taking values of e i from Appendix 2.3, log fH = 0.06 × 1 – 0.002 × 2 + 0.027 × 0.5 putting in values, WH = 4.69 × 10–4 percent = 4.69 ppm (Ans.) ©2001 CRC Press LLC

(b) For nitrogen, the reaction may be written as 1 [ N ] wt.pct. = --- N 2 ( g ) 2

(E4.5)

518 log K N = --------- + 1.063 T

(E4.6)

for which

Proceeding as for hydrogen, [hN] = 0.01, in 1 weight percent standard state Now, hN = fN · WN

(E4.7)

and log f N = e N ⋅ W C + e N ⋅ W Mn + e N ⋅ W Si C

Mn

Si

(E4.8)

Proceeding as before, WN = 0.0077 wt.% = 77 ppm (Ans.) (c) For oxygen, the reaction may be written as [C]wt.%. + [O]wt.%. = CO(g)

(E4.9)

1160 log K O = ------------ + 2.003 T

(E4.10)

P CO K O = -------------------[ hC ] [ hO ]

(E4.11)

for which

Again, at equilibrium,

hC = fC · WC = fC · 1 log f C = e C ⋅ W C + e C ⋅ W Mn + e C ⋅ W Si C

Mn

Si

(As in previous cases, assume interactions of H, N, and O on fC as negligible.) Putting in values, fC = 1.82, and hC = 1.82. So, P CO 0.1 –4 h O = ---------------------- = ------------------------- = 1.31 × 10 [ K O ] [ hC ] 419 × 1.82 ©2001 CRC Press LLC

(E4.12)

Again, WO = hO/fO and, log f O = e O ⋅ W C + e O ⋅ W Mn + e O ⋅ W Si C

Mn

Si

(E4.13)

C

Putting in values (taking e O = –0.421), WO = 4.1 × 10–4 wt.% = 4.1 ppm

(Ans.)

2.7 CHEMICAL POTENTIAL AND EQUILIBRIUM So far, we have followed the approach in which the overall free energy change had been employed as the criterion for assessing the feasibility of a process. There is an alternative approach based on chemical potential. Suppose that an element i is to be transferred from phase I to phase II. Then, we say that, for the transfer to be feasible thermodynamically, µ i (I) > µ i (II)

(2.59)

µ i (I) = µ i (II)

(2.60)

and for equilibrium,

where µ i (I) and µ i (II) denote the chemical potential of species i in phases I and II, respectively. µ i is identical with partial molar free energy of solute i in a solution ( G i ) . On the basis of Eq. (2.25), µ i (I) = µ i (I) + RT ln ai (I)

(2.61)

µ i (II) = µ i (II) + RT ln ai (II)

(2.62)

O

O

where µ i denotes the chemical potential of i at its standard state. If the standard state of i is the same in both the phases, then O

µ i (I) = µ i (II) O

O

(2.63)

so, ai(I) > ai(II), for transfer from phase (I) to (II), and, ai(I) = ai(II), for equilibrium

(2.64)

The chemical potential approach has the following advantages: 1. We can visualize a process better because of the similarity of the concept to some common physical processes. Just as heat flows from a higher heat potential (temperature) to a lower heat potential, and electricity flows from a higher electrical potential to a lower ©2001 CRC Press LLC

one, in the same way a chemical species i is transferred spontaneously from higher µ i to a lower µ i. 2. It is not necessary to bother about the overall reaction; it is sufficient to find out the chemical potential of the species concerned only. Suppose we know µ i(I). If another phase II is brought in contact with it, all we have to do is to calculate µ i(II) to find out direction of transfer of i.

2.7.1

CHEMICAL POTENTIAL

OF

OXYGEN

In refining processes, we are primarily concerned about the transfer of oxygen. For the reduction of a metal from its oxide, the reaction environment must have a lower chemical potential than oxygen. Similarly, if impurities in a metal are to be preferentially oxidized in refining, then the environment must have a higher chemical potential than oxygen as compared to that in the impure metal. The chemical potential of oxygen (O2) is expressed as µ O2 = µ O2 + RT ln a O2 = µ O2 + RT ln p O2 o

o

(2.65)

Since a O2 may be equated to p O2 for ideal gas, and µ O2 is set equal to zero, then o

µ O2 = RT ln p O2

(2.66)

Calculation of the oxygen potential in a gas phase is relatively simple. For liquid metals, one should consider the reaction O2(g) = 2[O]. For this, [ hO ] o ∆G h = – RT ln -----------= – 2RT ln [ h O ] + RT ln p O2 p O2

(2.67)

µ O2 ( metal ) = ∆G h + 2RT ln [ h O ]

(2.68)

2

Hence, o

For liquid slag in ironmaking, we consider the reaction 2 [ Fe ] + O 2 ( g ) = 2 ( FeO ); ∆G 69 o

(2.69)

( a FeO ) o ∆G 69 = – RT ln K 69 = – RT ln --------------------2 [ a Fe ] p O2 2

(2.70)

where p O2 is partial pressure of O2 in equilibrium with [Fe] and (FeO). Since aFe ≈ 1 in ironmaking and steelmaking, µ O2 ( slag ) = RT ln p O2 = ∆G 69 + 2RT ln ( a FeO ) O

(2.71)

If slag-metal equilibrium does not exist, Eq. (2.71) gives the µ O2 in slag, because it is not dependent on the composition of iron. If slag-metal equilibrium exists, then it is µ O2 in both slag and metal. ©2001 CRC Press LLC

Of course, primary steelmaking slags contain Fe2O3 (i.e., Fe3+ ions) also. There µ O2 is determined more by the following reaction: 4 (FeO) + O2(g) = 2(Fe2O3)

(2.72)

However, in secondary steelmaking, Eq. (2.71) is applicable, since the FeO concentration is low. The above analysis does not mean that oxygen is present as O2 in the slag or metal. As a matter of fact, it is far from being so, because oxygen exists as ions in slag and as dissociated atoms in metal. But, for thermodynamic calculations and concepts, this is unimportant. Example 2.5 Calculate the chemical potential of oxygen in a CO/CO2 gas mixture and slag as given in Example 2.2, and the chemical potential of nitrogen as per Example 2.4. Solution (a) Calculation of µ O2 in CO/CO2 gas mixture for the problem in Example 2.2: Consider the reaction, 2CO ( g ) + O 2 ( g ) = 2CO 2 ( g )

(E5.1) 2

p CO2 ∆G = – RT ln ---------------------2 p CO × p O2 o

equilibrium

p CO = RT ln ( p O2 ) e – 2RT ln ----------2 p CO where ( p O2 ) e is in equilibrium with CO and CO2. Since p CO 20 1 o --------- = ------ , µ o2 = RT ln ( p O2 ) e = ∆G + 2RT ln  ------  20 1 p CO2

(E5.2)

At 1300°C (1573 K), for the reaction of Eq. (E5.1),* ∆G = 2 [ ∆G f , CO2 – ∆G f ,CO ] o

o

o

(E5.3)

= 2[(–396.46 + 0.08 × 10–3 × 1573) + (118.0 + 84.35 × 10–3 × 1573)] = –291.32 kJ mol–1 Substituting in Eq. (E5.2), µ O2 = – 369.67 kJ mol O 2 –1

* From data in Appendix 2.1.


(Ans.)

(E5.4)

(b) Calculation of µ O2 in FeO-CaO-SiO2 slag for the problem in Example 2.2: Consider the reaction 2Fe(s) + O 2 (g) = 2 ( FeO )

(E5.5)

Since aFe = 1, aFeO = 0.45 (given), from Eq. (2.71), µ O2 = 2∆G FeO ( 1 ) + 2RT ln ( 0.45 ) o

(E5.6)

From Appendix 2.1, ∆G FeO ( 1 ) = – 238.07 + 49.45 × 1 0 T kJ mol o

–3

–1

At 1573 K, o

2∆G FeO ( l ) = – 320.57 kJ mol

–1

Therefore, from Eq. (E5.6), µ O2 (in slag) = –341.46 kJ/mol–1 O2

(Ans.)

µ O2 in slag is different from µ O2 in gas, because they are not at equilibrium. (c) Calculation of µ N2 for Example 2.4: Since p N2 = 0.05 atm in Example 2.4, p N2 = RT ln p N 2 = RT ln (0.05) At T = 1873 K, µ N2 = –46.65 kJ mol–1 N2 (Ans.) Since liquid steel and nitrogen in exit gas are at equilibrium, µ N2 in liquid steel also shall be the same.

2.8 SLAG BASICITY AND CAPACITIES Basicity of a slag increases with increased percentages of basic oxides in it. It is an important parameter governing refining. Steelmakers had always paid attention to it. In the early days, the numerical value of basicity was taken as the CaO/SiO2 ratio, modified ratio, or excess base. ©2001 CRC Press LLC

Since a basic oxide (e.g., CaO) tends to dissociate into a cation and oxygen ion (e.g., Ca2+, O2–), the concentration of free O2– increases with increasing basicity. Therefore, from a thermodynamic viewpoint, the activity of oxygen ion ( a O2 – ) may be taken as an appropriate measure of the basicity of slag. However, there is no method available for experimental determination of ( a O2 – ) .

2.8.1

OPTICAL BASICITY

A breakthrough came with the development of the concept of optical basicity (Λ) in the field of glass chemistry by Duffy and Ingram14 in 1975–76. It was applied to metallurgical slags first by Duffy, Ingram, and Somerville.15 From then on, numerous investigators have applied it to metallurgical slags for a variety of correlations. Experimental measurements of optical basicity in transparent media such as transparent glass and aqueous solutions were carried out employing Pb2+ as the probe ion. In an oxide medium, electron donation by oxygen brings about a reduction in the 6s–6p energy gap, and this in turn produces a shift in frequency in UV spectral band. υ free – υ sample Λ = ---------------------------υ free – υ CaO

(2.73)

where υfree, υCaO and υsample are frequencies at peak for free Pb2+, Pb2+ in CaO, and Pb2+ in a sample, respectively. Therefore, Λ = 1 for pure CaO by definition. Hence, Λ is an expression for lime character, even though there may not be any CaO in sample. Based on experimental measurements, the following empirical correlation was proposed by Duffy et al.14,15 1 ----- = 1.35 ( α i – 0.26 ) Λi

(2.74)

where αi is Pauling electronegativity of the cation in a single oxide i. This relationship has allowed estimation of Λi for a variety of oxides where experimental data are not available from the values of αi. The estimated Λi is known as theoretical optical basicity (Λth.i). For a multicomponent system such as slag or glass. Λth (for slag/glass) = Σ X i ′ Λth.i

(2.75)

where X i ′ = equivalent cation fraction of oxide i. Slags are opaque. The same is true of glasses containing oxides of transition metals. Hence, Λ is to be estimated for slags. The most widely employed method of estimation is on the basis of Eqs. (2.74) and (2.75). Other, lesser-known methods also have been employed.16 Optical basicity of individual oxides was estimated from Eq. (2.74), where experimental data were not available. This was tantamount to suggesting that each oxide is characterized by a unique value of Λi, irrespective of medium and temperature. However, it has not been accepted by recent investigators. Moreover, assignment of correct values of Λth to transition metal oxides such as FeO, MnO is controversial, since Eq. (2.74) is not applicable for these from theoretical considerations. Differing findings and opinions have been published in the literature. Estimated values of Λi have been questioned, and other methods of estimation based on refractive index, electronic polarizability, and electron density have been employed besides Pauling’s electronegativity.17–19 Some metallurgical slags contain fluorides or chlorides. Here, fluoride or chloride ions also ought to be considered, in addition to oxygen, for their contribution toward basicity. Considerable efforts have been made to evaluate Λi for common fluorides such as CaF2. ©2001 CRC Press LLC

One important problem facing application of the optical basicity concept has been differing values of Λi proposed by different investigators, especially for transition metal oxides such as FeO, MnO, TiO2, and others. The present status is shown in Appendix 2.4. The values of optical basicity recommended by Duffy and coworkers18,20 were on the basis of electronegativity, electronic polarizability, and refractive index, whereas those by Nakamura et al.21 were estimated from average electron density. It may be noted that there is both agreement as well as disagreement among various investigators. It is not presently possible to recommend one set over another. Example 2.6 Calculate the optical basicity of a slag of composition same as in Example 2.3. Solution X′ i = 2 × 0.0384 + 2 × 0.6058 + 3 × 0.2339 + 4 × 0.091 + 2 × 0.0284 = 2.414, for Example 2.3. Equivalent cation fractions of species are: 2 × 0.0384 X′ Mn2 + = ------------------------- = 0.031 2.414

2 × 0.6058 X′ Cn2 + = ------------------------- = 0.504 2.414

3 × 0.2339 X′ Al3 + = ------------------------- = 0.291 2.414

4 × 0.091 X′ Si4 + = ---------------------- = 0.151 2.414

2 × 0.028 X′ Fe2 + = ---------------------- = 0.023 2.414 Optical basicity of various species (considering data of Nakamura et al.21; Appendix 2.4): Λ MnO = 0.95, Λ CaO = 1.0, Λ Al2 O3 = 0.66, Λ SiO2 = 0.47, Λ FeO = 0.94 Substituting the values of X i ′ and Λi in Eq. (2.75), Λth,slag = 0.82 (Ans.)

2.8.2

SLAG CAPACITIES

Along with basicity, another concept, namely that of slag capacity, has evolved, and its use has become quite widespread. Richardson and Fincham10 in 1954 defined sulfide capacity (Cs) as the potential capacity of a melt to hold sulfur as sulfide. Mathematically, C s = ( wt. % S

2–

1⁄2

1⁄2

) ⋅ p O2 ⁄ p S2

(2.76)

where p O2 ⋅ p S2 are partial pressures of O2 and S2 in the gas at equilibrium with slag. Noting that the reaction is 1/2 S2 (g) + (O2–) = 1/2 O2 (g) + (S2–)

(2.77)

K 77 ⋅ a O2 – C s = ----------------------φ S2 –

(2.78)

it can be shown that


where K77 is the equilibrium constant for the reaction of Eq. (2.77) and φ S2 – is the activity coefficient of S2– in slag in an appropriate scale. Wagner22 has critically discussed the concept of basicity and various capacity parameters such as sulfide capacity, phosphate capacity, carbonate capacity, etc. He has discussed interrelationships among capacities. He also suggested use of carbonate capacity as a method of measurement of basicity. Many papers have been published on measurements and application of these capacities, and relationships among these.17,24 The reaction of phosphorus under oxidizing condition may be written as 1 5 3 2– 3– --- P 2 ( g ) + --- O 2 ( g ) + --- ( O ) = ( PO 4 ) 2 4 2

(2.79)

for which the phosphate capacity of slag may be defined as ( wt. % PO 4 ) C p = -------------------------------------1⁄2 5⁄4 ( P p2 ) ( p O2 ) 3–

(2.80)

From Eqs. (2.79) and (2.80), 3⁄2

[ a O2 – ] C p = K 79 ⋅ --------------------φ PO3–

(2.81)

4

where K79 is the equilibrium constant for the reaction of Eq. (2.79) and φ PO3– is the activity coefficient 3– 4 of PO 4 in slag in an appropriate scale. The combination of Eqs. (2.78) and (2.81) leads to the following relation: 3⁄2

φ S2 – 3 log C p = --- log C s + log K 82 + log ---------φ PO3– 2

(2.82)

4

K82 is an equilibrium constant term and depends only on temperature. At a constant temperature, 3⁄2 in the same slag system (e.g., Na2O – SiO2 system), ( φ S2– ) ⁄ φPO3– parameter is expected to be 4 22 constant, Hence, a single straight line with slope of 3/2 is expected. Figure 2.6 is based on review by Sano et al.24 It shows agreement with the above expectation. Of course, a corollary to this conclusion is that there is no universal correlation between log Cp and log Cs that will be applicable to all kinds of slag systems. Similar conclusions can be drawn about interrelationships of other capacities. Optical basicity concept is being utilized by industries as well. Equation (2.75) forms the basis of estimation of optical basicity in slags. Cs increases with increasing a O2– (i.e., increasing basicity) and hence ought to have a relation to optical basicity. Many workers have shown that log Cs = m Λ + n

(2.83)

where m and n are empirical constants. Figure 2.723 shows such an attempt for various slags at 1500°C. Young et al.18 have recently questioned applicability of a simple linear dependence of log Cs and log Cp on Λ. Capacities have been correlated with Λ, Λ2, temperature, as well as some ©2001 CRC Press LLC

FIGURE 2.6 Relationship between sulfide capacities and phosphate capacities for various fluxes. Source: Sano, N., in Proceedings of the Elliott Symposium, ISS, Cambridge, Mass., reprinted by permission of the Iron & Steel Society, Warrendale, PA.

composition parameters. However, it has also been shown that, at values of Cs less than 0.01, Eq. (2.83) as employed in Figure 2.723 is also all right. Since, in secondary steelmaking, Cs lies in this range, a linear relationship as in Eq. (2.83) would be adequate for industrial uses.

REFERENCES 1. Darken, L.S. and Gurry, R.W., Physical Chemistry of Metals, McGraw-Hill Book Co., New York, 1953. 2. Gaskell, D.R., Introduction to Metallurgical Thermodynamics, 2nd Ed., McGraw-Hill Book Co., New York, 1973. 3. Elliott, J.F. and Gleiser, M., Thermochemistry for Steelmaking, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass., USA, 1960. 4. Elliott, J.F., Gleiser, M. and Ramakrishna, V., Thermochemistry for Steelmaking, Vol. 2, AddisonWesley Publishing Co., Reading, Mass, USA, 1963. 5. Kubaschewski, O., Evans, E.L. and Alcock, C.B., Metallurgical Thermochemistry, 4th Ed., Pergamon Press, Oxford, 1967. 6. Wicks, C.E. and Block, F.E., Thermodynamic Properties of 65 Elements—Their Oxides, Halides, Carbides, and Nitrides, U.S. Bureau of Mines, United States Government Printing Office, Washington, 1963. 7. Committee for Fundamental Metallurgy, Slag Atlas, Verlag Stahleisen M.B.H., Dusseldorf, 1981. 8. Whiteley, J.H., Proc. Cleveland Inst. Engrs., 59 1922–23, p. 36. 9. Flood, H., Forland, T. and Grjotheim, K., in The Physical Chemistry of Melts, Inst. Mining and Metallurgy, London, 1953. 10. Turkdogan, E.T., Physical Chemistry of High Temperature Technology, Academic Press, New York, 1980. ©2001 CRC Press LLC

2.0

- LOG CS

3.0

Ca0 - Al203 Ca0 - Si02 Ca0 - AI203 - Si02 Ca0 - Mg0 - Al203 Ca0 - Si02 - B203 Ca0 - Mg0 - Si02 Ca0 - Mg0 - Al203Si02

4.0

5.0

0.55

o

T = 1500 C

0.60

0.65

0.70

0.75

0.80

0.85

0PTICAL BASICITY (Λ), ( - ) FIGURE 2.7 Logarithm of sulfide capacity vs. optical basicity at 1823 K.23 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Ban Ya, S., ISIJ Int., 33, 1993, p. 2. Iguchi, Y., in Proc. The Elliott Symposium, ISS, Cambridge, Mass., USA, 1990, p. 132. Ban Ya, S., and Chipman, J., Trans AIME, 242, 1968, p. 940. Duffy, J.A. and Ingram, M.D., J. of Non-Crystalline Solids, 21, 1976, p. 373. Duffy, J.A., Ingram, M.D. and Sommerville, I.D., J. Chem. Soc., Faraday Trans. 1, 74, 1978, p. 1410. Bergman, A. and Gustafsson, A., in Proc. 3rd Int. Conf. on Molten Slags and Fluxes, Inst. of Metals, London, 1989, p. 150. Proc. 3rd Int. Conf. on Molten Slags and Fluxes, Inst. of Metals, London, 1989. Young, R.W., Duffy, J.A., Hassall, G.J. and Xu, Z., Ironmaking and Steelmaking, 19,1992, p. 201. Nakamura, T., Yokoyama, T. and Toguri, J.M., ISIJ Int., 33, 1993, p. 204. Duffy, J.A., Ironmaking and Steelmaking, 17, 1990, p. 410. Nakamura, T., Ueda, Y. and Toguri, J.M., Trans. Japan Inst. Met., 50, 1986, p. 456. Wagner, C., Metall. Trans. B, 6B, 1975, p. 405. Sosinsky, D.J., Sommerville, I.D. and McLean, A., in Proc. 6th PTD Conference, ISS, Washington D.C., 1986, p. 697. Sano, N., in Proc. The Elliott Symposium, ISS, Cambridge, Mass., USA, 1990, p. 163. Table 2.1: Specific heats and enthalpies of transformation for iron.


3

Flow Fundamentals

The nature of fluid motion and the intensity of turbulence during liquid steel processing are of considerable importance to the success of secondary steelmaking due to their significant influence on mixing, mass transfer, inclusion removal, refractory lining wear, entrapment of slag, and reaction with the atmosphere. Therefore, several studies have been carried out in the past two to three decades or so on the subjects of fluid flow and mixing in ladles, tundishes, etc. Among these, molten steel in a ladle, stirred by argon gas, injected through porous and slit plugs located at the ladle bottom, constitutes the most commonly encountered situation in secondary steelmaking. Extensive fundamental studies on fluid flow have been carried out on this system. Hence, this chapter first of all briefly mentions the basics of fluid flow and then takes up flow in gas-stirred ladles. The motion of liquid steel arises from free convection due to temperature and composition gradient in the melt, and forced convection due to gas stirring, electromagnetic stirring, agitation by the pouring stream, and mechanical stirring. However, free convection has been found to be very mild and can be ignored in ladle metallurgy.

3.1 BASICS OF FLUID FLOW The fundamentals of fluid flow are available in standard texts.1–4 The following is a very brief introductory presentation for the sake of completeness. Figure 3.1 depicts the motion of a fluid element (dm), i.e. an infinitesimally small mass of a fluid, moving along a path. It moves under the action of some forces acting upon it. Such forces may be classified in the following two categories: 1. Body forces, which act throughout the volume of the fluid element. Gravitational force is the primary body force of relevance. However, there are the following other body forces:

dFB PATH

τ

dm dFS

σ

FIGURE 3.1 Forces and stresses acting on a fluid element.


• Thermal buoyancy force: density difference due to inhomogeneity of temperature in the fluid leads to this force, and it is expressed as ρgβt∆T, where g = acceleration due to gravity ßt = coefficient of volume expansion of the fluid due to temperature change ∆T = overall temperature difference in the fluid • Solutal buoyancy force: density difference due to composition inhomogeneity in the fluid leads to this force expressed as ρgβc∆C, where ßc = coefficient of volume expansion of the fluid due to concentration change ∆C = overall concentration difference of solute in the fluid • Electromagnetic force: sometimes known as Lorentz force, is expressed as J × B , where J is conduction current density and B is magnetic flux density. J and B are related through Maxwell’s equation. 2. Surface forces, which act at the surface of the fluid element due to contact with its surrounding. A surface force can be resolved into normal and shear forces. Applying Newton’s second law of motion to the fluid element, dF = dF B + dF s = a ⋅ dm

(3.1)

where a is acceleration of the element, dF B and dF S are, respectively, the body and surface forces acting on it. These are all vector quantities. Again, a ⋅ dm is nothing but the rate of change of momentum of the fluid element, dm. Force acting on a fluid element per unit area is designated as stress. There are two types of stresses: the one acting perpendicular to the surface of the fluid element is known as normal stress, and the other, acting parallel to the surface of the fluid element, is termed as shear stress. Fs ----- = ( – ∇ )p V

(3.2)

F where -----s is force per unit volume of fluid, ∇ is the symbol for gradient vector. V

3.1.1

VISCOSITY

Unlike a solid, a fluid cannot sustain a shear stress. In other words, it is completely deformable. If a shear stress is applied, then the fluid will undergo shear deformation continuously. Newton’s law of viscosity was the beginning of the quantification of the relationship between shear stress and shear deformation. The assumption is that shear stress is proportional to the rate of shear deformation. Newton’s law of viscosity, which is applicable to parallel and incompressible flow, is stated as ∂u τ yx = – µ --------x ∂y

(3.3)

With reference to Figure 3.2, τ yx denotes shear stress acting on the y-plane (i.e., a plane normal to the y-axis) along the x-direction. µ is a proportionality constant, known as coefficient of viscosity or simply viscosity. Figure 3.2 also shows the expected velocity profile in fluid adjacent to a solid surface for parallel flow. Velocity in the x-direction is zero at the surface, since the fluid layer just at the surface cannot slip past the solid. It increases as we move along the y-coordinate. ©2001 CRC Press LLC

FIGURE 3.2 Velocity profile in a fluid parallel to a flat plate.

An alternate form of Eq. (3.3) is µ∂ ( ρu x ) ∂ ( ρu x ) - = – ν ---------------τ yx = – ------------------ρ∂y ∂y

(3.4)

where ρ is density of the fluid. µ/ρ is known as kinematic viscosity (ν). Flow characteristics of a fluid are governed more by ν than µ. A velocity gradient in a fluid causes momentum transfer from higher to lower velocity due to its viscosity. Hence, shear stress in Figure 3.2 is also equal to the rate of transfer of x-momentum along y-direction per unit area normal to y-direction (i.e., xmomentum flux along y). Viscosities of gases can be estimated from the kinetic theory of gases. There are empirical rules available for estimation of the same for liquids. However, for the latter, it is by and large advisable to employ experimentally determined values. Viscosity, density, and some other physical and physicochemical properties of water, liquid iron, and some iron alloys, slags of interest in secondary steelmaking, are presented in Appendix 3.1. It may be noted that liquid iron has a value of ν comparable to water. Hence, it is as fluid as water. On the other hand, molten slags have much higher values of both µ and ν.

3.1.2

FLOW CHARACTERIZATION

Fluid motion can be categorized as follows: 1. 2. 3. 4. 5. 6.

Newtonian or non-Newtonian Viscous or nonviscous (ideal) Laminar or turbulent Incompressible or compressible Steady or unsteady Forced or free convection

The total characterization includes specifications on each of these points as well as the flow geometry. Liquids with high viscosity, such as viscous slags, do not have constant values of µ. Here, viscosity would be varying with the level of shear stress applied. These are known as nonNewtonian liquids. According to Eq. (3.3), viscous shear stress is proportional to µ as well as ©2001 CRC Press LLC

∂u x ⁄ ∂y (i.e., velocity gradient). If one of these is negligibly small, then τyx can be ignored, and the flow may be treated as nonviscous or ideal. Otherwise, it would be considered as viscous flow. Figure 3.3 depicts a typical velocity profile adjacent to a solid surface. Velocity of the fluid just at the surface is zero, and it increases rapidly to a constant value (uo) within a small distance. The region where the velocity is varying with distance is known as velocity boundary layer. Outside the velocity boundary layer is the bulk of the fluid. As an approximate general guideline, therefore, the motion of fluid in the boundary layer may be taken as viscous, and that in the bulk as nonviscous. As Figure 3.3 shows, the velocity profile approaches the bulk asymptotically. Hence, the thickness of the velocity boundary layer (δu) is theoretically infinity. To overcome this dilemma, δu is taken as the value of x where u = 0.99uo · δu is very small in a liquid—always less than a millimeter and even of the order of few microns. On the other hand, it is typically larger than a millimeter or even a centimeter in gases. If a tangent is drawn to the velocity profile at the solid–fluid interface, then its slope is equal to ( ∂u y ⁄ ∂x ) x = 0 . From this it follows that ∂u uo [ τ xy ] surface = – µ  --------y = – µ -------- ∂x  x = 0 δ u,eff

(3.5)

τxy acting on the surface would be opposite in sign and positive. δu,eff is known as the effective velocity boundary layer thickness. Laminar flow is obtained at low velocities, and turbulent flow at high velocities. The former is characterized by distinct streamlines with no cross mixing, whereas the latter is accompanied by extensive mixing. Suppose we are measuring the velocity of fluid at a particular location as a function of time. Let us also assume that the flow is steady, i.e., it does not vary with time. Then, Figure 3.4 shows schematically the pattern of velocity vs. time curves to be expected in both laminar and turbulent flow. In a steady turbulent flow, although the time-averaged velocity is constant, the instantaneous velocity exhibits random fluctuations. Imagine the fluid element A in Figure 3.5. Under turbulent flow, its instantaneous velocity fluctuates at random. Similar random fluctuations are exhibited by its neighbors. Since, in general, fluctuations of neighboring fluid elements are not in harmony with

FIGURE 3.3 Velocity profile in a fluid parallel to a flat plate.


FIGURE 3.4 (a) Schematic velocity vs. time curve illustrating the difference between laminar and turbulent flow, and (b) eddy fluctuations in water at Re = 6500, at the center of a pipe of 22 mm I.D., measured by a laser Doppler anemometer.2

B

A FIGURE 3.5 Eddy mechanism.

those of A, the latter is always receiving impacts from its neighbors, and vice versa. This occasionally will throw A out of its location to region B. In exchange, fluid from location B may be imagined to occupy location A. Such a process of exchange is visualized as an eddy. These eddy-like exchanges go on at random in all directions, leading to extensive mixing in turbulent flow. That is why turbulent flow is preferred in engineering. Exchange of eddy elements tends to impart a whirlpool-type motion ©2001 CRC Press LLC

in the eddy area. This is because the exchange takes place by small, jerky movements in a closed loop. Very large eddies can form even in laminar flow if the flow is disrupted. Examples are swirling motions behind spheres and cylinders at sudden enlargements of diameter in pipe flow. Actually, the term eddy traditionally had been employed for a large, macroscopic vortex flow. However, small ones, especially those of microscopic size, can be generated only in turbulent flow. When a fluid flows, considerable pressure differences may exist across the system. If the fluid is a gas, such pressure differences would lead to a variation of density, and under such a situation the flow is called a compressible flow. The motion of gases at a small pressure difference, and of liquids, is treated as incompressible. Forced convection refers to flow caused by an external agent such as a fan or pump. Free convection arises from thermal and solutal buoyancy forces. The results of an analysis of fluid motion depend on the geometry of flow. Standard texts mostly deal with the following two broad classes: 1. Flow in channel. The simplest example of this is flow through a straight circular pipe. 2. Flow around a submerged object. The simplest example of this is flow around a sphere. In the subsequent discussions, we shall be concerned with Newtonian, viscous, incompressible, and steady fluid motion only.

3.1.3

ANALYSIS

OF

FLUID FLOW

Standard texts1–4 have detailed derivations. There are basically two types of analysis, viz., differential analysis and integral analysis. Here, only a brief outline of differential analysis is presented. Equation of Continuity This is nothing but differential mass balance. Figure 3.6 presents a differential volume element of fluid in a rectangular coordinate system. From the principle of conservation of mass,

Rate of accumulation of mass in the volume element

=

[Rate of flow of mass into the volume element – Rate of flow of mass out of the volume element]

(3.6)

FIGURE 3.6 Region of volume ∆x∆y∆z fixed in space through which fluid is flowing.


∂ρ Rate of accumulation of mass in the control volume = ------ ( ∆x∆y∆z ) ∂t

(3.7)

where t is time. Along the x-coordinate, [Rate of flow of mass into the volume element – Rate of flow of mass out of the volume element]

= [ ( ρu x ) x – ( ρu x ) x + ∆x ]∆y∆z  ∂ ( ρu x )  = ( ρu x ) x –  ( ρu x ) x + ----------------dx  ∆y∆z ∂x   ∂ ( ρu x ) = ---------------∆x∆y∆z ∂x

(3.8)

where ux is velocity component along x-coordinate. Similar derivations can be made for rate of mass flow along y and z directions. Combining all these, Eq. (3.6) may be mathematically expressed as ∂ρ ∂ ( ρu x ) ∂ ( ρu y ) ∂ ( ρu z ) ------ + ---------------- + ---------------- + ---------------- = 0 ∂t ∂x ∂y ∂z

(3.9)

For an incompressible flow, ρ is constant, so Eq. (3.9) reduces to ∂u ∂u ∂u --------x + --------y + --------z = div.u = 0 ∂x ∂y ∂z

(3.10)

where div.u is divergence of velocity vector u . Equations (3.9) and (3.10) are known as the equation of continuity. Equation of Motion This is based on Newton’s second law of motion, i.e., Eq. (3.1). Let us consider the fluid element depicted in Figure 3.1. As it moves, its change of properties is a consequence of both change of position and of time. When it is considered this way, and the fluid element is followed in time and space, then the rate of change of a property is expressed by its substantial derivative. For a velocity vector, it would be ∂u ∂u ∂u Du ∂u a = ------- = ------ + u x --------x + u y --------y + --------z ∂x ∂y ∂z Dt ∂t Du Where ------- is substantial derivative of u. Dt ©2001 CRC Press LLC

(3.11)

Equation (3.1) may be restated as follows: Rate of change of momentum of the fluid element = (body forces) + (pressure forces) + (shear forces) acting on the fluid element

(3.12)

For an incompressible fluid and per unit volume of fluid, it may be derived, on the basis of concepts already presented, that Du 2 ρa = ρ ------- = ρF B – ∇P + µ∇ u Dt

(3.13)

where F B is body force per unit mass of the fluid. If gravitational force is taken as the only body force, then Du 2 ρa = ρ ------- = – ρgk – ∇ρ + µ∇ µ Dt

(3.14)

where k is the unit vector along the z-direction (i.e., vertical direction). ∇ is the Laplacian of u . For a rectangular coordinate system, it is given as 2

∂ u ∂ u ∂ u 2 ∇ u = ---------2-x + ---------2-y + ---------2-z ∂x ∂y ∂z 2

2

2

(3.15)

Equation (3.14) is the well known Navier–Stokes equation. For laminar flow and simple situations, analytical solutions are available in standard texts. For complex but laminar flow, resort to numerical methods. The Navier–Stokes equation, with certain modifications and empiricism, has been applied for fluid flow computations in turbulent flow as well. Of course, the procedures involve lengthy computer-oriented numerical methods. The vectorial equations can be resolved into three component equations along the three coordinates. These are available for the three standard coordinate systems, viz., rectangular, cylindrical, and spherical, in standard texts.

3.1.4

DIMENSIONLESS VARIABLES

The importance of dimensionless variables in momentum, heat, and mass transfer is well known. These are also known as dimensionless numbers and are widely employed. According to Buckingham’s π-theorem, the number of dimensionless variables is n – r, where n is the number of physical variables and r is the number of basic dimensions. In fluid flow, r = 3 (mass, length, time). For example, in laminar flow of a fluid past a sphere, the drag force (FD) exerted by the fluid on the sphere is a function of four variables, i.e. diameter of sphere (d), bulk flow velocity of fluid (uo), viscosity, and density of fluid (µ, ρ). In other words, FD = f(d, uo, µ, ρ)

(3.16)

According to the π-theorem, the number of dimensionless variables is (5 – 3), i.e., 2. Dimensional analysis leads to the following formulation: Eu = f(Re)

(3.17)

where Eu is Euler number, which is equal to F D ⁄ ρu o d , and Re is the Reynolds number (= ρuod/µ). 2


2

A reduction in the number of variables is a tremendous advantage of dimensionless numbers, as it greatly simplifies the experimental data collection program. Representation of data by equations, graphs, or tables is also enormously simplified. Another advantage is that the correlations become independent of the unit employed. Dimensionless numbers have physical significance as well. They are proportional to the ratios of forces. For example, resultant (i.e., inertia) force Re α ----------------------------------------------------------------viscous force

(3.18)

Because of this, they are employed as criteria for dynamic similarity in two different flow situations. An important application of similarity criteria and dimensionless numbers is in the area of physical modeling of processes in connection with design and development. For example, small models of aircraft are tested in a wind tunnel in connection with their design. In the area of steelmaking, room temperature, laboratory-size models, simulating liquid steel by water (i.e., water models), are very popular and have advanced our understanding of steelmaking processes significantly. Extrapolation of model results to actual prototypes is more reliable, provided there is dynamic similarity between the two through equality of relevant dimensionless numbers. Of course, the model and prototype have to be geometrically similar. Which dimensionless numbers would be relevant for simulation of a flow situation would depend on the significance of the forces. Table 3.1 lists the numbers important in secondary steelmaking. TABLE 3.1 Important Dimensionless Numbers in Secondary Steelmaking Name

Symbol

Definition

Force proportionality ratio

Reynolds number

Re

ρuL ---------µ

inertial -----------------viscous

Froude number

Fr

u -----gL

Modified Froude number

Frm

Weber number

Morton number

2

Application General fluid flow

inertial -----------------------------gravitational

In forced convection

ρg u --------------------------( ρ l – ρ g )gL

inertial -----------------------------gravitational

Gas–liquid system

Wb

ρµ L ------------σ

inertial -----------------------------------surface tension

Mo

gµ ----------3 ρl σ

2

2

4

( gravitational ) × ( viscous ) ---------------------------------------------------------------surface-tension

Gas bubble formation in liquid

Velocity of gas bubbles in liquid

L = characteristic length (such as diameter for a pipe or sphere) ρg =density of gas, ρl =density of liquid σ = surface tension of a liquid

3.1.5

TURBULENT FLOW

AND ITS

ANALYSIS

Fluid flow in metallurgical processes is turbulent in nature. As already stated in Sec. 3.1.2, the flow is characterized by random fluctuations in velocity at any location in the flow field as well as by random motion of eddies as a consequence. Section 3.1.2 also has discussed differences between laminar and turbulent flow. Laminar flow is characterized by well developed stable streamlines. On the other hand, a turbulent flow exhibits rapid mixing and no stable streamline. This was nicely demonstrated by Osborne Reynolds more than a century ago. When he injected some red dye in ©2001 CRC Press LLC

laminarly flowing water in a glass tube, the dye moved along a stable streamline, which was like a red thread that did not mix with surrounding water. But, in turbulent flow, the dye mixed up with water rapidly, thus making the entire water red. Turbulence is complex in nature and, in spite of a large number of studies, is not understood properly. In many typical situations, such as flow through a pipe, the disturbance caused by the presence of a solid surface in contact with the fluid causes the onset of turbulence. Laminar flow is observed at small Reynolds number values, e.g., 2100 for flow through a straight pipe. At values of Re larger than this critical Re, the flow is turbulent. Sometimes, a laminar-to-turbulent transition is characterized by a transitional flow regime, as in the case of flow through a pipe. Value of critical Re depends on the flow geometry. For example, in a flow around a sphere, Recrit is approximately 0.1. Eddies exhibit a large size range. The largest one may be comparable to the size of the vessel and the smallest one less than a millimeter. The smaller an eddy, the higher is its jump frequency and hence consequent frequency of velocity fluctuation at a point, which may be as high as approximately 1000 per second. Large eddies derive their energy from the main flow and may contain as much as 20% of the kinetic energy of the turbulent motion. The interaction among larger eddies generates smaller eddies, and so on. The smaller the size of an eddy, the higher its specific kinetic energy (i.e., kinetic energy per unit mass or volume). Also, smaller eddies are isotropic, whereas larger eddies tend to exhibit anisotropy. Dissipation of the kinetic energy of an eddy into heat can occur only through viscous forces. With decreasing eddy size, viscous forces increasingly resist further disintegration of eddies. The smallest eddy size (lmin) may be estimated using Kolmogorov’s equation5 as ν l min = ----εd

3 0.25

(3.19)

where ν is kinematic viscosity and εd is the rate of total kinetic energy dissipation of the turbulent motion. Under secondary steelmaking conditions, lmin is on the order of fraction of a millimeter. Figure 3.7 schematically shows the eddy size distribution of a fully developed turbulent flow as a function of the inverse of eddy size (le). The size distribution in large-sized eddies depends on the flow pattern and vessel geometry. However, for a fully developed, steady turbulent motion,

FIGURE 3.7 Diagrammatic scheme of the energies of eddies of different sizes relative to the reciprocal of the eddy length.


the size distribution for small eddies attains equilibrium and is probabilistic. It is independent of flow conditions and is known as Universal equilibrium range. In turbulent motion, the velocity at any instant of time (u) at a location may be considered as consisting of three components (ux, uy, and uz) along x, y, and z coordinates, respectively. u x = u x + u′ x ; u y = u y + u′ y ; u z = u z + u′ z

(3.20)

where u is the time-averaged (i.e., mean) velocity of the fluid, and u´ is the fluctuating component of velocity. Again, to

1 u x = --- ∫ u x ( t )dt to

(3.21)

o

Similarly, u y and u z may be defined. Here, to is the time over which averaging is done. u is the quantity measured by ordinary instruments. As already stated, u is independent of time at steady state. u´ can be measured only by special fast-response instruments such as a hot film anemometer or a laser Doppler anemometer. Intensity of turbulence is measured by the relative magnitudes of u´ and u . Since the value of u´ at a location fluctuates at random, some kind of averaging of u´ is required. However, u′ = 0 by definition. Hence, intensity of turbulence (I) has been defined as: 1⁄2 1 2 2 2 --- ( u′ x + u′ y + u′ z ) 3 I = ----------------------------------------------------u

(3.22)

For isotropic turbulence, u′ x = u′ y = u′ z = u′ and hence Eq. (3.22) reduces to Eq. (3.23), i.e., 2 1⁄2

[ u′ ] I = -----------------u

(3.23)

The numerator in Eq. (3.23) is root mean square of u´ and is a non-zero quantity. Another fundamental parameter is a set of Reynolds stresses. These are ρu′ x u′ y , ρu′ x u′ z , and ρu′ y u′ z . These have dimensions of stress. They can also be shown to be related to the rate of momentum transport by eddies along a velocity gradient, which is another physical interpretation of shear stress. Hence, Reynolds stresses have physical existence and are not just conceptual quantities. An important quantity, commonly used for characterizing turbulent flow behavior, is turbulent kinetic energy per unit mass of fluid (k), defined as 1 2 2 2 k = --- [ u′ x + u′ y + u′ z ] 2

(3.24)

3 2 k = --- [ u′ ] 2

(3.25)

For isotropic turbulence,


Momentum transfer by eddy mechanism has one similarity to transfer by molecular mechanism. Motions of both eddies and molecules are random in all directions. Only when a velocity gradient is present does net momentum transfer occur. Therefore, in analogy with Newton’s law of viscosity, we may write ∂u ( τ yx ) t = – µ t --------x ∂y

(3.26)

µ t is called eddy viscosity or turbulent viscosity and was first introduced by Bousinesq. Therefore, in turbulent motion, µ eff = µ t + µ

(3.27)

where µ eff is the effective viscosity of flow. Since an eddy contains trillions or more molecules, µ t » µ, and µ eff is approximately equal to µ t. The difficulty with the eddy viscosity concept is that µ t is not a property of the fluid but depends on the nature and intensity of turbulence as well. Hence, for quantitative analysis, µ t is to be correlated with measurable variables governing fluid flow. The “one equation model” based on Prandtl’s concept of mixing the length of eddies (lm) proposes the following relationship: µ t = ρ lm k1/2

(3.28)

However, it has been found to be inadequate. Among the “two equation” models, the one by Launder and Spalding,6 popularly known as k-ε model, proposes ρk µ t = C D -------ε 2

(3.29)

where ε is rate of dissipation of k, and CD is an empirical constant equal to 0.09. Numerical computations of turbulent flow are done by solution of the turbulent Navier–Stokes equation with the help of the eddy viscosity concept. These are complex calculations requiring large computer time. The basic procedure will be presented in Chapter 11. Calculated results will be presented in subsequent chapters wherever required.

3.2 FLUID FLOW IN STEEL MELTS IN GAS-STIRRED LADLES Recently, Mazumdar and Guthrie7 have reviewed fluid flow, mixing, and mass transfer in gas stirred ladles. Stirring may be achieved by purging the steel melt with argon and also by gases liberated from the melt during processing. The latter is important for successful processing during vacuum degassing of liquid steel, when the evolution of CO, H2, and N2 causes vigorous stirring in the melt and consequent droplet ejection into the evacuated chamber. Electromagnetic stirring is also employed in secondary steelmaking. However, the most common situation is stirring by the purging of inert gas. Hence, so far as fundamental studies on fluid flow are concerned, principal attention has been focused on turbulence and agitation in the ladle due to argon purging. The gas is introduced into the melt mostly through porous plugs fitted at the ladle bottom (i.e., bottom purging) or by a lance immersed vertically into the melt from the top. Fluid motion and turbulence in both these arrangements have some similarities but have differences as well. It is the bottom purging geometry that has been most widely studied. Hence, this section is restricted to fluid flow in liquid due to bottom purging of inert gas. Liberation of gases from the ©2001 CRC Press LLC

melt due to reactions is ignored. In industry, small ladles are fitted with one bottom plug and large ones with two plugs. Again, these are typically not axisymmetric with the ladle but are fitted in eccentric fashion. However, fundamental investigations have been mostly conducted in the laboratory with transparent room-temperature water models. In these studies, mostly axisymmetric nozzles were employed for the sake of simplicity and basic interpretations. The presentations in this section are aimed at elucidation of fundamentals. Our understanding of the same will be better if the above background information is kept in mind. Fluid flow in industrial situations that differ significantly in the mode of stirring and melt geometry is taken up in later sections when necessary and possible. Even there, the information contained in this section provides the basics. Table 3.2 presents some values of gas flow rates per unit volume of liquid in ladle refining of liquid steel as well as other situations involving bottom purging. TABLE 3.2 Gas Flow Rates (Qv) per m3 Bath Volume Situation Ar-stirred ladle

Qv , m3 s–1 × 104 per m3 of bath

Reference

44–1800

8

-do-

3–54

9

-do-

25–330

10

Water model

17–240

11

OBM converter

25000

12

For calculation of Qv, gas temperatures were assumed as 300 K for the water model and 1873 K for the steel melt. The overall range of Qv for steel melt may be taken as 10 × 10–4 to 100 × 10–4 s–1. The table also demonstrates that Qv is 2 to 3 orders of magnitude lower in ladle refining than that in an OBM converter. In a combined top and bottom blowing converter, the bottom gas injection rate is few percent (say at least 2%) of the oxygen blowing rate through top lance. Hence, Qv would be at least 500 × 10–4 s–1. Even then, it is a few times larger than typical values of Qv in ladle refining. In converter steelmaking, tuyeres (i.e., nozzles) are typically employed for bottom gas injection. It is a must if oxygen is introduced, either singly or mixed with inert gas. The high velocity of gas issuing from tuyeres causes jetting flow and prevents back attack of tuyeres of molten metal. In ladle refining, on the other hand, the flow rates and velocities of gas are low. Hence, a jetting regime cannot be obtained, and porous or slit refractory plugs are more suitable than tuyeres. Moreover, bubbles are statistically smaller in size in porous plug systems than for nozzles. This enhances the refining rate due to a larger gas–liquid interfacial area. Hammerer et al.13 have reviewed latest developments in gas-purging plugs. Figure 3.8 shows sketches of them. These may be classified as • plugs made of porous refractory material • segment-purging plugs (i.e., slit plugs) with gas flow by round channels or predominantly straight slits in refractory shapes Besides permeability, durability, and economy, operational safety is important. Formation and release of gas bubbles cause pressure fluctuations in the gas line, leading to back attack of plugs by steel melt. Plugs get damaged by penetration of liquid metal into them through back attack, as well as by peeling at the surface in contact with the melt. Both of these are less common for slit plugs, which are therefore more popular. Porous plugs are employed for gas bubbling at moderate rates only (less than 0.01 Nm3s–1). ©2001 CRC Press LLC

FIGURE 3.8 Kinds of purging plug systems.

3.2.1

GAS BUBBLES

IN

LIQUID

The behavior of gas bubbles in liquid has been extensively studied. Szekely1 has reviewed it and may be referred to for more details. Growth and motion of single bubbles in liquid are reasonably well understood. In gas flow through submerged nozzles, discrete bubbles form at a low flow rate, and gas issues at a high flow rate as a jet from the nozzle. At low flow rates, the bubble diameter (dB), upon detachment from the nozzle, is determined by a balance between surface tension and buoyancy force. For an air-water system, the following correlations have been proposed: 6d n σ d B = ----------------------g ( ρl – ρg )

1⁄3

, for Re < 500

(3.30)

and 1⁄2

1⁄3

d B = 0.046d n Re n , for 500 ≤ Re ≤ 2100

(3.31)

where dn is the nozzle diameter, and other symbols are as noted in Table 3.1. The nozzle Reynolds number (Ren) is given as ρ g d n u n ⁄ µ g , where un is linear velocity of gas in the nozzle. Liquid metals are non-wetting to a refractory nozzle or plug. Hence, dn in Eqs. (3.30) and (3.31) is to be taken as the outside diameter of nozzle rather than the inside diameter. A porous plug may be considered as a collection of fine tubes (straight or zigzag). At low gas flow rates, discrete bubbles form at the plug exit, but at higher flow rates bubbles coalesce at the plug exit itself and assume a large size before detachment. Anagbo and Brimacombe,14 in their water model study, found coalescence above a flow rate of 0.4 m3 s–1 per m2 of plug area. In Table 3.2, noting that the argon stirred ladles have a 60-tonne capacity, and assuming 0.15 m as the diameter of the porous plug, the critical value of Qv turns out to be 8.3 × 10–4 m3 s–1 per m3 bath volume. This is indeed low. Moreover, bubble detachment would be more difficult in liquid metal due to the non-wetting nature of the liquid. Hence, it may be concluded that coalescence before detachment is expected. It is shown schematically in Figure 3.9, which depicts a simplified situation only. For example, with segment-purging plugs, fine streams of gas would be issuing out through parallel channels. There, coalescence is expected only at a higher gas flow rate. Hence, smaller bubble sizes and faster gas-liquid reaction are expected even at a higher flow rate. Moreover, ©2001 CRC Press LLC

Wetted

Non-wetted Nozzle

Orifice

Porous plug

FIGURE 3.9 Bubble formation at wetted and nonwetted nozzle, orifice, and porous plug.

pulsations would be much less in the absence of coalescence. This explains why there is less back attack in segment-purging plugs as compared to that in porous brick plugs. An additional refinement in the calculation of dB has been the incorporation of the effect of the volume of the antechamber (Vc), which is defined as the volume between the last location for a large pressure drop (i.e., a valve) and the actual nozzle or orifice. Due to gradual rather than instantaneous buildup of pressure in the antechamber, the final volume of the bubble, upon detachment, would be larger than calculated from Eqs. (3.30) and (3.31). Sano and Mori29 have proposed an empirical correlation relating dB to dn,o´, dimensionless antechamber volume (capacitance number, Nc), and other variables. Experimental measurements in liquid iron have demonstrated good 15 agreement with the above. Figure 3.10 shows significant dependence of dB on N c . A great deal of theoretical work has been done on the rising velocity and shape of gas bubbles in liquids, and in most cases the theory is in quite good agreement with measurements. There are

FIGURE 3.10 A comparison of experimentally measured bubble diameters in molten metals with predictions based on the analysis of Guthrie and Irons, demonstrating the effect of the capacitance number.15


several dimensionless numbers controlling bubble shape.13 An important one is the bubble Reynolds number, defined as d B uB ρl Re B = --------------µl

(3.32)

where uB = linear rise velocity of bubble. Small bubbles (ReB < 1) are spherical in shape. Large bubbles (ReB > 1000), under some other restrictions, are spherical cap shaped. To satisfy various conditions, they have to be larger than 1 cm in diameter in liquids of low viscosity (e.g., water or liquid metals). Ellipsoidal and other shapes are obtained at an intermediate Reynolds number. Bubbles rise by pulsation and can be swirling, too. Considerable circulation is also present in the gas inside. Small spherical bubbles behave like rigid spheres, and the terminal velocity (ut ) is given by Stokes’ law, viz., 2

dBg - ( ρ – ρg ) u t = ---------18µ l l

(3.33)

ut for spherical cap bubbles can be expressed as gd u t = 1.02 --------e 2

1⁄2

(3.34)

where de is diameter of a sphere whose volume is equal to that of the bubble. Experimental measurements in water as well as in molten metals indicate that a better approximation is obtained if the coefficient is taken as 0.9 rather than 1.02 for de less than 3 cm. In industrial situations, we are concerned with assemblages of interacting bubbles, known as bubble swarms, rather than single bubbles. In such systems, bubbles behave differently from single bubbles, and our knowledge is much more limited due to the complexity of the situation. An important parameter characterizing a bubble swarm is gas hold up (α), where volume of gas in gas/liquid mixture α = ------------------------------------------------------------------------------------total volume of mixture

(3.35)

Based on this parameter, essentially three regimes may be identified. 1. Bubbling regime (α < 0.4) 2. Froth (α ≅ 0.4 – 0.6) 3. Foam (α ≅ 0.9 – 0.98) In ladle refining, we are concerned with the bubbling regime only. Almost all studies on bubble swarms have been conducted in room-temperature systems with water or other transparent liquids. The upward velocity of a rising bubble may be considerably larger than what Eqs. (3.33) or (3.34) would predict, because the upward motion of the liquid assists bubble motion.

3.2.2

THE PLUME

IN A

GAS-STIRRED LIQUID BATH

Figure 3.11 shows a sketch of a simplified situation when the gas is introduced into a cylindrical vessel containing liquid via a nozzle located at the bottom and placed along the axis of the vessel. ©2001 CRC Press LLC

FIGURE 3.11 Sketch of the situation in a gas-stirred melt.

Around the axis of the vessel, there is a two-phase region consisting of gas bubbles and liquid. This is known as the plume. Upward movement of bubbles in the plume leads to circulation of liquid in the vessel. Such a liquid motion is called recirculatory flow, and it is turbulent as well. At low gas velocities, discrete bubbles form at the nozzle/plug tips. The resulting plume is known as an ordinary plume. At higher velocities, a continuous gas stream issues out into the liquid. It subsequently disintegrates into discrete bubbles at a short distance after exiting from the nozzle or plug. It is called a forced plume. Henceforth, both will be referred to simply as plume. If the plume is idealized as a truncated cone, its profile can be characterized in terms of its cone angle (θp), as shown in Figure 3.11. The plume profile and its dimensions depend on various operating variables. The most critical of these is the gas velocity at the nozzle tip. Measurement of the plume cone angle in liquid steel is obtained indirectly from the plume diameter as it emerges at the top surface. This is not reliable, since the profile deviates from that of a cone near the top surface. Photographic measurements in water models have indicated a range of 20 to 30 degrees. Krishnamurthy et al.16 made comprehensive measurements of θp as a function of gas flow rate (Q), bath height (H), vessel diameter (D), and nozzle diameter (dn). They employed an axisymmetric nozzle at bottom of the vessel and, through regression fitting of data, obtained the correlation: – 0.441 – 0.254 θp 0.12 H  d-----n -------- = 0.915Fr m  ----  D  D 180

(3.36)

The definition of modified Froude number (Frm) was provided in Table 3.1, where u is to be taken as the velocity of gas issuing out of the nozzle, and L means H. Xie et al.17 determined plume width in molten Wood’s metal at 100°C and compared this with measurements in a water model and mercury. Statistically speaking, no difference could be obtained, indicating that we may employ Eq. (3.36) for estimating θp in liquid steel, of course, after incorporating the actual value of Q upon exit from the nozzle at bath temperature. Although the above discussions would provide a simple approach to estimation of θp, it is to be kept in mind that the plume diameter, at least near the top surface, is likely to be larger in steel melt than in water as a result of the greater density of steel and consequently more bubble expansion. ©2001 CRC Press LLC

Significant experimental data have been collected on physical characteristics of the plume in water as well as low-melting metals over the last five years. Electrical resistivity probes have been employed to determine dispersion of gas bubbles as characterized by local time-averaged gas fraction, bubble size, bubble frequency, and bubble rise velocity. The use of hot film anemometers and laser Doppler velocimeters has allowed the measurement of liquid velocity in water models. Since all these depend on the height above the nozzle tip as well as the horizontal radial distance from the nozzle axis, data have been collected as a function of both. In a horizontal section, maximum gas fraction (αmax) is obtained at axial location, i.e., at radius (r) = 0. Several investigators17,18 have suggested Gaussian distribution, i.e., r α ---------- = exp  – -----2  b α max 2

(3.37)

where b is an empirical constant. Comprehensive measurements by Castillejos and Brimacombe19 in water and mercury have yielded the following regression-fitted empirical correlation for bottom blowing through an axisymmetric nozzle in a water model: r 2.4 α ---------- = exp – 0.7  ----  r 1 α max

(3.38)

At r = r1, α max α = --------2 Equation (3.38) demonstrates that the decay of α/αmax with increasing r is more than that predicted by the Gaussian curve. Data of others17,18 also seem to suggest the same, although they have fitted with the Gaussian curve. Castillejos and Brimacombe19 proposed further that g r 1 ------2 Qn

1⁄5

 gd 5 = 0.275  -------2-n  Qn 

0.155

ρ1   ---- ρg

0.11

Z  --- d n

0.51

(3.39)

and, αmax = 0.815 N–0.1, for N < 1.35 = 1.069 N–1, for N ≥ 1.35

(3.40)

where Qn is volumetric gas flow rate at the temperature and pressure prevailing at nozzle exit, and N is a dimensionless parameter equal to  gd 5n  -------2-  Qn 

0.26

ρ -----1  ρ g

0.13

Z  --- d n

0.94

where Z is the vertical distance from the nozzle exit. ρg is the density of gas at nozzle exit. Xie et al.17 carried out similar investigations with Wood’s metal at 100°C (ρ1 = 9.4 × 103 kg –3 m ) and have proposed a Gaussian distribution of α as in Eq. (3.37). On the basis of regression fitting of experimental data, they have proposed the following relations: ©2001 CRC Press LLC

b = 0.28 ( Z + H o )

7 ⁄ 12

( Qz ⁄ g )

α max = 0.65 [ Q n ( ρ 1 ⁄ σ 1 g ) 2

2

1⁄2 1⁄4

]

1 ⁄ 12

/( Z + H o)

(3.41) (3.42)

where Ho is the axial distance of the hypothetical origin of conical plume from the nozzle exit, 0.1 1⁄2 2 given as 4.5d n ( Q n ⁄ g ) . σ1 is the surface tension of the liquid, and Qz is the gas flow rate at pressure and temperature at a height from the nozzle exit. For Wood’s metal, σ1 is 0.46 N m–1. Figure 3.12 compares some calculated values of α as a function of r for water and Wood’s metal based on the above equations. Xie et al.17 also made measurements with gas blowing through eccentric nozzles, and they proposed the following correlations:  r – r 2m α ecc,r ----------- = exp –  ------------2  α max  br 

(3.43)

2 α ecc,z Z ----------- = exp –  ----2- b  α max z

(3.44)

and

where rm is the radial distance of the nozzle from the center of the vessel. It was observed that, despite the stable lateral deflection mentioned, the distribution of α was symmetric around the axis of the nozzle. bz = 1.17br on the average, meaning that the plume was an elliptic cross section. From measurements of α, the mean velocity of the gas stream (ug) in a cross section of the plume can be determined as the ratio of gas flow rate to the total gas fraction in the cross section, i.e.,

FIGURE 3.12 Comparison of void fraction vs. radial position data for different investigators.


–1 u g = Q  ∫ α d A  

(3.45)

A

Based on Eqs. (3.41), (3.42), and (3.45), and assuming radial symmetry of the plume around the axis for a centric nozzle, ug = 75.73 [Q/(Z + Ho)]1/6

(3.46)

A measure of the extent of total gas holdup may be taken as

∫A α d A A refers to plume cross-sectional area. It was found that this parameter was the same for centric and eccentric gas blowing. Some measurements of time-averaged upward velocity of liquid in the plume ( u 1 p ) are available in the literature for a nozzle-fitted water model.9,10,18,20 ( u 1 p ) is at maximum along the axis. Values ranged between 0.2 and 0.3 ms–1, depending on value of Qn. Radial distribution of velocity is Gaussian, with the maximum value along the axis, i.e., u1 p 2 2 --------------= exp ( – r ⁄ b u ) u 1 p,max

(3.47)

where bu is a constant that depends on gas flow rate, height above the nozzle, etc. Quantitative correlation of experimental data has been proposed by Oeters et al.10 for centric blowing of air in water as – 0.12

(3.48)

bu = 0.38 Q0.15 Z0.62

(3.49)

u 1ρ,max = 3.37Q

0.25

Z

It shows that the axial velocity, u 1 p,max , does not vary much with vertical distance, Z. This has been corroborated by other investigators as well. This is in contrast to a free gas jet where the axial velocity decreases rapidly as Z increases. The difference lies in the fact that the rising bubbles impart upward momentum to the entrained liquid throughout the plume volume due to buoyancy force, whereas the momentum of a free jet is derived solely from its momentum upon exiting from the nozzle. The buoyant plume may be visualized as a pump, making the liquid flow upward. The volumetric flow rate of liquid at any horizontal section Ql may be obtained from Eqs. (3.47), (3.48), and (3.49) as Q1 =

∫A ulp dA

0.55

= 1.52Q n Z

1.13

(3.50)

Oeters et al.10 also proposed that the above correlations may be employed for liquid steel as well, provided that the gas flow rate (Qn) is calculated for the actual pressure and temperature of gas at the nozzle exit. Moreover, a broadening of the plume in liquid steel may be taken care of by substituting Z with Z · ψ, where ©2001 CRC Press LLC

1 1 ψ = --- ln ----------------β (1 – β)

(3.51)

Z β = ---------------ha + H

(3.52)

and

where ha = the height of liquid steel equivalent to the atmospheric pressure. However, Eqs. (3.49) through (3.52) would require verification and may be treated as approximate guidelines only. Time-averaged bubble rise velocities in the plume ( u B ) have been measured in water models18–20 and Wood’s metal,17 and ( u B ) did not vary significantly along the axis of the nozzle. The variation was much less in the radial direction as compared to α and ( u lp ) . For example, ( u B ) was more than 60% of its axial value in all cases, and sometimes the profile was almost flat. In water, the overall range obtained by the investigators ranged from 0.2 to 2 ms–1 depending on Q and Z. In Wood’s metal, ( u B ) along the axis ranged between 0.5 and 0.7 ms–1, and it was found to be proportional to ug. It has been proposed17 that Eq. (3.46) can be used to estimate ( u B ) as well, except that the coefficient should be 54.51 in place of 75.73. Comparison of ( u B ) and ( u lp ) in water model studies revealed that ( u B ) was always larger than ( u lp ) . This is due to bubble slip. Time–averaged bubble slip velocity ( u s ) = u B – u lp

(3.53)

( u s ) varied approximately between 0.2 and 0.4 ms–1. On the basis of their experimental data and comprehensive analysis of the same, Sheng and Irons20 have shown that ( u s ) was approximately the same as the rise velocity of a bubble of equivalent diameter in stagnant water. The spout region of the plume occupies only 3 to 4 percent of its total volume but is of importance in connection with processes occurring at melt surface, such as gas absorption, slag metal reaction, etc. Sahajwalla et al.21 investigated this region using an electroresistivity probe in a water model. The gas fraction was at minimum at the axis and increased with radial distance from the axis to a value of 0.95. This is in contrast to what has been found in the rest of the plume. Photographic measurements of plume dimensions corresponded to α = 0.82 to 0.86. Variation of spout radius (rs) with other parameters was expressed by the following relationship: g r s  ------2 Q  n

1⁄5

= 0.48Z + 8.3 ( Fr ) *

– 0.18

(3.54)

2 1⁄5

where Z = Z ( g/Q n ) *

The bubble frequency ranged between 14 to 16 s–1 along the axis and decreased to 1 to 4 s–1 toward periphery. The plume oscillation frequency was 2 to 4 s–1. For gas-stirred industrial ladles, the purging time is often not too long. A relevant question there is whether the flow is unsteady (i.e., transient) or steady. One mathematical modeling work22 found that a steady state could be attained in three minutes. This was about the time required to obtain good thermal homogenization as well. It was also found that the time for homogenization was approximately the same regardless of whether the flow was taken as transient or steady. As already stated, fundamental studies with gas blowing through porous plugs are limited. Two water-model investigations14,23 have provided some information. These were porous glass-disc and not segment-purging plugs. Gas fraction measurements by electroresistivity probe yielded the correlation for axial location14 as ©2001 CRC Press LLC

Z α max = 0.71 ---------------------– 0.2 2 ( Q n /g )

– 0.9

(3.55)

Radial variation of α was found to obey Gaussian distribution. The mean bubble diameter showed a sudden increase with the onset of coalescence. Axial and radial velocity components of the liquid varied along the radial direction in a similar fashion as for gas flow through the nozzle. In the design of a water model, besides geometric similarity, the most important dimensionless number considered for simulation is the modified Froude number (Frm) as defined in Table 3.1. In connection with a gas-purged ladle, it is to be rewritten as un ρg Fr m = ------⋅ -------------------gH ( ρ 1 – ρ g ) 2

(3.56)

where un is the linear velocity of gas issuing through a nozzle. For porous plugs, un does not have any meaning and cannot be determined from the gas flow rate. For argument’s sake, let us take the example of a 60 t ladle in Table 3.2 with a porous plug diameter of 0.15 m, and assume entire plug surface area as a nozzle. For Qv = 330 × 10–4 m3s–1 per m3 bath volume, nominal value of un would be 16 ms–1 and Frm = 0.57. This may be compared with Frm >10 if a tuyere was employed, and Frm >100 in bottom-blowing converters. Hence, the inertial force is too small to significantly influence the flow for a porous plug, and Frm should not be a relevant criterion for simulation from this point of view. Mazumdar and Guthrie7 have also questioned relevance of Frm in gas injection through a porous plug. Forces that are expected to govern the nature of flow are • • • •

3.2.3

buoyant force of the rising plume inertial force due to liquid motion surface forces at the top of the bath viscous shear forces at ladle wall

FLOW FIELD

IN

LIQUID OUTSIDE

THE

PLUME

As Figure 3.11 shows, the flow induced by the plume is recirculatory in liquid outside the plume. Velocities have been measured in water models by laser Doppler velocimeter (LDV) for axisymmetric nozzles10,20 as well as porous plugs.14,23 Figure 3.13 shows a typical velocity field.20 Sahai and Guthrie24 were among the earliest to attempt characterization of the recirculatory flow. They summarized that hydrodynamic conditions near an axisymmetric nozzle or plug are not critical to flow recirculation in large cylindrical vessels. This view is considered to be valid even now. This is because the flow is primarily driven by the buoyant force of rising gas bubbles. Hence, we may assume the velocity fields to be fairly similar for a porous plug as for a nozzle, provided that other conditions (viz., gas flow rate and bath dimensions) remain the same. Figure 3.14 shows flow patterns for air injection into water for various locations of a porous plug.25 The existence of dead zones near the bottom of vessel, especially at the bottom corners, is well established. The main flow torus has a chance to come close to the bottom only at a high gas flow rate and with the H/D ratio ranging between 0.4 and 0.8.10 The geometry and size of the dead zone are dependent also on the gas-purging arrangement. Figure 3.15 shows this schematically for various arrangements.26 Figure 3.13 demonstrates considerably higher velocity in the plume region than in the bulk liquid. Velocities are very small near the wall and bottom of a vessel. Quantitatively, the axial velocities ranged from 0 to 0.4 ms–1 and radial velocities less than 0.1 ms–1 for gas flow rates ranging from 10–4 to 10–3 Nm3 s–1, which covers the ladle refining conditions, generally speaking. ©2001 CRC Press LLC

FIGURE 3.13 Flow pattern of the mean liquid velocities in the model ladle produced with the flush-mounted nozzle at Q = 10–4 Nm3 s–1.20

Measurement by LDV is also capable of determining values of RMS of the fluctuating component of velocity, i.e., 2 1⁄2

( u′ )

Figure 3.16 shows a plot of this, obtained in axisymmetric blowing through the nozzle in the water model.20 Qualitative similarity with velocity field (Figure 3.13) may be noted. Turbulent kinetic energy (k) as defined in Eq. (3.25) is another parameter of importance in characterizing turbulence. Figure 3.17 shows isopleths of k20 for the same experimental conditions as for Figures 3.13 and 3.16. Very low values of k in the bulk and high values in plume region may again be noted. A fundamental parameter characterizing turbulence is the intensity of turbulence (I) as defined in Eq. (3.23). Sheng and Irons20 found I to be approximately 0.2 for bulk flow and larger than 0.5 in the plume region, and turbulence was isotropic. Ballal and Ghosh27 simulated a bottom-blown steelmaking converter process using a water model. They were interested in stresses on the bottom and side wall due to fluid motion. Air flow rates were higher than those employed in the simulation of ladle flow by other investigators. The number of nozzles was 1, 3, 6, and 12. The single nozzle was axisymmetric, and multinozzles were either symmetrically or asymmetrically located around the vessel axis. Tiny platinum electrodes were flush mounted at various locations on the bottom and wall of the vessel. The electrochemical technique was employed to determine saturation current density, which yielded concentration and velocity gradients at the surface and hence wall shear stress ( τ ) . Specially designed electronic instruments allowed the determination of both mean shear stress ( τ ) and RMS of a fluctuating component. It was found that, for a certain nozzle arrangement, in the entire gas flow range, 2 1⁄2

( τ′ )

= Cτ

(3.57)

where C is a constant. Noting that C may be taken as I, values of I ranged from 0.33 to 0.53. ©2001 CRC Press LLC

FIGURE 3.14 Different positions of porous plugs and the resulting flow patterns for bottom gas injection.25

Mazumdar et al.28 carried out measurements of fluctuating and mean velocities of liquid at several locations of the bath by LDV in a water model with centric gas injection by nozzle. They employed four gas flow rates and three arrangements, viz., free bath surface, surface covered by a floating wooden block, and surface covered with 15 mm thick oil layer. Averaging over the bath yielded values of the average speed of bath circulation ( u av ) and the averaged RMS of the fluctuating velocity component, 2 1⁄2

[ u′ ] av

The I obtained by taking the ratios of these ranged between 0.2 and 0.31, with a master average value of 0.25. Summarizing all these findings, it may be concluded that, under ladle refining conditions, I may be taken as 0.3 or somewhat less in the bulk liquid, on an average. The total energy input through gas (E) is given as E = Eb + Ek + Eexp ©2001 CRC Press LLC

(3.58)

FIGURE 3.15 Flow patterns of liquid in a bath generated by blowing gas at 2.5 × 10–4 Nm3 s–1 through a porous plug.26

FIGURE 3.16 Distribution of the RMS component of liquid velocity in a model ladle at Q = 10–4 Nm3 s–1.20

where Eb = buoyancy energy of the gas bubble in liquid, Ek = kinetic energy of the gas at exit from the nozzle/plug, and Eexp = expansion energy of the bubble during its rise through the liquid. The rate of energy input (ε) is a more relevant parameter. A modified form of Eq. (3.58) is ε = εb + εk + εexp

(3.59)

Krishnamurthy et al.11 tried to assess the contribution of εk to mixing in the bath, which is related to energy utilization for bath stirring. They found εb to be negligible as compared to ε at low gas flow rates. This agrees with observations by others.29 εexp is theoretically equal to εb, and ©2001 CRC Press LLC

FIGURE 3.17 Contour map of the distribution of turbulent kinetic energy in a large vessel, produced with the flush-mounted nozzle for Q = 10–4 Nm3 s–1.20

it was included in calculation of ε in the classic work of Nakanishi et al.30 However, it seems that only a small fraction of this is really utilized in inducing bath motion. In one of the earliest analyses of this, Bhavaraju et al.31 also ignored it. Hence, for gas-stirred ladles and many other situations, investigators take ε = εb for the sake of avoiding complications. It may be an approximation, but it has provided the basis for further advancement in the area of process dynamics in secondary steelmaking and elsewhere. This is because εb can be estimated from experimental conditions easily and reliably. In simple terms, ε = (ρ1gH) · QM

(3.60)

with ρ1gH being the buoyancy force per unit volume of gas. Due to expansion of the bubble as it rises, the value of QM is to be a mean value of volumetric gas flow rate. Bhavaraju et al.31 employed a logarithmic mean value where P Tl Q M = Q. ------o- . -------P M 298

(3.61)

where Q is the gas flow rate in Nm3 s–1, Po is atmospheric pressure, Tl is the temperature of liquid in Kelvins, and PM is the logarithmic mean pressure, given as PH – Po P M = -------------------------ln ( P H ⁄ P o )

(3.62)

PH = Po + ρ1gH

(3.63)

where


Combining Eqs. (3.60) through (3.63), and putting in values, 340QT ε m = -------------------1 ln ( ( 1 + 0.707 )H /P o ) M

(3.64)

where εm is rate of energy input in watts per kilogram of liquid steel. M is the mass of the steel in kg, H is in meters, and Po is in bar. For water, 0.707 in Eq. (3.64) is to be replaced by ρ water - = 0.099 0.707 × -----------ρ steel As stated in Sec. 3.1.3, there are basically two methods of analysis of fluid flow-differential and integral. Differential analysis requires the solution of equation of continuity (Eq. 3.10) and the equation of motion, i.e., Eq. (3.13) or its simplified form, viz., the Navier–Stokes equation, Eq. (3.14). This approach is the most rigorous one and is very popular for numerical computation of fluid flow problems. The Navier–Stokes equation was originally applied to laminar flow. Nowadays, it is employed for turbulent flow as well. This calls for certain modification and empiricism, and it involves computer-oriented numerical methods. As discussed in Sec. 3.1.5, turbulent viscosity (µ t) is not a property of the fluid but depends on nature and intensity of turbulence as well. As already stated in this connection, the k-ε model of Launder and Spalding6 is popular [Eq. (3.29)]. Szekely and his coworkers pioneered this approach for the analysis of fluid flow in metallurgical processing.32,33 One of the most recent papers is by Joo and Guthrie.34 Very useful information has been obtained. However, this is a specialized topic that has been well reviewed by Szekely.11 Moreover, it will be briefly dealt with in Chapter 11 of this book. Hence, further discussion is not presented here. Integral analysis of flow in gas-stirred ladle was initiated by Chiang et al.9 and Sahai and Guthrie.24 One may either go for macroscopic momentum balance or macroscopic energy balance. The latter has given some useful conclusions. Integral analysis has certain limitations, the most important being its inability to predict spatial variation of velocity and turbulence parameters. However, both are well suited for macroscopic predictions and understanding of phenomena and analysis at an elementary level. Hence, it is briefly presented below. Based on their water-model investigations, Sahai and Guthrie24 employed the correlation u av 1⁄3 ------- . ( R ) = 0.18 up

(3.65)

where u av is the average bulk velocity of liquid, and R is the radius of the vessel. Mazumdar et al.,28 from their velocity measurements, also determined the total specific kinetic energy of recirculating liquid. Figure 3.18 shows these as a function of a specific buoyancy input energy rate. It may be noted that the kinetic energy content of recirculating liquid was only a fraction of the total energy input. The fraction was 0.235 for a free bath surface with a floating wooden block. But it was only 0.12 with a slag layer. According to the authors, this was due to the energy required to create oil-water emulsion, and it demonstrates the likelihood of a significant retarding effect of top slag on recirculatory flow of steel melt in a ladle. This demonstrates that only 10 to 30 percent of the input energy is dissipated by turbulence in the bulk liquid, with the rest getting lost due to bubble slippage in the plume, formation of waves and droplets at the surface of the bath, and friction at the vessel wall. It seems that bubble slippage is the dominant one. What this means is that the plume should be treated as two-phase flow rather ©2001 CRC Press LLC

FIGURE 3.18 Plot of total kinetic energy contained in a recirculating aqueous phase vs. energy input per unit of mass liquid for various upper phase conditions.28

than a quasi-single phase flow. Some recent papers have attempted modeling on this basis. But even then, the loss of energy at the free surface, especially in the presence of a slag layer, remains a source of uncertainty in energy balance. The average plume velocity has been correlated with other variables as follows for a water model:26 1⁄3

u p = 4.5 ( Q n H

1⁄4

)/R

1⁄3

(3.66)

Combining Eq. (3.65) with (3.66) yields 1⁄3

u av = 0.79 ( Q n H

1⁄4

)/R

2⁄3

(3.67)

Krishnamurthy et al.35 employed a modified approach taking into account bubble slip in the plume and also employed their data of plume cone angle measurement Eq. (3.36). The following correlations were obtained: u lp = 0.446ε

0.174

(3.68)

and Q1 = 2.81 × 10–3 ε0.625 H0.942 dn0.119

(3.69)

where u lp is the upward average liquid velocity in plume, H is the bath height and Q1 is the volumetric flow rate of liquid in the plume. It may be further noted that Q1 is the volumetric flow ©2001 CRC Press LLC

rate in bulk liquid as well, and it is a measure of rate of liquid circulation. Noting that ε ≅ 104 Q1, dn ≅ 0.01 m, and H ≅ 1 m in the water model, Eq. (3.69) agrees reasonably well with Eq. (3.50) as proposed by Oeters et al.10

REFERENCES 1. Szekely, J., Fluid Flow Phenomena in Metals Processing, Ch. 8, Academic Press, New York, 1979, p. 305. 2. Guthrie, R.I.L., Engineering in Process Metallurgy, Oxford Science Publications, Oxford University, New York, 1989. 3. Bird, R.B., Stewart, W.E. and Lightfoot, E.N., Transport Phenomena, John Wiley & Sons Inc., New York, 1960. 4. Szekely, J. and Themelis, N.J., Rate Phenomena in Process Metallurgy, Wiley-Interscience, John Wiley & Sons Inc., New York, 1971. 5. Davies, J.T., Turbulence Phenomena, Academic Press, New York, 1972. 6. Launder, B.E. and Spalding, D.B., Computer Methods in Applied Mechanics and Engineering, 3, 1974, p. 269. 7. Mazumdar, D. and Guthrie, R.I.L., ISIJ International, 35, 1995, p. 1. 8. Asai, S., Kawachi, M. and Muchi, I., SCANINJECT III, MEFOS, Lulea, Sweden, 1983, p. 12:1. 9. Chiang, H.T., Lehner, T. and Kjellberg, B., Scand. J. Met., 9, 1980, p. 105. 10. Oeters, F., Plushkell, W., Steinmetz, E. and Wilhelmi, H., Steel Research, 59, 1988, p. 192. 11. Krishnamurthy, G.G., Mehrotra, S.P. and Ghosh, A., Metall. Trans., 18B, 1988, p. 839. 12. Etienne, A., CRM Rep., 43, 1975, p. 15. 13. Hammerer, W., Raidl, G. and Barthel, H., Proc. Steelmaking Conf., ISS, Toronto, 75, 1992, p. 291. 14. Anagbo, P.E. and Brimacombe, J.K., Metall. Trans., 21B, 1990, p. 367. 15. Guthrie, R.I.L. and Irons, G.A., Metall. Trans., 9B, 1978, p. 101. 16. Krishnamurthy, G.G., Ghosh, A. and Mehrotra, S.P., Metall. Trans., 19B, 1988, p. 885. 17. Xie, Y., Orsten, S. and Oeters, F., ISIJ International, 32, 1992, p. 66. 18. Iguchi et al., ISIJ International, 32, 1992, p. 857. 19. Castillejos, A.H. and Brimacombe, J.K., Metall. Trans., 18B, 1987, p. 659. 20. Sheng, Y.Y. and Irons, G.A., Metall. Trans., 23B, 1992, p. 779. 21. Sahajwalla, V., Castillejos, A.H. and Brimacombe, J.K., Metall. Trans., 21B, 1990, p.71. 22. Castillejos, A.H., Salcudean, M.E. and Brimacombe, J.K., Metall. Trans., 20B, 1989, p. 603. 23. Johansen, S.T., Robertson, D.G.C., Woje, K. and Engh, T.A., Metall. Trans., 19B, 1988, p. 745. 24. Sahai, Y. and Guthrie, R.I.L., Metall. Trans., 13B, 1982, p.203. 25. Krishnamurthy, G.G. and Mehrotra, S.P., Ironmaking and Steelmaking, 19, 1992, p. 377. 26. Narita, K., Tomita, A., Hiroka, Y. and Satoh, Y., Tetsu-to-Hagane, 57, 1971, p. 1101. 27. Ballal, N.B. and Ghosh, A., Metall. Trans., 12B, 1981, p. 525. 28. Mazumdar, D., Nakajima, H., and Guthrie, R.I.L., Metall. Trans., 19B, 1988, p. 507. 29. Sano, M. and Mori, K., Trans. ISIJ, 23, 1983, p. 169. 30. Nakanishi, K., Fujii, T. and Szekely, J., Ironmaking and Steelmaking, 3, 1975, p. 193. 31. Bhavaraju, S.M., Russel, T.W.F. and H.W. Blanch, H.W., AIChE J., 24, 1978, p. 454. 32. Szekely, J., Wang, H.J. and Keiser, K.M., Metall. Trans., 7B, 1976, p. 287. 33. El-Kaddah, N. and Szekely, J., SCANINJECT III, MEFOS, Lulea, Sweden, 1983, p. 3:1. 34. Joo, S. and Guthrie, R.I.L., Metall. Trans., 23B, 1992, p. 765. 35. Krishnamurthy, G.G., Ghosh, A. and Mehrotra, S.P., Metall. Trans., 20B, 1989, p. 53.


4

Mixing, Mass Transfer, and Kinetics

4.1 INTRODUCTION Since chemical reactions occur during steelmaking, the vessels are reactors according to general terminology. Steelmaking, including secondary steelmaking, is concerned with liquid-state processing. The reactors are semi-batch types, with the exception of the tundish, which is close to a continuous stirred tank reactor. In semi-batch reactors, the liquids are added and withdrawn in batches, whereas the gases flow in and out of the reactors continuously. Solid reagents are either added in batches or injected continuously as powder. Besides chemical reactions, some physical and physico-chemical processes are of importance in secondary steelmaking, i.e., homogenization of composition and temperature, separation of nonmetallic particles from steel melt, loss and gain of heat content of the melt, and dissolution of alloying elements. The rate of processing would be governed by the rates of these processes. The rate of processing, which includes refining of the steel melt, is controlled by one or more of the following • • • •

kinetics of reactions among phases mixing in the melt feed rate of reactants rate of heat supply to the reaction zone

The above listing excludes external factors such as shop logistics. Temperature control is as important as composition control in secondary steelmaking. However, the issue of temperature control is dealt with in Chapter 8. Homogenization of temperature of the steel melt is primarily dependent on convective heat transfer, which is akin to convective mass transfer. Hence, knowledge of one can be utilized for the other. Rates of specific reactions and processes are discussed in later chapters. Dissolution of alloying additions in molten steel is partly controlled by rate of heat supplied to the cold addition. There are other minor examples. However, all other reactions in secondary steelmaking are not limited by the rate of heat supplied to the reaction zone. Therefore, we are primarily concerned with reaction kinetics, mixing, and the feed rate of reactants. In this connection, the ternary diagram in Figure 4.1, proposed by Robertson et al.,1 in connection with powder injection processes in a secondary steelmaking ladle, is quite illustrative. Near corner 1, reactions are close to chemical equilibrium, and the liquid is well mixed. Hence, feeding rate of powdered reagents is going to control the process rate. Near corner 2, powder mixing and feeding are fast. Hence, control is by reaction rate, which is slower. Near corner 3, mixing is the slowest, and therefore is rate controlling.


1

2 3

tf /Σt

1

0 =

reaction + mixing

d

at

mixing rate controls

xe

feeding feeding + + reaction mixing feeding, reaction and mixing

mi

eq uil ibr ium

y

xx

ctl

rfe

pe

feeding rate controls

reaction rate controls

3

2

tmix / Σ t

powder

dumping

tr / Σt

Σt = tf + tr + tmix

FIGURE 4.1 Ternary diagram showing the influence of the three possible rate-determining processes during power injection refining.1

The present chapter is a brief presentation of the fundamentals of kinetics, mixing, and mass transfer with specific reference to steel melt in a ladle, stirred by inert gas through a nozzle/porous plug from the bottom. The kinetics of specific reactions and processes is dealt with in later chapters, as already stated. So far as basics are concerned, standard texts are available.2–4 Ghosh and Ray5 also have briefly presented topics relevant to extractive metallurgy.

4.2 4.2.1

MIXING IN STEEL MELTS IN GAS-STIRRED LADLES FUNDAMENTALS

OF

MIXING

Section 3.2 dealt with fluid flow in steel melts in gas-stirred ladles. This section is concerned with mixing in steel melts in gas-stirred ladles. Mixing is dependent on the nature and intensity of fluid motion and turbulence in the melt. As in Section 3.2, here also we consider only inert gas purging by porous plugs/nozzles fitted at the bottom of the vessel. Also in the area of mixing, experimental investigations with steel melts have not been done often. Fundamental investigations have been ©2001 CRC Press LLC

conducted mostly in laboratory water models. Again, a majority of the studies employed axisymmetric nozzles. In this section, only the fundamentals are emphasized. Mixing in various industrial situations is presented in other sections wherever information is available. There have been numerous physical and mathematical modeling studies of mixing in the last 15 to 20 years. Some good review papers have also been published6–8 in recent years. Comprehensive literature is available in the chemical engineering field.9–10 In view of these considerations, the number of references has been kept limited. Mixing occurs by convection (i.e., bulk flow), turbulent (i.e., eddy) diffusion, and molecular diffusion. Experimentally, the speed of mixing is measured by pulse-tracer technique. A small quantity of tracer is suddenly added into the liquid at some location. The concentration of the tracer is monitored at some other location in the liquid using a measuring probe. In water models, aqueous solution of KCl or HCl are popular tracers. Dissolved KCl or acid increases electrical conductivity of water. Hence, its concentration at any location can be measured as a function of time by the electrical conductivity probe. Imagine the sudden addition of a tracer into a liquid. Bulk motion would transport the tracerrich liquid region, known as clump, to other regions. It also causes disintegration of the clump into smaller and smaller eddies as it moves through the liquid. Dispersion of eddies by eddy diffusion causes further mixing. The disintegration of clumps, however, can not continue indefinitely. As discussed in Section 3.1.5, with decreasing eddy size, viscous forces increasingly resist further disintegration of eddies. There is a smallest size beyond which there will be no further disintegration. At this stage, macromixing of the tracer is complete. However, the liquid is still not perfectly mixed, and concentration inhomogeneities exist on a microscopic scale. Further homogenization of composition (i.e., micromixing) is possible by molecular diffusion only. Molecular diffusion is an extremely slow process. Hence, micromixing is unattainable in industrial processing as well as in studies on mixing. Therefore, perfect mixing would mean complete macromixing only, and it occurs by a combination of bulk motion and turbulent diffusion. Equation (3.27) has defined turbulent viscosity (µ t). In an analogy with this, turbulent diffusivity (Dt) can be defined. Combining contributions of bulk flow and turbulent diffusion to mixing, the vectorial form of the equation is: ∂C i -------- = u ⋅ ∇C i + ∇ ⋅ ( D t ∇C i ) ∂t

(4.1)

∂C where Ci is concentration of tracer i at time t after tracer addition. --------i refers to rate of change of ∂t concentration at a location (say, the probe location). Figure 4.2 presents a typical recorder voltage-time curve for addition of KCl solution into a cylindrical water bath stirred by blowing air from the bottom using an axisymmetric nozzle.8,11 The change in the concentration of KCl at the probe location was proportional to the change of recorder voltage. Hence, the curve represents the variation of concentration of KCl over time at the probe location. Major oscillations in the recorder trace are due to recirculatory flow. The peak-to-peak interval for major peaks is the approximate time for one recirculation (tC) and was about 8 s in Figure 4.2. The amplitude of oscillation decreases rapidly with time due to the progressive disintegration of clumps as the bulk liquid recirculates. Experimentally, mixing speed is determined by measuring the mixing time (tmix) of a small quantity of tracer added into a liquid suddenly. It is difficult to measure tmix for 100 percent macromixing. Hence, some standardization is desirable. In this connection, degree of mixing (Y) has been defined as: o

Ci – Ci Y = -----------------o f Ci – Ci ©2001 CRC Press LLC

(4.2)

FIGURE 4.2 A typical recorder voltage-time trace showing mixing time at two different degrees of mixing.11 o

where C i is the instantaneous average concentration at any time t, C i is the uniform initial f concentration before tracer addition, and C i is the uniform final concentration at the end of mixing. Y = 0.95 (i.e., 95 percent mixing) has been generally accepted for defining mixing time. However, Krishnamurthy et al.,11 in their fundamental investigation, employed Y = 0.995. (Figure 4.2). The statistical theory of mixing10 predicts that tmix = a constant × log(1 – Y)

(4.3)

Krishnamurthy12 has shown that, for various experiments,8,11 Equation (4.3) tends to predict a somewhat lower value of tmix than experimental measurement. It is an indication that the mixing process is controlled both by bulk convection and turbulent diffusion. The higher the gas flow rate, the more intense would be stirring and consequent mixing, resulting in lower tmix. This is well established. Krishnamurthy et al.8 made comprehensive measurements of tmix at several combinations of tracer injection and probe locations in their water model and obtained a single value of tmix under a given experimental condition. However, several investigators6 found that tmix is dependent on tracer addition and monitoring point locations. Such a phenomenon was observed at relatively low specific gas flow rates, typical of ladle metallurgy. Figure 4.3 presents mixing time vs. gas flow rate at different measuring positions in a water model with centric bottom gas injection, showing significant dependence of tmix on probe location.13 The degree of mixing was taken as 95 percent.

4.2.2

VARIABLES INFLUENCING MIXING

It has been already mentioned, in Section 3.2.3, that hydrodynamic conditions near the nozzle or plug are not critical to flow recirculation in large cylindrical vessels. A similar comment is applicable to mixing in the liquid bath. So, it does not matter whether one employs a nozzle or porous plug. However, the location and number of nozzles/plugs have significant influence on mixing.7,8,14 Figure 4.4 presents the results of a water model study by Joo and Guthrie.14 They employed a porous plug. It shows tmix as a function of nondimensional radial coordinate (r/R), where r = 0 at center and r = R at the vessel wall. The minimum value of tmix was obtained at mid-radius (i.e., r/R = 0.5). It is occasionally necessary to bubble an industrial ladle with two or more plugs to achieve gentle but rapid mixing to promote slag-metal reaction, but to avoid explosive bubble bursting. This is achieved by employing a multiplug/multinozzle gas purging arrangement. Minimum mixing ©2001 CRC Press LLC

FIGURE 4.3 Mixing time vs. gas flow rate (centric nozzle, tracer addition in dead zone). 1, 2, and 3 = locations of concentration measurement.13

FIGURE 4.4 Plot of mixing time vs. radial position for a single plug for various gas flow rates.14

time with dual plugs was obtained if the two plugs were located at diametrically opposite positions at mid-radius (Figure 4.4). Modern industrial gas-stirred ladles are fitted with plugs this way. It was mentioned in Sec. 3.2.3 that a slag layer at the top of a steel melt is expected to cause a loss of energy and slow down the recirculatory flow.15 Evidence for this has been gathered from water model experiments (Figure 3.18). Hence, it is expected that mixing would be slower in the presence of top slag, in contrast to that in a bath with free surface. A significant increase of tmix due to the presence of an oil layer on top of the water bath has been confirmed by investigators.6,16 As shown in Figure 4.3, increasing gas flow rate (Q) promotes mixing and decreases tmix. It has also been found that tmix decreases as bath height (H) increases, provided H/D < 2, where D is the vessel diameter. In steel ladles, H/D < 2, and hence the above conclusion is applicable. It has ©2001 CRC Press LLC

also been found to depend on D as well as nozzle diameter (dn). Properties of a liquid such as viscosity, density, and surface tension are also expected to influence tmix. Under a specified condition, Qn is proportional to the rate of buoyancy energy input (εb) as given in Eq. (3.62). As already stated, εb has been accepted as a measure of the rate of energy input into the bath due to gas flow. εb per unit mass of liquid, i.e., εm, as defined by Eq. (3.64), is the popular parameter employed. Several quantitative relations have been proposed about the dependence of tmix on εb, H, and D. These are semi-empirical, based partly on mathematical analysis and partly on experimental data. Measurements in molten steel are very limited. Experimental data have been collected primarily in water models. Mathematical analysis of the mixing process is based on the following models. The Turbulence Model This was first developed by Nakanishi, Szekely, and Chiang17 for turbulent recirculatory flow. The assumption was that mixing was solely controlled by eddy diffusion. Besides the equations of continuity and motion, another differential equation for eddy diffusion of the tracer was set up. The eddy diffusivity was set equal to eddy kinematic viscosity. The empirical correlation of Nakanishi et al.18, fitted with measurements from an argon stirred ladle, RH degasser, water model, etc., is based on the concept of a turbulence model, which tends to suggest that tmix should depend only on εm. The correlation of Nakanishi et al. was as follows: – 0.4

t mix = 800e m

(4.4)

As explained in Section 3.1, εm is the rate of buoyancy energy input per unit mass of the liquid. However, Nakanishi et al.18 also included bubble expansion energy in εm, which gave a value twice as large as that of buoyancy energy. Moreover, εm was in watts/tonne. Correcting for a factor of two and taking εm in watts/kilogram, Eq. (4.4) may be rewritten as – 0.4

t mix = 38.3e m

(4.5)

However, this turbulence model tended to predict that mixing time was independent of vessel size, vessel geometry, and mode of stirring. Hence it could not explain experimental observations. Equation (4.3) may be rewritten as 1 – Y = exp(–tmix/to)

(4.6)

For 95 percent mixing, 1 – Y = 0.05, and tmix/to ≈ 3. Murthy and Szekely,19 on the basis of postulations made by some earlier workers, argued that the energy dissipation rate due to turbulence is only a fraction of εm, and they related to to H and D. Combining all these, they tried to explain the –1 ⁄ 3 dependence of to on H and D. It was also predicted that t mix α ε m . Another difficulty with the turbulence models is their inability to explain the oscillating nature of concentration vs. time curves upon addition of a tracer (Figure 4.2). Again, the natures of these curves are dependent on probe location. Mazumdar and Guthrie20 carried out extensive mathematical and physical modeling and arrived at the conclusion that all experimental behavior patterns can be explained only if it is assumed that mixing is controlled by both convection and turbulent diffusion. Prediction of mixing times by numerical solution of differential equations also has been carried out recently.14,21 ©2001 CRC Press LLC

Circulation Models These models assume that the circulation rate of liquid in the bath controls mixing. Here, to is taken as equal to the time required for one circulation of the liquid (tc). Sano and Mori22 were the first to employ this approach, which explained the dependence of tmix on H and D, as observed experimentally. Krishnamurthy et al.,23 through their macroscopic energy balance model, proposed equations for calculation of tc. Combining that with their experimental values of tmix (Y = 0.995), they found that t mix H d = f  Fr m , ----, -----n Circulation number ( C i ) = ------ D D tc

(4.7)

The above approaches could not quantitatively explain the concentration versus time curves (Figure 4.2), which was attributed to existence of dead zones and different flow regimes in the vessel. Hence, these were subsequently refined by dividing the vessel into several tanks and assuming complete mixing within any tank. These have successfully explained the concentration versus time curves.12,13 Several mixing time correlations have been proposed by various investigators.6 In the early studies, no standardized degree of mixing was employed. It is necessary to assess how the different correlations compare with one another. For this, in Table 4.1 we have selected only those in which Y = 0.95 and which have been arrived at or tested against experimental data of water model with centric gas injection through a nozzle. TABLE 4.1 Mixing Time Correlations for 95 Percent Degree of Mixing Reference Mazumdar and Guthrie20 Stapurewicz and Themelis24 Neifer, Rodi, and Sucker21

Correlation – 0.33

t mix = 12.2ε m

H

– 1.0

– 0.39

t mix = 11.1ε m t mix = 3.2Q

– 0.38

H

H

D

1.66

0.39

– 0.64

D

2.0

Figure 4.5 presents calculated curves of tmix vs. εm by various correlations of Table 4.1 for a water model of 0.5 m diameter and 0.4 m height at a temperature of 298 K, and atmospheric pressure of 1 bar. εm was calculated by Eq. (3.64). For comparison, the empirical correlation of [Eq. (4.5)] has also been included. Although some investigators18,21 have tried to suggest that their correlations may be applied even to liquid iron, others have proposed a scale factor. An examination of the literature21 tends to suggest that experimental values of tmix in liquid steel are somewhat larger than those in a water model at same values of εm, D, and H. Asai et al.25 have suggested that, for design purposes, tmix should be measured in carefully designed water models and then be multiplied by a factor of 1.9 for liquid steel (i.e., [ρFe/ρw]1/3).

4.3 KINETICS OF REACTIONS AMONG PHASES Metallurgical reactions are almost exclusively heterogeneous in nature, where reactions occur among phases. Examples are solid–liquid reactions, slag–metal reactions, and gas–metal reactions. Consider the following reaction occurring between molten steel and molten slag: ©2001 CRC Press LLC

FIGURE 4.5 Comparison of tmix vs. mixing energy plots obtained from various correlations.

[S] + (O2–) = (S2–) + [O]

(4.8)

S and O denote sulfur and oxygen, [ ] indicates metal phase, and ( ) indicates slag phase. Since slag is ionic in nature, S and O are presumed to exist there as S2– and O2–, respectively. The above exchange reaction actually takes place as a coupled electrochemical half-cell reactions as follows at the slag–metal interface. [S] + 2e– = (S2–)

(4.9)

(O2–) = [O] + 2e–

(4.10)

Reactions (4.9) and (4.10) occur at different sites at the slag–metal interface. Electron transfer from one site to another takes place via liquid metal, which is an electrical conductor. This is shown schematically in Figure 4.6. The overall process consists of several steps, known as kinetic steps, shown in Figure 4.6. They are as follows:

FIGURE 4.6 Electrochemical mechanism of slag–metal interfacial reaction.


1. 2. 3. 4. 5.

Transfer of sulfur from the bulk of the metal phase to the slag-metal interface Transfer of O2– from bulk of the slag phase to the interface Chemical reaction at the interface Transfer of oxygen from the interface to the bulk metal Transfer of S2– from the interface to the slag phase

Step (3) is a chemical reaction and is governed by the laws of chemical kinetics. Other steps are governed by laws of mass transfer. The above kinetic steps for the reaction shown in Eq. (4.8) are all in series. If any one of them is prevented, the overall reaction ceases to occur. It is also to be noted that the slowest kinetic step would tend to influence the rate predominantly, and it is typically termed the rate-controlling or rate-limiting step. The conclusions drawn above would be just the reverse if the kinetic steps were in parallel, where the fastest step would influence the overall rate the most. However, it is impossible to conceive of a process where all the steps would be in parallel. Hence, it may be concluded that the slowest step in the series would be the primary rate-controlling step. From the above viewpoint, the slowest step is the most important one. A major objective of all kinetic studies is to find out what the slowest step is. Of course, other kinetic steps would also influence the overall process rate to some extent. At times, two or more steps may have comparable rates. However, owing to the complexity of the steelmaking processes, one kinetic step is often assumed to control the rate, and others are assumed to be infinitely fast and thus at virtual equilibrium. With this simplification, the rate of a process estimated on the basis of only the slowest kinetic step would be the highest, and larger than the actually observed rate. Such an estimate therefore is termed as virtual maximum rate (VMR). VMR calculations often provide great insight.

4.3.1

INTERFACIAL CHEMICAL REACTION

The fundamentals of gas–metal, slag–metal, and metal–gas–slag reactions in steelmaking can be best understood on the basis of the findings of laboratory experiments carried out over the past several decades. Here, the system is isothermal and each phase is well mixed. Hence, studies have provided information on reaction kinetics exclusively. The laboratory findings have been that steelmaking reactions are generally controlled by mass transfer at the phase boundary, and not by interfacial chemical reaction. This is expected from theoretical considerations as well. However, there are exceptions, the most notable being absorption and desorption of nitrogen by molten steel, which is a case of mixed control kinetics, i.e., both interfacial reaction and mass transfer partially controlling the rate of overall reaction. With the above background in mind, very little is written here on the kinetics of interfacial chemical reaction. The kinetics of a nitrogen reaction is discussed more elaborately in Chapter 6. Suppose the heterogeneous reaction is A+B=C+D

(4.11)

Then, according to the law of mass action, the rate of reaction (r) may be related to concentrations of reactants (A, B) and products (C, D) as C C D D r = Ak c  C A C B – ------------ K 

(4.12)

where A is area of interface of the phases involved, CA etc. denote concentrations of respective species per unit volume (i.e., mol/vol or mass/vol.), kc is the chemical rate constant, and K is the equilibrium constant for Reaction (4.11). ©2001 CRC Press LLC

Equation (4.12) is the rate expression for a reversible reaction, where both forward and backward rates are significant. For an irreversible process, the backward rate is much smaller than the forward rate and can be ignored. Then, r = Akc CA CB

(4.13)

Actually, theoretical predictions of rate expressions are either impossible or very difficult. Hence they are determined experimentally. For example, for Reaction (4.11), suppose the experimentally determined rate expression is α

β

r = Ak c C A C B

(4.14)

Then, α + β = the order of reaction. Again, kc increases with an increase in temperature. Experimentally, it has been found that the following relationship holds true in a limited range of temperatures: B ln k C = A – --T

(4.15)

where T is temperature and A, B are empirical constants. Arrhenius attempted to explain this observation through his famous equation as E k C = A exp  – -------  RT 

(4.16)

where E is known as activation energy, R is Universal gas constant, and A is a preexponential factor. Although, in principle, A and E can be estimated with the quantum mechanical approach, it is difficult and unreliable. Hence, A and E are determined experimentally. The Arrhenius equation is theoretically also valid for other molecular transport processes such as diffusion and viscous flow. Hence, it is not restricted to chemical reactions. E has a clear-cut theoretical meaning only if one kinetic step exclusively controls the rate. Wherever that is not true, E is not the true activation energy but is just a temperature coefficient of some sort.

4.3.2

MASS TRANSFER

Mass transfer is concerned with the transfer of a chemical species from higher to lower concentration. Mixing, already discussed in Section 4.2, is also a process of mass transfer. It is a question of terminology only. By mixing, we mean mixing in the bulk fluid. By perfect mixing, only macromixing was meant, and molecular diffusion was ignored. In contrast, the motion of individual atoms and molecules is our ultimate concern in mass transfer. Hence, molecular diffusion is also important. In Figure 4.6, transport of O, S, O2– and S2– are mass transfer processes adjacent to the slag–metal interface. In general, it is known as phaseboundary mass transfer. Actually, it is these transports in connection with heterogeneous reactions that constitute the principal application of the subject of mass transfer in science and engineering. The general equation for mass transfer of species i along the x-direction may be written as: ∂C ∂C m˙ i, x J i, x = -------- = D i --------i + C i u x – D t --------i ∂X ∂X Ax molecular + bulk + turbulent diffusion convection diffusion


(4.17)

where

Ji,x = flux of species i along the x-direction dm m˙ i = ---------i = mass rate of transport of species i along x dt Ax = area normal to the x-direction Di = molecular diffusivity of species i in the fluid Ci = concentration of species i in mass per unit volume u x = time-averaged fluid velocity along the x-direction Dt = turbulent or eddy diffusivity

Phase-boundary mass transfer processes may be classified as • Mass transfer at the solid-fluid interface • Mass transfer between two fluids Mass Transfer at the Solid-Fluid Interface In solids, molecular diffusion is the only mechanism of mass transfer. It is extremely slow. Hence, in steelmaking vis-a-vis extraction and refining processes in general, we ignore it and assume the composition of solid to be constant. On the fluid side of the interface, the existence of a velocity boundary layer has already been discussed (Figure 3.3). In a similar fashion, a concentration boundary layer develops in the fluid adjacent to the solid surface (Figure 4.7). Just at the interface, we may assume u x = 0 and, due to laminar flow, Dt = 0. This simplifies Eq. (4.17) as m˙ i, x -------Ax

at x = 0

∂C = – D i  --------i  ∂x  x = 0

(4.18)

Noting that Ax is solid surface area (A), and through the geometric construction shown in Figure 4.7, Di s 0 s 0 - ( C – C i ) = Ak m,i ( C i – C i ) ( m˙ i ) at interface = A ---------δ c, eff i

S Ci

(Fluid) (Solid)

Ci

Concentration profile

o

Ci

δC, eff x=0

x

FIGURE 4.7 Concentration boundary layer in fluid adjacent to a solid surface during mass transfer.


(4.19)

δc,eff is known as the effective concentration boundary layer thickness and km,i is the mass transfer coefficient for species i ( D i ⁄ δ c, eff ) . δc, eff depends on fluid flow. The more intense the flow, the smaller is δc,eff with larger km,i and m˙ i . This is how fluid flow influences rate. km,i also depends on the transport properties of the fluid (µ, ρ, D), and the geometry and size of the system. A measure of size is characteristic length (L), which, for example, is the diameter for a pipe or sphere, as already mentioned in Table 3.1. To estimate mass transfer rates, km,i is to be estimated or determined. Experimental measurements have been carried out on a variety of systems. Dimensionless correlations are very advantageous, and this is how km,i is correlated with fluid flow, etc. Table 4.2 presents the dimensionless numbers for convective mass transfer. The symbols have already been defined in Chapter 3 in connection with Table 3.1. TABLE 4.2 Common Dimensionless Numbers in Convective Mass Transfer Dimensionless Number Item

Definition

Name

Symbol

Lu -----ν

Reynolds no.

Re

gL ∆ρ -------2 ------ν ρo

3

Grasshof no.

Gr

Mass transfer properties

ν ---D

Schmidt no.

Sc

Mass transfer coefficient

kmL ⁄ D

Sherwood no.

Sh

Fluid flow (forced convection)

Fluid flow (free convection)

Note: ν = µ/ρ =kinematic viscosity of the fluid.

In general, Sh = B + D Rem Scn, for forced convection

(4.20)

Sh = B´ + D´ Grm´ Scn´, for free convection

(4.21)

and

For fixed geometry, B, B´, D, D´, m, n, m´, and n´ are constants within ranges of Re, Sc, and Grm. They are mostly the same for analogous heat transfer situations, and some typical mass transfer correlations have been obtained from analogous heat transfer correlations. Prandtl’s number (Pr) should be replaced by Sc, and Nusselt’s number (Nu) by Sh for this purpose. Such dimensionless correlations are available for several geometries and flow regimes in standard texts.2,3,5 They are mostly empirical. Appendix 3.1 contains values of µ, ρ, and ν for some liquids of interest in secondary steelmaking. Appendix 4.1 presents some values of the diffusion coefficient. Mass Transfer between Two Fluids The reaction between two fluids is exemplified by those of molten metal with molten slag, molten salt, or gas. The boundary layer theory of convective mass transfer has been highly successful at ©2001 CRC Press LLC

solid–fluid interfaces. Attempts have been made to extend the same to the two-fluid situation by assuming the existence of a concentration boundary layer on both sides of the interface. It is all right in some cases. But, by and large, there is a problem. At a solid–fluid interface, the fluid layer at the interface sticks to the solid. Therefore, it is stagnant and not renewed. Moreover, turbulence cannot reach the interface. However, these assumptions are not valid at fluid–fluid interfaces. Davies26 has dealt with various aspects of turbulence phenomena, behavior of eddies, and their role on mass transfer. Turbulence at the interface of two fluids tends to get damped due to the resistive action of surface tension. The damping effect is pronounced in the direction perpendicular to the surface, but not so much in parallel direction. Levich considered this damping and proposed the following correlation: 1⁄2 3⁄2

k m,i = 0.32D i u o ρ

1⁄2

–1 ⁄ 2

σ equiv

(4.22)

where uo is the fluctuating RMS velocity in the bulk of the liquid, and σequiv is equivalent surface tension, defined as σ equiv = σ + ( l ρg ⁄ 16 ) 2

(4.23)

where σ is the surface/interface tension and l is the eddy mixing length. However, the boundary layer approach is incapable of taking into account continuous renewal of the interface layer due to fluid motion. This led to the development of the various surface renewal theories of mass transfer between two fluids. Out of these, only two are popular. Higbie assumed that eddies penetrate into the interfacial layer and renew the interface periodically. Each eddy is exposed for the same time before replacement by a fresh eddy arriving from the bulk. During this period, mass transfer is by unsteady diffusion. Higbie’s surface renewal theory is also applicable when the flow at the interface is laminar. If the viscosity of one fluid is much larger than that of the other, then the former exhibits a negligible velocity gradient near the interface and flows like a rigid solid. For example, in a gas–liquid system, the liquid near the interface would flow like a rigid body, since it has a much higher viscosity as compared to that of the gas. Similarly, in a slag–metal system, the slag would tend to flow like a rigid body. In such situations, mass transfer at the interface in the high viscosity phase would be exclusively by molecular diffusion. Since the surface gets renewed continuously due to flow at the interface, such diffusion is unsteady, and it was derived that D 1⁄2 k m,i = 2  -------i   πt e

(4.24)

where te is the exposure time, i.e., the time spent by a fluid element at the interface. In turbulent flow also, application of Higbie’s model is straightforward, provided that we assume the same value of te for all eddies and te is known. However, behavior of eddies is more probabilistic, and not all eddies are expected to spend the same time at the interface before replacement by a fresh eddy from the bulk. Danckwerts made a more realistic assumption that a fraction of surface renewal in time t is equal to [1 – exp(–St)]. Figure 4.8 shows the difference between Higbie’s and Danckwerts’ models schematically. On the basis of the above model, Danckwerts derived that km,i = (Di S)1/2

(4.25)

S is known as surface renewal factor. It is the rate of renewal of the surface in terms of the fraction of surface renewed per second. In this model, S is to be determined experimentally and is, therefore, ©2001 CRC Press LLC

FIGURE 4.8 Distribution of turbulence eddies on the surface according to the postulates of Higbie (1935) and Danckwerts (1951).

a source of uncertainty. S varies between 5 to 25 per second for mild turbulence, and up to 500 per second for violent turbulence.26 All the above models predict that km should be proportional to D1/2. This is in contrast to boundary layer theory, which predicts a dependence on D0.7–1. In chemical engineering, many investigators attempted to verify validity of this for mass transfer in liquid at gas-liquid and liquidliquid interfaces. The proportionality of km on D1/2 has been verified, and surface renewal mode is generally accepted now.26 In the metallurgical field, one of the earliest investigations was by Boorstein and Pehlke.27 They measured dissolution rates of hydrogen and nitrogen in inductively stirred liquid iron. Stirring intensity was varied. It was found that k m, H D ---------- = ------Hk m, N DN in quiescent melt, and D 1⁄2 k m, H ---------- =  ------H-  D N k m, N for well-stirred melts, thus indicating the applicability of boundary layer theory for the former, and surface renewal theory for the latter. More studies in the metallurgical field will be presented later, at appropriate places. It would suffice to summarize here that surface renewal theory is generally employed for correlation of experimental results for a two-fluid situation. The rate of surface renewal increases with jump frequency of eddies, which varies from a few per second for large eddies to about 1000 per second or more for the smallest (i.e., Kolmogorov eddies). At normal and gentle turbulence, S ranges between 5 to 25 per second, and hence surface renewal is expected to be primarily by large (i.e., Prandtl) eddies. Visual observations also have confirmed this.26 This is because smaller eddies tend to get damped considerably near the interface unless turbulence is intense. One of the earliest attempts to determine S in a metallurgical situation was the water model study by Kumar and Ghosh,28 who measured rate of absorption of CO2 in water by the pH method and also estimated liquid–gas bubble surface area by a photographic technique. They found S to range from 40 to 100 per second. Investigations in the chemical engineering field have established that there is no significant difference between gas–liquid and liquid–liquid mass transfer, either in basic mechanisms or even in some quantitative relationships (e.g., in a stirred tank with one liquid stirred).26 ©2001 CRC Press LLC

Robertson and Staples29 proposed the following empirical correlation for the metal-phase mass transfer coefficient for a mercury-water and molten lead-molten salt system stirred from the bottom by inert gas: km = 172 D1/2 Q1/2 R

(4.26)

where R is the vessel radius in meter, km is in ms–1, D is in m2 s–1, and the volumetric gas flow rate Q is in m3 s–1. Taniguchi et al.30 measured the rate of CO2 absorption at the free surface of a water bath stirred by nitrogen from the bottom and proposed the following relationship for km in the water phase: km = 138 D1/2 Q1/2 R

(4.27)

The resemblance of Eqs. (4.26) and (4.27) may be noted, although the former is for liquid–liquid and latter for liquid–gas reaction. The presence of surface active species on the surface would retard the motion of fresh eddies coming from the bulk liquid, as shown in Figure 4.9a. This would lead to a lowering of the value of km as compared to that for a clean surface. A decrease of km by a factor of two for gas–liquid situation, and even by a factor of four for liquid–liquid mass transfer, has been observed.26 A reverse situation is shown in Figure 4.9b, where fresh eddies bring surface active species from the bulk. Here, surface flow is enhanced. This is known as Marangoni effect, after Marangoni, who first discovered it in the 19th century. It may even cause spontaneous interfacial turbulence. Richardson and coworkers at the Imperial College conducted several studies on this.31,32

FIGURE 4.9 Interfacial (a) retardation or (b) enhancement of movement induced by surface pressure.


4.4 MASS TRANSFER IN A GAS-STIRRED LADLE 4.4.1

SOLID-LIQUID INTERACTIONS

The addition of lump solids or the injection of solid powders is required in ladle refining and elsewhere in secondary steelmaking. Addition of ferroalloys is an example. Melting-cum-dissolution of these alloys is controlled by rates of heat and mass transfer. Section 4.3.2 briefly presented the basics of mass transfer in fluid adjacent to a solid–fluid interface. The dimensionless correlations are of the type presented in Eqs. (4.20) and (4.21). For example, mass transfer correlation around a solid sphere in forced convection is given by the famous Ranz–Marshall equation as follows:

Sh = 2 + 0.6 Re1/2 Sc1/3

(4.28)

where the characteristic length is the diameter of sphere (d), and the characteristic velocity is the time-averaged bulk velocity. For convective heat transfer around the sphere, the analogous equation is: Nu = 2 + 0.6 (Re)1/2 (Pr)1/3 where

(4.29)

hL Nu = -----λ ν µC Pr = --- = ------α λ h = surface heat transfer coefficient, analogous to k λ = thermal conductivity α = thermal diffusivity C = specific heat of the fluid

Experiments have been done by several investigators in water models of a ladle, stirred by gas from the bottom. Gas injections were centric. The rates of the melting of ice and the dissolution of benzoic acid in water were studied. Solids were immersed in various locations, too. A few studies in molten steel are also available. Mazumdar and Guthrie6 have reviewed these. In a recent investigation, Iguchi et al.33 employed an electrochemical technique. Several dimensionless correlations have been proposed in the literature. Controversy exists about their relative merits and reliability. Hence, a detailed presentation is omitted here. The first difficulty in the determination of a dimensionless correlation of experimental data is that there is no characteristic bulk velocity in a gas-stirred liquid. Hence, local velocity and local Reynold’s number were employed. This requires solution of the turbulent Navier-Stokes equation to obtain the velocity field. It also has been established that an equation of the type of Eq. (4.20), such as Eq. (4.28) for a sphere, is obeyed provided intensity of turbulence (I) is less than 0.2 or so. At higher values of I, these equations tend to give a better fit with experimental data by incorporating I in the corelations. All investigators are in agreement on this. For example, Mazumdar et al.34 measured the rates of dissolution of vertical cylinders of benzoic acid in a gas-stirred water model and proposed the following relationship: Sh = 0.73 (Reloc,r)0.57 (I)0.32 (Sc)0.33 where ©2001 CRC Press LLC

(4.30)

2

Re loc,r

2 1⁄2

( ux + uy ) ⋅ D = ----------------------------------ν

(4.31)

and x and y are horizontal coordinates, and D is the diameter of the cylinder. On the other hand, for a sphere, Iguchi et al.33 proposed a modified version of the Ranz–Marshall equation as follows: Sh = 2 + 0.6 Re(0.5+0.1 I) · Sc1/3

(4.32)

For 103 < Re < 104, 0.3 < I < 0.5. At I < 0.3, Eq. (4.28) was found to be adequate. It may be further noted30 that Eq. (4.30) predicts that km ∝ Q0.2 approximately, where Q is the volumetric gas flow rate. This is in agreement with experimental data for dissolution of steel cylinders in carbonsaturated iron melts.35

4.4.2

LIQUID–LIQUID INTERACTIONS

Kinetic studies carried out in connection with various ladle metallurgy operations include the reaction between a gas bubble and liquid, slag–metal reactions, and absorption of gases at a free surface and in a plume’s eye. Fundamental studies have been primarily conducted in room temperature models. Not all of these are discussed in this chapter. Reactions and mass transfer between a gas bubble and liquid is dealt with in Chapter 6, on degassing. Absorption of gases at a plume surface is taken up in connection with deoxidation kinetics and clean steel. There have been plant studies of the kinetics of specific reactions such as desulfurization. These also will be taken up in later chapters. In this section, we are briefly concerned with some fundamental laboratory investigations on mass transfer between two liquids in a vessel stirred by bubbling inert gas from bottom. Obviously, the metallurgical objective is to understand slag–metal reaction in bubble-stirred systems. Basic open hearth steelmaking, which was the dominant primary steelmaking process up to the 1960s, constituted the principal target for early workers across the World. Richardson32 and Turkdogan36 have reviewed them. A principal difficulty of fundamental study is that the actual slag–metal interfacial area (A) is larger than the geometrical surface area due to unevenness of the interface formation of slag–metal emulsion. A is a function of gas flow rate, etc. Moreover, it is a difficult task to properly determine it. Hence, experimental rate measurement yields the value of the kA parameter, where k is specific rate constant. When mass transfer is rate controlling, then k = km and we obtain the km A parameter on the basis of Eq. (4.19). km A has a dimension of (ms–1 × m2), i.e., m3 s–1. Sometimes the kmA/V parameter of dimension s–1 is preferred, where V is the volume of the concerned liquid. However, kmA is a more fundamental parameter as compared to kmA/V. In gas-stirred systems, broadly speaking, the kmA parameter has been found to be proportional to Qn.Value of n has been found to be different in different ranges of gas flow rate. This is demonstrated by the study of Kim et al.37 as shown in Figure 4.10. Inert gas was injected from bottom axisymmetrically. Oil simulated the slag phase, and water simulated the metal phase. The equilibrium partition coefficient of thymol between oil and water is very large—somewhat like the partition of sulfur between slag and metal. This made the kinetics unambiguously controlled by mass transfer in the metal phase. Figure 4.10 shows three regimes in ln(kA) vs. Q curve, with different values of n. At a low flow rate, n = 0.6. Some other investigators have also reported n = 0.529,30. Visual observations did not reveal any perturbation in the oil layer. In the middle regime n = 2.51, and the oil layer near the edge of the plume eye continuously formed ligaments and disintegrated into droplets. In regime III, the entire oil layer was found to be dispersed in water as droplets. A lower value of n there is explained by the large residence time of droplets in water in this regime and their consequent ©2001 CRC Press LLC

FIGURE 4.10 kA vs. gas flow rate for mass transfer between two liquids in a vessel stirred by axisymmetric gas injection from the bottom.37 Reprinted by permission of Iron & Steel Society, Warrendale, PA, U.S.A.

saturation with thymol, retarding further transfer. Since Q is proportional to εm according to Eq. n (3.64), k m Aαε m . Formation of oil-in-water and slag-in-metal emulsion increases the liquid–liquid interfacial area by even a factor of about 100. Consequent large enhancement in kA parameter has been well established. The mechanism of drop formation is shown schematically in Figure 4.11.38 When the inertia force due to liquid circulation exceeds surface tension and buoyancy force at slag layer on plume edge, slag droplets form. Mietz et al.38 have also demonstrated an almost proportionate increase of the mass transfer rate with the extent of slag emulsification. An important parameter in this context is the critical (i.e., minimum) gas flow rate, Qcr, required for emulsion formation. Kim et al.37, based on dimensional analysis and their experimental data, proposed the following empirical equation: Q cr = 0.035H

1.81

0.35 ms ∆ρ  σ -----------------2  ρ 

(4.33)

s

where H is the height of metal bath, ρs is the density of the slag (upper) phase, ∆ρ is the density difference between the two liquids, and σms is interfacial tension between metal and slag.

FIGURE 4.11 Principles of slag emulsification in steel ladles. (a) Scheme of the detaching process, and (b) equilibrium between inertia force Fc, buoyancy force Fg cos α, and surface force Fσ at the point of droplet detachment. Source: from F. Oeters, Metallurgie der stahlherstellung, Verlag stahleisen, Düsseldorf, 1989.


Iguchi et al.39 have performed comprehensive cold model experiments with several liquids simulating slag to determine the critical condition for entrapment of slag in metal, and they proposed the following empirical correlation: u cr ,c ⁄ V = 1.2 ( ν s ⁄ ν m )

0.068

(Hs ⁄ D)

– 0.11

(4.34)

where, u cr ,c = 1.2ur p–0.28 V = (σms g/ρs)1/4 ur = (g Qcr/Hm)1/3 p = [Qcr2/(gHm5)]1/5 The subscripts s and m denote slag and metal, respectively. u cr ,c denotes critical centerline velocity. D is vessel diameter. Here also, gas injection was through a centric nozzle. Agreement with earlier investigators’ results was reasonable. Sahajwalla et al.40 have reviewed some of these studies, including their own experimental work. They found that εm in watts/kilogram, as given in Eq. (3.64), ranged between 0.065 and 0.13 at Qcr for various investigators. Attempts have been made to examine the validity of surface renewal theory for liquid–liquid reactions in gas-stirred ladles. It has already been mentioned that Robertson et al29 and Taniguchi et al30 verified km α D1/2, and some other aspects (Section 4.3.2). Hirasawa et al41 applied the theory of turbulent mass transfer phenomena26 to their own investigation of the reaction of silicon dissolved in molten copper with molten slag containing FeO at 1250°C, as well as to the experimental data of Robertson et al.29 on mercury amalgam-aqueous solutions and molten lead–molten salt systems. Figure 4.12 shows a typical k′ si vs. Q plot for a molten Cu-slag system. Comparison of Figures 4.10 and 4.12 reveals that, in a molten Cu–slag system, region II has a lower dependence on Q, in contrast to a room-temperature model study. This was explained with the help of flow patterns in slag and metal (Figure 4.13). At low Q, the slag was stagnant due to

FIGURE 4.12 Relationship between apparent mass transfer coefficient ( k′ si ) and gas flow rate.41


FIGURE 4.13 Schematic representation of the flow patterns in a slag–metal bath.41

its higher viscosity. At higher Q (region II), slag flow, as shown in Figure 4.13, retarded interfacial flow of liquid metal, causing this behavior pattern. Region III, of course, was due to an increase of the slag–metal interfacial area. Therefore, only region I could be analyzed by the turbulence theory and dimensionless correlations developed. Ogawa et al42 simulated a gas-stirred ladle as well as induction stirring in their water model 1⁄2 studies. Using KCl and Benzoic acid as solute, they found that k m,i ∝ D i approximately. Figure 1⁄2 –2/3 4.14 shows k m,i ∝ D i as a function of εm · V , where V is bath volume. Calculation based on Eq. (4.25) yielded values of S in the range of 15 to 8000 per second. While the lower value is all

km/

Di, min

√

- 2

Gas bubbling

2000 1000

Induction stirring (upward) Induction stirring (downward)

Ar bubbling (85 ton)

500

200

ASEA-SKF (85 ton)

100 50

20 10

20

50

ε

100

500 1000

V -2/3 , watt t-1 m -2

FIGURE 4.14 Relationship between km/(Di)1/2 and ε V–2/3.42


200

right, the higher values are abnormally large and not expected (Section 4.3). Presumably, these are based on the kmA parameter, rather than km and, hence, not indicators of true values of s.

4.5 MIXING VS. MASS TRANSFER CONTROL The pragmatic approach to kinetic calculations in steelmaking is to take the rate equation as that of a first-order process, estimate the rate constant from experimental data, and then use it judiciously. Consider removal of an impurity element (i), dissolved in liquid steel. Then, material balance for i leads to dC e – V --------i = Ak ( C i – C i ) dt

(4.35)

for a first-order reversible reaction. Here, V and A are the volume and interfacial area of the melt. e Ci is the instantaneous concentration of i in the melt, and C i is the concentration in equilibrium with the phase in contact with steel (slag or gas). o Integrating Eq. (4.35) between t = 0, C i = C i (i.e., initial concentration) and t = t, Ci = Ci, e

Ci – Ci -e = 1 – X = – kat ln ----------------o Ci – Ci

(4.36)

where a = A/V = specific interfacial area e

For irreversible processes, C i is very small, and Eq. (4.36) reduces to C ln ------oi = 1 – X = – kat Ci

(4.37)

Convective mass transfer at the phase boundary may be treated as a first-order reversible process s e o s e with k m = k , C i = C i , and C i = C i in Eq. (4.35). The justification for taking C i = C i is the assumption that the rate of interfacial reaction is very fast and not rate controlling. Hence, chemical equilibrium at the interface may be assumed. The term (1 – X) in Eqs. (4.36) and (4.37) is a measure of the extent of impurity removal. For example when X = 0.05, 1 – X = 0.95, which means that 95% of solute i has been eliminated from the steel melt. Based on some literature values of kma, Ghosh43 plotted log X vs. t for a few reactions in steelmaking (Figure 4.15). It may be noted that 95% refining required only 40 to 260 seconds, demonstrating very high rates of steelmaking reactions. Statistical theory also treats mixing as a first-order reversible process [Eqs. (4.2) and (4.3)]. Here, Cf is equivalent to the final equilibrium concentration. Y = 0.95 means a 95 percent degree of mixing and is attained when t = tmix according to convention. A scan of the literature revealed that tmix ranged between 50 and 500 seconds for a variety of processes in steelmaking. Inhomogeneity in the melt due to dead volumes may show mixing times as high as 103 s in some locations. Thus, mixing time and 95 percent conversion time for mass transfer controlled reactions are in the same overall range in steelmaking processes. Since both have rate expressions as for firstorder reversible processes, it is often difficult to say whether a process is controlled by slow mixing or slow mass transfer. In the context of slag–metal reaction in a gas-stirred ladle, it has been concluded by all that, when stirring is vigorous and slag–metal emulsion forms, mass transfer is faster than the rate of mixing. ©2001 CRC Press LLC

1.0 0.5

Slag - metal reaction (LD)

e

C _-O _ reaction (BOH)

Ci - Ci

o e Ci - Ci 0.1

0.05

0.01

H _ removal (Purging ladle, immersed lance)

50

100

150

200

250

TIME, sec

FIGURE 4.15 Estimated X vs. time plots for some steelmaking reactions.43

Szekely et al.44 considered the modified Biot number (Bim), defined as (km A) ⋅ H Bi m = ----------------------D eff

(4.38)

Their sample calculation for typical gentle stirring in a ladle yielded Bim of approximately 10–1 to 10–2. This they considered as low and concluded that the desulfurization reaction would be mass transfer controlled. Mietz and Bruhl45 carried out model calculations for mass transfer with mixing metallurgy in a ladle for sulfur removal. Their principal conclusion was that mixing would be the slower process if dead volumes are not avoided for both gentle and strong stirring. Equations (4.36) and (4.37) have been derived by considering rate control either by interfacial chemical reaction or mass transfer in one phase only. However, there are situations when we may have to take into account the control of a reaction rate jointly by mass transfer in both phases. In that case, Eq. (4.19) is to be employed for both phases (I and II) as follows. m˙ i = ( m˙ i ) at interface = Ak m,i ( C i – C i ) I

S,I

o,I

= Ak m,i ( C i – C i ) II

II

S,II

(4.39)

Again, assuming interfacial equilibrium, S,II

Ci --------- = Li S,I Ci

(4.40)

where Li is the equilibrium partition coefficient of species i between phase II and phase I. Combining Eqs. (4.39) and (4.40), o,II

o,II

Ci  Ci  A o,I I – II o,I m˙ i = ----------------------------  C i – --------- = Ak m,i  C i – --------   1 1 Li Li  ------+ -----------I II k m,i k m,i L i ©2001 CRC Press LLC

(4.41)

REFERENCES 1. Robertson, D.G.C., Ohguchi, S., Deo, B., and Willis, A., Proc. SCANINJECT III, Part 1, MEFOS, Lulea, Sweden, 1983, p. 8.1. 2. Szekely, J., and Themelis, N.J., Rate Phenomena in Process Metallurgy, Wiley-Interscience, John Wiley & Sons Inc., New York, 1971. 3. Geiger, G.H., and Poirer, D.R., Transport Phenomena in Metallurgy, Addison-Wesley Publishing Co., Reading, MA, 1980. 4. Levenspiel, O., Chemical Reaction Engineering, John Wiley & Sons, Inc., New York, 1962. 5. Ghosh, A., and Ray, H.S., Principles of Extractive Metallurgy, Wiley Eastern Limited, New Delhi, 1991. 6. Mazumdar, D., and Guthrie, R.I.L., ISIJ International, 35, 1995, p. 1. 7. Oeters, F., Plushkell, W., Steinmetz, E., and Wilhelmi, H., Steel Research, 59, 1988, p. 192. 8. Krishnamurthy, G.G., and Mehrotra, S.P., Ironmaking and Steelmaking, 19, 1992, p. 377. 9. Danckwerts, P.V., Applied Science Research, 3, 1953, p. 279. 10. Nagata, S., Mixing Principles and Applications, published jointly by Kodansha Ltd. Tokyo and John Wiley & Sons Inc., New York, 1975. 11. Krishnamurthy, G.G., Mehrotra, S.P., and Ghosh, A., Metall. Trans.,19B, 1988, p. 839. 12. Krishnamurthy, G.G., ISIJ Int., 29, 1989, p.49. 13. Mietz, J., and Oeters, F., Steel Research, 59, 1988, p.52. 14. Joo, S., and Guthrie, R.I.L., Metall. Trans., 23B, 1992, p.765. 15. Mazumdar, D., Nakajima, H., and Guthrie, R.I.L., Metall. Trans., 19B, 1988, p.507. 16. Haida, O., Emi, T., Yamada, S., and Sudo, F., Proc. SCANINJECT II, MEFOS, Lulea, Sweden, 1980, p. 20.1. 17. Nakanishi, K., Szekely, J., and Chiang, C.W., Ironmaking and Steelmaking, 3, 1975, p. 115. 18. Nakanishi, K., Fujii, T., and Szekely, J., Ironmaking and Steelmaking, 3, 1975, p. 193. 19. Murthy, A., and Szekely, J., Metall. Trans., 17B, 1986, p. 487. 20. Mazumdar, D., and Guthrie, R.I.L., Metall. Trans., 17B, 1986, p.725. 21. Neifer, M., Rodi, S., and Sucker, D., Steel Research, 64, 1993, p. 54. 22. Sano, M., and Mori, K., Trans. ISIJ, 23, 1983, p. 169. 23. Krishnamurthy, G.G., Ghosh, A., and Mehrotra, S.P., Metall. Trans., 20B, 1989, p.53. 24. Stapurewicz, T., and Themelis, N.J., Can. Met. Quarterly, 26, 1987, p. 123. 25. Asai, S., Okamoto, T., He, J., and Muchi, I., Trans ISIJ, 23, 1983, p. 43. 26. Davies, J.T., Turbulence Phenomena, Academic Press, New York, 1972. 27. Boorstein, M., and Phelke, R.D., Trans. AIME, 245, 1969, p. 1843. 28. Kumar, J., and Ghosh, A., Trans. Indian Inst. Metals, 30, 1977, p. 39. 29. Robertson, D.G.C., and Staples, B.D., Process Engg. Of Pyrometallurgy, ed. M. Jones, Inst. Min. Met. London, 1974. 30. Taniguchi, S., Okada, Y., Sakai, A., and Kikuchi, A., Proc. 6th Int. Iron and Steel Cong., Nagoya, 1990, 1, p. 394. 31. Brimacombe, J.K., Proc. Richardson Conference, eds. J.H.E. Jeffes and R.J. Tait, Inst. of Min. and Met., London, 1973. 32. Richardson, F.D., Physical Chemistry of Melts in Metallurgy, 2, Academic Press, London, 1974. 33. Iguchi, M., Tomida, H., Nakajima, K., and Morita, Z., ISIJ International, 33, 1993, p. 728. 34. Mazumdar, D., Kajani, S.K. and Ghosh, A., Steel Research, 61, 1990, p. 339. 35. Mazumdar, D., Verma, V., and Kumar, N., Ironmaking and Steelmaking, 19, 1992, p. 152. 36. Turkdogan, E.T., Physical Chemistry of High Temperature Technology, Academic Press, New York, 1980. 37. Kim, S.H., Fruehan, R.J., and Guthrie, R.I.L., Steelmaking Proceedings, Iron and Steel Soc., USA, 1987, p. 107. 38. Mietz, J., Schneider, S., and Oeters, F., Steel Research, 62, 1991, p. 1. 39. Iguchi, M., Sumida, Y., Okada, R., and Morita, Z., ISIJ International, 34, 1994, p. 164. 40. Sahajwalla, V., Brimacombe, J.K., and Salcudean, M.E., Steelmaking proceedings, Iron and Steel Soc., USA, 72, 1989, p. 497. 41. Hirasawa, M., Mori, K., Sano, M., Shimatani, Y., and Okazaki, Y., Trans. ISIJ, 27, 1987, p. 277. ©2001 CRC Press LLC

42. 43. 44. 45.

Ogawa, K., and Onoue, T., ISIJ International, 29, 1989, p. 148. Ghosh, A., Tool and Alloy Steels, 25, 1991, Silver Jubilee issue, p. 65. Szekely, J., Carlsson, C., and Helle, L., Ladle Metallurgy, Springer Verlag, New York, 1989. Mietz, J., and Bruhl, M., Steel Research, 61, 1990, p. 105.


5

Deoxidation of Liquid Steel

Steelmaking is a process of selective oxidation of impurities in molten iron. During this, however, the molten steel also dissolves some oxygen. Solubility of oxygen in solid steel is negligibly small. Therefore, during solidification of steel in ingot or continuous casting, the excess oxygen is rejected by the solidifying metal. This excess oxygen causes defects such as blowholes and nonmetallic inclusions in castings. It also has significant influence on the structure of the cast metal. Therefore, it is necessary to control the oxygen content in molten steel before it is teemed. Actually, the oxygen content of the bath in the furnace is high, and it is necessary to bring it down by carrying out deoxidation after primary steelmaking and before teeming the molten metal into an ingot or continuous casting mold. This chapter is concerned with thermodynamics and kinetics of deoxidation, and finally on industrial deoxidation. 5.1

THERMODYNAMICS OF DEOXIDATION OF MOLTEN STEEL

The dissolution of oxygen in molten steel may be represented by the equation 1 --- O 2 ( g ) = [ O ] 2

(5.1)

where [O] denotes oxygen dissolved in the metal as atomic oxygen. For the above reaction,  hO  K O =  --------1 ⁄ 2  p O2 

(5.2) equilibrium

where KO is equilibrium constant for Reaction (5.1), p O2 denotes partial pressure of oxygen in the gas phase in atmosphere, and hO is the activity of dissolved oxygen in liquid steel with reference to the 1 wt.% standard state. KO is related to temperature as1 6120 logK O = ------------ + 0.15 T

(5.3)

hO = [ f O ] [ W O ]

(5.4)

Again,

where WO denotes the concentration of dissolved oxygen in weight percent, and fO is the activity coefficient of dissolved oxygen in steel in 1 wt.% standard state. In pure liquid iron,


log f O = 0.17 [ W O ]

(5.5)

The above relations would allow us to estimate WO in liquid iron at any value of p O2 with which the molten iron would be brought to equilibrium. This value of WO is nothing but solubility of [O] at that p O2 . However, oxygen tends to form stable oxides with iron. Therefore, molten iron becomes saturated with [O] when the oxide starts forming, i.e., when liquid iron and oxide are at equilibrium. This oxide, in its pure form, is denoted as FexO, where x is approximately 0.985 at 1600°C. For the sake of simplicity we shall take x equal to 1 often and designate this compound as FeO. For the reaction FexO(1) = xFe(1) + [O]wt.%., 6150 logK Fe = – ------------ + 2.604 T

(Ref. 1)

(5.6)

where  [ h O × [ a Fe ] x  - K Fe =  ---------------------------a Fe x O   equilibrium

(Ref. 1)

(5.7)

Here, aFe = the activity of Fe in the metal phase in the Raoultian scale (approximately 1), and a F e x O denotes the activity of FexO in oxide phase. If the FeO is not pure and is present in an oxide slag, then aFeO < 1, and h (i.e., solubility of [O] in equilibrium with the slag) would be less. Example 5.1 Calculate the concentration of oxygen in molten iron at 1600°C in equilibrium with (a) pure FexO, and (b) a liquid slag of FeO-SiO2 containing 40 mol.% SiO2. Solution [ hO ] [FO][W O] K Fe = -------------- = -----------------------( a FeO ) ( a FeO ) or, logK Fe = log f O + logW O – loga FeO = – 0.17 + logW O – loga FeO

(E1.1)

Again, at 1600°C, from Eq. (5.6), log KFe = –0.672

(E1.2)

(a) In pure FeO, aFeO = 1, and hence combining Eqs. (E1.1) and (E1.2) and solving, WO = 0.233 wt.%

(Ans.)

(WO at 1550°C and 1650°C are 0.185 and 0.29 wt.%, respectively) (b) In liquid FeO-SiO2 slag at 1600°C and at X SiO2 = 0.4 ( X SiO2 denotes mole fraction of SiO2 in FeO-SiO2), aFeO = 0.43. ©2001 CRC Press LLC

Solving Eqs. (E1.1) and (E1.2), we obtain: WO = 0.10 wt.%

(Ans.)

The traditional method of determination of oxygen in steel samples is chemical analysis by vacuum fusion or inert gas fusion apparatus. Here, a sample of solidified steel is taken in a graphite crucible and then heated to approximately 2000°C under vacuum or under a highly purified inert atmosphere. The steel sample melts and the oxygen contained in it reacts very fast with the crucible and generates carbon monoxide. The quantity of CO is measured by a sensitive instrument such as an infrared analyzer, and from it the quantity of oxygen in the sample is estimated. This apparatus has been made quite accurate and reasonably fast. Analyses of alloying elements in steel are done very quickly and conveniently using an emission spectrometer. Commercial development of the instrument has recently been reported wherein the optical wavelength range has been extended to the ultraviolet region, enabling the determination of total oxygen as well. This would eliminate the need for separate sampling and analysis. However, the author is not aware of relative precision and reliability of these two techniques. In industrial melts, the bath not only contains dissolved oxygen but also oxide particles. During freezing, solidifying steel rejects most of its dissolved oxygen, which forms additional oxide particles, and these are also retained by the solid as inclusions. The above methods of determination give the total oxygen content, which is the sum of dissolved O and oxygen in inclusions. This hampered progress of our understanding about the behavior of oxygen in steelmaking and deoxidation until the development of immersion oxygen sensors based on ZrO2 and doped with CaO or MgO during the decade of the 1960s. Thereafter, this has become quite a popular tool for the measurement of dissolved oxygen content in molten steel, both in the laboratory and in industry. Excellent reviews are available in the literature on the principles and details of such sensors.2–5 For the sake of illustration, Figure 5.1 shows the sensor employed by Fruehan et al.3 schematically. The ZrO2 (CaO) or ThO2 (Y2O2) disk served as the solid electrolyte, and at high temperature it is an ionic conductor with O2– as the only mobile ionic species. The Cr + Cr2O3 mixture is the

FIGURE 5.1 Sketch of an oxygen sensor.3


reference electrode. This assembly is immersed into liquid steel. Molten steel constitutes the other electrode. A molybdenum-Al2O3 cermet was dipped into it, and the electrical circuit was completed by platinum lead wires connected to the measuring circuit. These sensors can be used only once, i.e., they are a disposable type. Immersion time required is less than a minute. Efforts are going on to develop sensors that can be continuously immersed in liquid steel for a longer period. Laboratory successes have been reported. Such sensors behave as reversible galvanic cells. Since the solid electrolyte conducts oxygen ions only, the cell electromotive force (EMF) is related only to the difference of the chemical potentials of oxygen at the two electrodes. µ O2 (liquid steel) – µ O2 (reference) = – Z FE

(5.8)

where µ O2 designates the chemical potential of oxygen, F is Faraday’s constant, Z is valence (4, here) and E is cell EMF. The galvanic cell in Figure 5.1 may be represented as Cr ( s ) + Cr 2 O 3 ( s ) ZrO 2 + CaO [ O ] (reference)

(in liquid steel)

(solid electrolyte)

(5.9)

With reference to Section 2.7, µ O2 (reference) = RT ln p O2 (reference) = ∆G f for formation of Cr 2 O 3 ( s ) per mole O 2 o

2 o = --- ∆G f ( Cr 2 O 3 ) 3

(5.10)

and µ O2 (liq. steel) = RT ln p O2 (in equlibrium with liq. steel) hO - (from Eq. 5.2) = 2RT ln -----KO

(5.11)

Combining the above equations, hO 2 o - – --- ∆G f ( Cr 2 O 3 ) = – 4FE 2RT ln -----KO 3

(5.12)

Therefore, knowing (Cr2O3) and KO from the literature, the cell EMF allows us to calculate [hO]. With reference to Section 2.6.2, [fO] can be estimated from chemical analysis of steel. Therefore, the content of dissolved oxygen (i.e., WO) can be obtained from Eq. (5.4). Several designs of commercial oxygen sensors are now on the market. A popular one is CELOX, marketed by Electro-Nite n.v., Belgium. It has been jointly developed by CRM, Belgium, and Hoogovens Ijmuiden B.V., along with Electro-Nite.6 The cell is Mo Cr + Cr 2 O 3 ZrO 2 ( MgO ) liq. steel Fe The solid electrolyte is in the form of a tube with one end closed. ©2001 CRC Press LLC

(5.13)

All such sensors also contain immersion thermocouples as well so that the temperature of molten steel is also recorded simultaneously. At steelmaking temperatures, the solid electrolyte exhibits partial electronic conduction, especially at a low level of dissolved oxygen. The measured cell voltage of the cell of type illustrated by Expression (5.13) would also include thermo-EMF due to use of dissimilar leads, viz., Mo and Fe. The manufacturer provides correction terms for it. In pure liquid iron, the solubility of oxygen is governed by either Eq. (5.2) or (5.7). However, in molten steel, there are other more reactive alloying elements such as C, Si, and Mn. The oxygen solubility is governed by reaction with one or more of these elements. It has been well established that the carbon content of steel has a considerable influence on bath oxygen content at the end of heat in steelmaking furnaces. The reaction is p CO [ C ] + [ O ] = CO ( g ); K CO = ------------------[ hc ] [ ho ]

(5.14)

The value of equilibrium constant (KCO) is given as.1 1160 logK CO = ------------ + 2.003 T

(5.15)

Figure 5.2 shows the relationship between dissolved carbon and dissolved oxygen in a molten steel bath in a 100 kVA induction furnace. The equilibrium line corresponds to pCO = 1 atm at 1600°C. Dissolved oxygen contents were measured by a solid electrolyte oxygen sensor with two types of reference electrodes.

5.1.1

THERMODYNAMICS

OF

SIMPLE DEOXIDATION

Deoxidation of liquid steel is carried out mostly via ladle, tundish, and mold. Even in a furnace, deoxidizers are often added directly into the metal bath. In all these cases, the product of deoxidation, which is an oxide or a solution of more than one oxide, forms as precipitates. Deoxidation never occurs at a constant temperature. The temperature of molten steel keeps dropping from furnace to mold. The addition of a deoxidizer also causes some temperature change due to heat of reaction. However, we shall consider it as isothermal. This will not affect our

FIGURE 5.2 Dissolved oxygen content of liquid iron as a function of bath carbon at 1873 K in a 100 kVA induction furnace.4


considerations of deoxidation equilibria, since only the final temperature at which the equilibrium is supposed to be attained is of importance. Thermodynamically, it would not make any difference if the process were presumed to take place at that temperature. Deoxidation may be carried out by addition of one deoxidizer only. This is known as simple deoxidation. In contrast, we may use more than one deoxidizer simultaneously and, in that case, it will be termed a complex deoxidation. In this section, we will discuss simple deoxidation. A deoxidation reaction may be represented as x[M] + y[O] = (MxOy)

(5.16)

where M denotes the deoxidizer, and MxOy is the deoxidation product. The equilibrium constant ( K′ M ) for reaction (5.16) is given as  ( a M x Oy )  -y  K′ M =  ------------------------x  [ h M ] [ h O ] equilibrium

(5.17)

Again, on the basis of Eq. (2.46), hM = fM · WM and hO = fO · WO. If the deoxidation product is pure, then a M x O y = 1. Also, in very dilute solutions, fM and fO may be taken as 1. Hence, Eq. (5.17) may be rewritten as 1 x y [ W M ] [ W O ] = --------- = K M K′ M

(5.18)

where KM is known as deoxidation constant. Obviously, WM and WO, as already understood, are weight percentages of [M] and [O], respectively, at equilibrium with pure oxide. It may be noted that KM is like the solubility product in an aqueous solution. It is a measure of solubility of the compound MxOy in molten steel at the temperature under consideration. As in Eq. (5.6), variation of KM with temperature may be represented by an equation of the type: A logK M = – --- + B T

(5.19)

where A and B are constants. Equation (5.19) shows that as T increases, log KM and hence KM also increase. In other words, the solubility of MxOy in molten steel increases with temperature. Since, in deoxidation, we are interested in lowering the concentration of oxygen with the addition of as little deoxidizer as possible, an increase in temperature would adversely affect the thermodynamics of the process. Experimental determination as well as thermodynamic estimation of KM for various deoxidizers have been going on for the last four or five decades. With advancements in science and technology, more accurate values are being found with the passage of time. This has led to a number of compilations, some old and some new, where efforts have been made to record the most acceptable values. The exercise is still going on, and discrepancies still exist, especially with more reactive elements such as Al, Zr, Ce, Ca, etc. Appendix 5.1 presents such a compilation taken from that of the 19th Steelmaking Committee of the Japan Society for Promotion of Science,1 as well as from other sources.7–9 It may be noted that all oxide products are definite compounds except for deoxidation by manganese, where the product is either a solid or a liquid solution of FeO-MnO of variable composition. The underlying reason for this behavior is the fact that manganese is a weak deoxidizer, ©2001 CRC Press LLC

since the stability of MnO, although greater than that of FeO, is not drastically different from that of the latter (Figure 2.1). For deoxidation by Mn, it is in a way more appropriate to consider the reaction (MnO) + [Fe] = [Mn] + (FeO)

(5.20)

Fe and Mn form an ideal solution (i.e., one that obeys Raoult’s law). The same is true of the MnO-FeO slag. Therefore, aMnO = XMnO, aFeO = XFeO, and hMn = WMn. Noting that aFe = 1, the equilibrium constant for Reaction (5.20) is [ h Mn ] × ( a FeO ) [ W Mn ] ( X FeO ) K Mn – Fe = ---------------------------------- = ------------------------------( a MnO ) ( X MnO )

(5.21)

where X denotes mole fraction. Equation (5.21) shows that X MnO ⁄ X FeO in the deoxidation product would be proportional to WMn at constant temperature. Figure 5.3 shows the relationship. The oxide product is liquid at low and solid at high values. Example 5.2 Consider deoxidation by addition of ferromanganese (60 percent Mn) to molten steel at 1600°C. The initial oxygen content is 0.04 wt.%. It has to be brought down to 0.02 wt.%. Calculate the quantity of ferromanganese required per tonne of steel. The manganese content of steel before deoxidation is 0.1 wt.%. Solution Consider the following reaction: [ h Mn ] [ h o ] (MnO) = [Mn] + [O]; KMn = ---------------------( a MnO )

(E2.1)

FIGURE 5.3 Composition of liquid or solid FeO-MnO solution in equilibrium with liquid iron containing manganese and oxygen.9


As noted earlier, hMn may be taken as WMn, and aMnO as XMnO. Assuming also that hO = WO, K Mn [ W Mn ] ---------------- = -----------( X MnO ) [W O]

(E2.2)

From Appendix 5.1, –11070 logK Mn = ------------------ + 4.536 T i.e., at 1600°C (1873 K), KMn = 0.041. Noting that the final WO = 0.02 wt.%, [ W Mn ] --------------- = 2.05 ( a MnO )

(E2.3)

K Mn – Fe [ W Mn ] ---------------- = ----------------( X MnO ) ( X FeO )

(E2.4)

Now from Eq. (5.21),

From Appendix 5.1, 6980 logK Mn – Fe = – ------------ + 2.91 (assuming the product to be solid MnO – FeO) T that is, at 1873 K, KMn-Fe = 0.15 Therefore, combining Eqs. (E2.3) and (E2.4), X FeO = 0.073 or, X MnO = 1 – X FeO = 0.927 or, W Mn = 1.90 wt.% Now, the total quantity of Mn required = Mn required to form MnO + Mn required to increase the Mn-content of bath from 0.1 to 1.90 wt.% ©2001 CRC Press LLC

Now, the Mn required to form MnO per tonnne of steel Quantity of oxygen removed per tonne of steel = ---------------------------------------------------------------------------------------------------------------- × X MnO × Atomic mass of Mn Atomic mass of oxygen ( 0.04 – 0.02 ) × 10 × 10 × 0.927 × 55 = ---------------------------------------------------------------------------------------------16 –2

3

= 0.64 kg/t The Mn required to increase the Mn content of bath = (1.90 – 0.1) × 10–2 × 103 = 23.7 kg/t steel. Total Mn required = 18.64 kg. 100 Total ferromanganese required = 18.64 × --------- = 31.1 kg/ton steel . 60

(Ans.)

From Figure 5.3, it is confirmed that the assumption of solid FeO-MnO as the deoxidation product is correct. Taking the activity coefficients, viz., fO and fM, as 1, one would be able to calculate the relationship between [WM] and [WO] using Eq. (5.18) and Appendix 5.1 for many deoxidizers. Such calculations would be all right at very low values of WM. At higher ranges, it would give approximate values only, since fM and fO (especially fO) may deviate somewhat from 1. For more precise j calculations, therefore, first-order interaction coefficients e M ⋅ W j are to be considered. The relationship between activity coefficients and interaction coefficients follow from Chapter 2 and are as noted below. log f M =

∑j e M ⋅ W j

(5.22)

log f O =

∑j eO ⋅ W j

(5.23)

j

j

where j denotes all the alloying elements present in liquid steel. For example, if the steel contains C and Mn, then log f O = e O ⋅ W O + e O ⋅ W C + e O ⋅ W Mn O

C

Mn

(5.24)

Some values of interaction coefficients are tabulated in Appendix 2.3. Taking the logarithm of Eq. (5.18), we have 1 x logW O = --- logK M – --logW M y y

(5.25)

In a log WO vs. log WM plot, Eq. (5.25) would yield a straight line with a slope of –x/y. However, such linearities are not always expected if rigorous equations such as Eqs. (5.16), (5.17), (5.22), and (5.23) are employed. Calculated log WO vs. log WM curves for various deoxidizers in Figure 5.4 demonstrate such nonlinearities. Immersion oxygen sensors are nowadays widely employed in oxygen control during secondary steelmaking, especially for deoxidation control. An associated use is estimation of dissolved alu©2001 CRC Press LLC

FIGURE 5.4 Deoxidation equilibria in liquid iron at 1873 K.9

minum in molten steel.4,6 Since residual dissolved aluminum content is very low (0.005 to 0.05%), its determination by an emission spectrometer was unreliable in view of interference from Al2O3 inclusions. The analysis is time consuming, too. However, today’s commercially available spectrometers are useful for the determination of dissolved aluminum content in steel. Measurements are made on several spots of the sample. The minimum value is assumed to be from an inclusionfree spot and, in principle, acceptable as a measure of dissolved aluminum content. The principle of the determination of [Al] using an oxygen sensor follows from the equilibrium of the following reaction, viz., Al2O3(s) = 2[Al] + 3[O]

(5.26)

K Al = [ h Al ] [ h O ] , since a Al2 O3 = 1

(5.27)

log KAl = 2 log hAl + 3 log hO

(5.28)

2

3

i.e.,

With the value of log KAl from Appendix 5.1, and measured hO, the value of hAl can be obtained. Evaluation of fAl on the basis of Eq. (5.22) allows the determination of WAl. Example 5.3 Consider the determination of dissolved oxygen in liquid steel using an oxygen sensor with a CrCr2O3 reference electrode at 1600°C. What would be the value of hO if the EMF of the cell is –153 mV? Also calculate dissolved aluminum content. Ignore solute–solute interactions. Solution Combining Eqs. (5.8), (5.10), and (5.11), ©2001 CRC Press LLC

hO O 2 ∆G f  --- Cr 2 O 3 , – 2RT ln ------ = – ZFE 3  KO

(E3.1)

From Appendix 2.1, at 1600°C (1873 K) ∆G f ( Cr 2 O 3 ) = – 422.7 × 10 J/mol; O 2 O

3

with R = 8.314 J mol–1K–1, Z = 4, F = 96,500 J volt–1 gm. equiv.–1, E = –0.153 V, from Eq. (5.3), KO = 2615 at 1873 K. Putting in the values and solving, hO = 0.0005. From Appendix 5.1, For 2Al + 3O = Al2O3(s); KAl = 2.51 × 10–14 = [WAl]2 [WO]3 WO is nothing but hO in the above equation, since solute–solute interactions have been ignored in arriving at it. Putting in the values in Eq. (5.12) and solving, WAl = 0.0141 wt.%

(Ans.)

Figure 5.4 shows that Mn is the weakest deoxidizer of all, and Al, Zr, etc. are very powerful. Deviation from the straight line for Mn deoxidation is caused by the variable composition of the deoxidation product as well as the fact that the liquid FeO-MnO changes to solid FeO-MnO with higher manganese content. Deoxidizers such as Al, Ti, Zr, etc. exhibit a minimum in the solubility M of oxygen. This behavior is due to the large negative value of e C (see Appendix 2.3) for these elements. The situation has been analyzed by several authors, such as Ghosh and Murty,9 and such minima have been quantitatively explained. Differentiating Eq. (5.28) with regard to WAl, d ( logh Al ) d ( logh O ) d ( logK Al ) - + ---------------------- = ----------------------2 ---------------------- = 0 dW Al dW Al dW Al

(5.29)

d ( logW O + log f O ) d ( logW Al + log f Al ) - + 3 -------------------------------------------- = 0 2 ---------------------------------------------dW Al dW Al

(5.30)

d ( log f Al ) d ( log f O ) 2 3 1 dW O 1 ------------- ⋅ --------- + ------------- ⋅ -------- ------------ = 0 + 2 ----------------------+ 3 --------------------2.303 W Al 2.303 W O dW Al dW Al dW Al

(5.31)

log f Al = e Al ⋅ W Al + e Al ⋅ W O

(5.32)

log f O = e O ⋅ W Al + e O ⋅ W O

(5.33)

or,

or,

Now, Al

Al


O

O

Substituting these in Eq. (5.31) and noting that interaction coefficients are constant, 2 1 Al Al ------------- ⋅ ---------- + 2e Al + 3e O 2.303 W A1 dW O ------------- = – ------------------------------------------------------------dW A1 3 1 O O ------------- ⋅ -------- + 2e Al + 3e O 2.303 W O

(5.34)

dW O = 0 At minimum oxygen solubility, -----------dW Al Hence, 1 • W Al = – -----------------------------------------3 Al Al 2.303  e Al + --- e O   2 

(5.35)

•

where W Al = WAl at oxygen minimum. Equation (5.35) provides a simple relationship between WAl at oxygen minimum and the Al • interaction coefficients. As Appendix 2.3 shows, e O has a large negative value. This makes W Al positive and small in magnitude and substantiates the statement made above that large negative M values of e O are responsible for these minima. On the basis of their exercise, the following analytical equation was proposed by Ghosh and Murty9 to describe the curves: logK M = x ( logW M + e M ⋅ W M ) + y ( logW O + e O ⋅ W M + r O ⋅ W M ) M

M

M

2

(5.36)

M

where r O is the second-order interaction coefficient. Unlike conventional deoxidizers, the alkaline earths, viz., Ca and Mg, are gaseous at steelmaking temperatures (pMg = 25 atm, and pCa = 1.8 atm at 1600°C). Moreover, they are sparingly soluble in molten steel. The solubility of Mg is 0.1 wt.% at pMg = 25 atm, and that of Ca is 0.032 wt.% at pCa = 1.8 atm at 1600°C. As a result of poor solubility, as well as the extremely reactive nature of these elements, the equilibrium relationships between them and dissolved oxygen are difficult to determine experimentally, and there are uncertainties. Experimental measurements and assessment exercises of data are still continuing.10,11 Example 5.4 Consider deoxidation of molten steel by aluminum at 1600°C. The bath contains 1% Mn and 0.1% C. The final oxygen content is to be brought down to 0.001 wt.%. Calculate the residual aluminum content of molten steel assuming that [Al] – [O] – Al2O3 equilibrium is attained. Also take into account all interaction coefficients. Solution log KAl = 2 log hAl + 3 log hO

(5.28)

= 2 [ logW Al + e A1 × W Mn + e A1 × W C + e A1 × W O + e A1 × W A1 ] Mn

C

O

A1

+ 3 [ logW O + e O × W Mn + e O × W A1 + e O × W C + e O × W O ] Mn


A1

C

O

(E4.1)

Noting that WMn = 1, WC = 0.1, and WO = 0.001, and substituting the values in Eq. (E4.1) and taking KAl value from Appendix 5.1 and values of e from Appendix 2.3, we obtain log [2.5 × 10–14] = 2 log WAl + 3 log 0.001 – 3.42 WAl – 0.06

(E4.2)

Taking a first guess as WAl = 0.01, a trial-and-error solution yields WAl = 5.36 × 10-3 wt.% as the residual aluminum in the bath (Ans.)

5.1.2

THERMODYNAMICS

OF

COMPLEX DEOXIDATION

As already stated, if more than one deoxidizer is added to the molten steel simultaneously, it is known as complex deoxidation. Some important complex deoxidizers are Si-Mn, Ca-Si, Ca-Si-Al, etc. Complex deoxidation offers the following advantages and is being employed increasingly for a better quality product. 1. The dissolved oxygen content is lower in complex deoxidation as compared to simple deoxidation from equilibrium considerations. Consider deoxidation by silicon. [ h Si ] [ h O ] [ W Si ] [ W O ] - = ---------------------------K Si = ----------------------( a SiO2 ) ( a SiO2 ) 2

2

(5.37)

If only ferrosilicon is added, then the product is pure SiO2, i.e., a SiO2 = 1. On the other hand, simultaneous addition of ferrosilicon and ferromanganese in a suitable ratio leads to the formation of liquid MnO-SiO2. Consequently, a SiO2 is less than 1, and hence [WSi] [WO]2 is less than that obtained by simple ferrosilicon addition. At a fixed value of WSi, therefore, WO, would be less in complex deoxidation. 2. The deoxidation product, if liquid, agglomerates easily into larger sizes and consequently floats up faster, making the steel cleaner. This is what happens in many complex deoxidation such as in the example presented above. 3. Properties of inclusions remaining in solidified steel can be made better by complex deoxidation, thus yielding a steel of superior quality. This will be discussed again later, in an appropriate place. Equilibrium calculations involving complex deoxidation require data on activity vs. composition j relationships in the binary or ternary oxide systems of interest, besides values of KM and e i . These are available for many systems.12 Figure 5.5 presents the activity-composition data for a MnO-SiO2 system. The activities are in Raoultian scale, whereas the composition has been expressed in terms of weight percent of SiO2. Figure 2.3 has presented isoactivity lines for silica in the ternary CaOSiO2-Al2O3 system at 1550°C. Figure 5.6 shows the same for CaO and Al2O3. The activities were determined in the liquid slag region only. For activity in oxide (i.e., slag) systems, the general discussions in Chapter 2 may be consulted. For complex deoxidation, the desired product should be within this liquid field. Thermodynamic calculations involving complex deoxidation should aim at the following: • Estimation of weight percentages of deoxidizing elements and oxygen remaining in molten steel when equilibrium is attained • Estimation of the composition of the deoxidation product in equilibrium with the above Rigorous calculations pose difficulties for two reasons. First of all, the activity vs. composition data in oxide systems are not available in the form of equations. Secondly, interaction of more than ©2001 CRC Press LLC

FIGURE 5.5 Activity vs. composition relationship in MnO-SiO2 melts; standard state are pure solid MnO and pure β-crystobalite. Source: Elliott et al., Ref. 4 of Chapter 2.

FIGURE 5.6 Activities of CaO and Al2O1.5 in CaO-Al2O3-SiO2 system at 1823 K. Source: Elliott et al., Ref. 4 of Chapter 2.


one deoxidizer calls for an iterative procedure for the solution of Eqs. (5.22) and (5.23). Therefore, it is necessary to use a computer-oriented method. A major challenge is the minimization of calculation errors. Turkdogan13 has carried out thermodynamic analysis for complex deoxidation by Si-Mn. Bagaria, Deo, and Ghosh14 have carried out thermodynamic analysis of simultaneous deoxidation by Mn-Si-Al. Ghosh and Naik15 have done the same for deoxidation systems: Ca-SiAl and Mg-Si-Al. Readers may refer to those works for details. Some salient findings by Ghosh and Naik are presented below. Calculations were performed in the range where the deoxidation product is liquid CaO-SiO2Al2O3 slag in the ternary diagram (Figure 2.3) at two temperatures. Figure 5.7 presents some results of calculations for a Ca-Si-Al system as log WO vs. log WM (M = Si or Al) curves for three compositions of liquid deoxidation products. The dotted curves are based on rigorous calculations, taking into consideration all interaction coefficients. For the solid curves, h values were taken to be the same as weight percent, i.e., the interaction coefficients were ignored. The two curves differ by about 20%. Thermodynamically, the complex deoxidizer was found to be, at most, an order of magnitude more powerful than simple deoxidation by Al or Si. The above exercise is important from the point of view of industrial application. Ignoring of interactions, i.e., taking hi = Wi, simplifies the calculation procedure in a significant way. The above analysis shows that the kind of error one may encounter is tolerable for many applications. It is also possible to predict thermodynamically the sequence of precipitation of deoxidation product, provided the process is treated as reversible. This issue is pertinent for deoxidation, where the product composition varies with time. An example of this approach is the work by Wilson et al.16 on a Fe-O-S-Ca system. Another is the analysis of a Fe-O-Ca-Al system by Faulring et al.17 Here, the hCa/hAl ratio in liquid iron determined the nature of the deoxidation product. This topic is taken up in Chapter 9 again in connection with inclusion modification. Example 5.5 Consider deoxidation of molten steel by the simultaneous addition of ferromanganese and ferrosilicon at 1600°C. If the residual WO and WSi are, respectively, 0.01 and 0.1 wt.%, determine the composition of the deoxidation product and WMn in steel at equilibrium with the above conditions. Ignore interactions among elements in molten steel.

FIGURE 5.7 Some [wt.% O] vs. [wt.% M] and [hO] vs. [hM] relationships for [Al]-[O]-(Al2O3) and [Si][O]-(SiO2) equilibria for deoxidation by Ca-Si-Al at 1873 K.15


Solution Consider deoxidation by Si (Eq. 5.37). Now, KSi = 2.11 × 10–5 at 1600°C (Appendix 5.1), WSi = 0.1, and WO = 0.01 This yields a SiO2 = 0.47 . Assuming the deoxidation product as MnO-SiO2 and using Figure 5.5, the weight percent of SiO2 in the deoxidation product is 41%, and therefore, MnO content is 59%. Also, aMnO is approximately 0.25. Again, consider the reaction of Mn as in Example 5.2. [ W Mn ] [ W O ] K Mn = 0.053 = ---------------------------( a MnO ) Substituting values for aMnO and WO, WMn becomes 1.33 wt.%. Therefore, • The deoxidation product contains 41 wt.% SiO2 and 59 wt.% MnO. • The weight percent of Mn in steel = 1.33. (Ans.) 5.2

KINETICS OF THE DEOXIDATION OF MOLTEN STEEL

In Section 5.1, we were concerned with dissolved oxygen only. However, in industrial deoxidation practice, dissolved and total oxygen both are of importance. Even if the former is low, the presence of entrapped deoxidation products gives rise to inclusions in solidified steel. The products of deoxidation should be separated out from the molten steel before the latter solidifies, if a clean steel is desired. Therefore, the subject of deoxidation kinetics is concerned with deoxidation reaction as well as separation of deoxidation products. Studies of deoxidation kinetics started seriously in 196018 and are still continuing. Factors controlling the rates have been established reasonably well from theoretical considerations as well as from experiments conducted in laboratories and plants. The availability of new equipment and techniques has been of considerable help. In almost all studies prior to 1970, only total oxygen could be determined by sampling and vacuum or inert gas fusion analysis. Later investigators also employed immersion oxygen sensors for the determination of dissolved oxygen in molten steel. The advent of electron probe microanalyzers allowed the rapid and easy determination of chemical compositions of inclusions in steel. Development of the Quantimet brought about a method for rapid determination of inclusion size, number, etc. By 1970, and even earlier, thermodynamic parameters for important deoxidation reactions were available that could provide a fair degree of confidence. The basic behavior pattern of oxygen and inclusions from a furnace to solidification during steelmaking may be visualized with the help of Figure 5.8 from Plockinger and Wahlster.18 The dissolved oxygen content decreases rapidly upon deoxidation in the ladle and keeps on decreasing all the way. Inclusion content in liquid steel becomes quite high in the ladle upon deoxidation, followed by decrease due to separation of the deoxidation product. Since steel has negligible solubility for oxygen, the dissolved oxygen in liquid steel also, upon solidification, could give rise to more inclusions. Therefore, the expected inclusion content in steel would always be higher in the solid than in the liquid in a mold. To gain a greater understanding of the factors influencing the rates, a number of fundamental investigations have been carried out from 1960 onward. These have been done mostly in the laboratory under controlled conditions. Therefore, we shall first discuss the findings of laboratory experiments. Laboratory investigations have been carried out mostly with a small melt (on the order of a few kilograms of steel) and under inert atmosphere. High-frequency induction furnaces were ©2001 CRC Press LLC

FIGURE 5.8 Change of oxygen and inclusion content of steel from furnace to ingot.

normally employed for maintaining steel molten at the desired temperature. If the steel is directly heated by a high-frequency power source, then an eddy current flows through it. This eddy current and the magnetic field of the induction coil generate force, which causes the flow and circulation of molten steel. This is known as induction stirring. On the other hand, if the ceramic crucible containing the steel is surrounded by a hollow cylinder of graphite or molybdenum, then the eddy current flows primarily through the latter and heats it up. Then, the steel is heated up indirectly by radiative and convective heat transfer from the hot cylinder. In this case, induction stirring of molten steel may be made negligibly small, and the bath would be a quiet one. Discussions of deoxidation kinetics as conducted in the laboratory may be subdivided into: 1. kinetics of deoxidation reaction 2. kinetics of elimination of deoxidation products from liquid steel

5.2.1

KINETICS

OF

DEOXIDATION REACTION

The kinetics of a deoxidation reaction consists of the following steps (or stages): 1. dissolution of deoxidizer into molten steel 2. chemical reaction between dissolved oxygen and deoxidizing element at phase boundary or homogeneously 3. nucleation of deoxidation product 4. growth of nuclei, principally by diffusion Rates of deoxidation reaction have been followed by many investigators13,19 by monitoring the change in the dissolved oxygen content of molten steel over time. Figure 5.9 shows dissolved [O] as well as total oxygen content of molten steel as a function of time for deoxidation by electrolytic ©2001 CRC Press LLC

FIGURE 5.9 Change of oxygen content of molten steel following the addition of manganese.19

manganese. It shows that the dissolved [O] decreases much more rapidly than the total oxygen. This behavior pattern has been found by all investigators. Turkdogan et al13 found deoxidation reaction with Si to be complete more or less within two minutes. Recent investigations by Kundu et al.20 and Patil et al.21 have demonstrated that Si-O-SiO2 equilibrium is attained within five minutes of the addition of ferrosilicon into an induction stirred laboratory melt. But it takes almost 20 minutes for the total oxygen content to achieve a steady state.21 Olette et al.19 carried out deoxidation by the addition of aluminum shots as well as by the injection of liquid aluminum (Figure 5.10) and found most of the reaction to be complete within one minute.

FIGURE 5.10 Effect of the method of introduction of aluminum on dissolved oxygen in molten steel.19


Scrutiny of Figures 5.9 and 5.10 grossly reveals two stages. Initially, the dissolved [O] decreases rapidly in first 30 s or so. This is followed by the second stage, which exhibits a much slower rate in the decrease of [O]. This behavior pattern has been explained resulting from the very high speed of three kinetic steps 2, 3, and 4 as listed above, which is responsible for the rapid initial rate, and reasonably high speed of step 1. Let us now try to present the supporting logic as well as experimental evidence for the above explanation. The reaction x[M] + y[O] = (MxOy)

(5.16)

would take place mostly either on the surface of the added deoxidizer or on the surfaces of (MxOy) particles. Therefore, it is primarily a phase-boundary reaction. No one has been able to determine the rate of the actual phase boundary reaction step (step 2). However, at high temperatures, it is mostly very fast and has been assumed to be so here. Homogeneous reactions, of course, are even faster. The mechanism of the dissolution of deoxidizer would depend on its melting point. The common deoxidizers, viz., ferromanganese, ferrosilicon and aluminum, all melt below 1500°C. Therefore, they melt and dissolve. The melting rate depends on the heat requirement as well as rate of heat transfer to the deoxidizer. Dissolution of ferromanganese is endothermic. On the other hand, dissolution of ferrosilicon is slightly exothermic, but its melting point is higher than that of ferromanganese. Therefore, on overall count, both would perhaps melt at about the same rate. Aluminum is expected to melt faster due to its much lower melting point. Guthrie22 has reviewed addition kinetics in steelmaking. As soon as a cold solid addition such as ferroalloy or aluminum is made, a layer of steel freezes around it and forms a solid crust. From then on, the mechanism of dissolution would depend on the melting point of the addition. If it is lower than that of steel, it may become molten, with the crust of solid steel intact as an extreme case. If the melting point of the addition is higher than that of steel, such as ferrotungsten, then the crust of steel will remelt, exposing the alloy to the melt and leading to its dissolution by simultaneous heat and mass transfer. The effect of the formation of a steel shell was illustrated through sample calculation for melting 10 cm dia. ferrosilicon sphere under some assumed conditions. Melting time was estimated as 1200 seconds if we consider the formation of a steel shell, but only 45 seconds if no shell is formed. Similarly, for Al, it was also the melting of the frozen shell that took most of the time. Factors that govern the rate of dissolution are bath hydrodynamics, density, melting point and thermal conductivity of the addition, size of the addition, and melt superheat. Of course, as stated earlier, if the addition is a deoxidizer, then the heat effect of the reaction is also important. Approximations point out a time of melting of at least 1 minute or so22 due to formation of the solid crust. Therefore, addition of liquid aluminum would enhance the rate of deoxidation as compared to that for solid. This is borne out by Figure 5.10. A detailed mathematical treatment is available.23,24 After melting, the dissolution of the deoxidizer requires its mixing and homogenization in the molten steel bath. This would depend on intensity of the fluid convection due to density differences (free convection) as well as stirring from other sources (such as induction stirring). However, the whole process of dissolution may be delayed if tenacious oxide films form around the dissolving deoxidant. Some observed slowness in the deoxidation reaction may be attributed to formation of stable oxide films formed on the interface of regions with a high content of deoxidizing agent and a high content of oxygen. Grethen and Phillippe25 have presented a photomicrograph of such a film of MnO · Al2O3 in a deoxidation system: Al-Fe-Mn. Deoxidation involves the formation of a new phase (i.e., the deoxidation product) as a result of Reaction (5.16). New phases form by what are known as processes of nucleation and growth. Nucleation refers to formation of a small embryo of the new phase that is capable of growth. Such ©2001 CRC Press LLC

an embryo (also called a critical nucleus) consists of a small number of molecules and has a dimension on the order of 10 Å or so. Again, two mechanisms of nucleation are possible: 1. homogeneous nucleation, which occurs in the matrix as such 2. heterogeneous nucleation, which occurs with the aid of a substrate According to the classical theory, the work required to form a spherical nucleus homogeneously is 4 3 ω = 4πrσ + --- πr ( ∆G ⁄ v ) 3 where

σ r ∆G v

= = = =

(5.38)

interfacial tension between liquid steel and deoxidation product radius of the nucleus change in free energy for Reaction (5.16) per mole molar volume of the deoxidation product (i.e., the new phase)

σ is positive, whereas ∆G is negative. This results in the type of ω vs. r curve shown in Figure 5.11. At r > r*, the nucleus grows spontaneously, and hence r* is the radius of the critical nucleus. * At r = r , 2σv dω * ------- = 0, and hence, r = – ---------∆G dr

(5.39)

Combining Eqs. (5.38) and (5.39), ω* = (16 πσ3v2)/3(∆G)2

(5.40)

The rate of formation of the nucleus in terms of number of critical nuclei per unit volume per second ( N˙ ) is –ω N˙ = Aexp  ------------  nk B T  *

(5.41)

where kB is Boltzmann’s constant (i.e., R/NO, where NO is Avogadro’s number), and n is the number of atoms in a critical nucleus.

FIGURE 5.11 Energy barrier for homogeneous nucleation.


N˙ increases if ω* decreases, which again happens if (∆G)2 increases [Eq. (5.40)]. ∆G, i.e., the free energy of reaction, would actually have to be negative for the deoxidation reaction to proceed in the forward direction. Therefore, an increase in (∆G)2 actually means that ∆G becomes more and more negative. Now, ( a M x Oy ) Q --------- = ------------------------------------------------y x K′ M [ hM ] [ ho ] ------------------------------------------------- ( a M x Oy )  -y   -----------------------x  [ h M ] [ h o ] equilibrium

(5.42)

From Eq. 2.8, Q o ∆G = ∆G + RT ln Q = – RT ln K′ M + RT ln Q = RT ln --------K′ M

(5.43)

For pure deoxidation product, in accordance with Eq. (5.18), we may write [[W M ] [W O] ] Q - equilibrium --------- = ----------------------------------x y K′ M [W M ] [W O] x

y

(5.44)

Since ∆G is negative, Q ⁄ K′ M < 1 . The inverse of Q ⁄ K′ M (i.e., K′ M ⁄ Q ) is known as the supersaturation ratio (X) and is larger than 1. The more negative ∆G is, the higher X is. Therefore, for higher rate of nucleation, the supersaturation is also going to be higher. Bogdandy et al.,26 Turpin and Elliott,27 and Turkdogan28 tried to estimate supersaturations required for a reasonable rate of nucleation for common deoxidation systems. Turpin and Elliott took N˙ = 1 3 nucleus/cm3/s as a reasonable rate, whereas Turkdogan took N˙ = 10 nuclei/cm3/s. However, it hardly matters, since the supersaturation changes very little with a change of N˙ , even by several orders of magnitude, due to the nature of the equations already presented. For estimation purposes, the value of A was taken as approximately 1027 cm–3 s–1, which is the maximum theoretical collision frequency. Values of interfacial tension, σ, were not known that precisely and therefore constituted a source of uncertainty. Values of this critical supersaturation as estimated by different workers ranged from 103 to 108 for strong deoxidizers (Al, Zr, and Ti), 500 to 4000 for manganese silicate, and 200 to 20,000 for silica. However, a reexamination of these calculations is called for. Such supersaturations are attainable in the initial stages of deoxidation by strong deoxidizers, but not so much with weak deoxidizers. According to Sano et al.,30 rapid homogeneous nucleation is possible during the initial stage of deoxidation even by Mn and Si. Moreover, in the melt, exogenous oxide particles are likely to be present in all cases. These particles would serve as substrates for heterogeneous nucleation, which is easier because less energy is required. Anyway, to sum up the situation, it has been concluded by various researchers that rapid nucleation of deoxidation product is possible when the deoxidizer is added. This has made the rapid initial decrease of dissolved oxygen content possible. However, as a result of reaction, the supersaturation in the melt also comes down drastically. Therefore, nucleation eventually ceases. Growth of deoxidation products occurs by a number of mechanism. However, growth by diffusion alone can contribute to the reaction and consequent lowering of dissolved oxygen. It has been analyzed by Turkdogan,28 Sano et al.,29 and Lindberg and Torsell.30 The essential conclusion ©2001 CRC Press LLC

is that growth by diffusion also is expected to be extremely rapid, taking barely a few seconds for completion. This is in view of very large number of nuclei formed—of the order of 105 to 107 nuclei/cm3 of melt. From the above discussions, it is evident that the deoxidation reaction accompanied by simultaneous nucleation and growth should be complete, the attainment of equilibrium is expected within a few seconds, and dissolution and homogenization of the deoxidizer are also instantaneous. This is not expected if dissolved oxygen decreases more slowly (Figures 5.9 and 5.10). Dissolution is perhaps more or less complete during the initial stage, but the mixing and homogenization, even in laboratory experiments, take a few minutes, and this seems to be the primary cause for a slow decrease in dissolved oxygen content during the second stage—although dissolution of the solid may have some role to play here as well (Figures 5.9 and 5.10). Example 5.6 Calculate the absolute maximum size of the deoxidation product as a result of the growth of critical nuclei by diffusion alone. Assume the deoxidation product to be silica and the number of critical nuclei (z) per cm3 to be 106. Initially, the melt contains 0.15 wt.% silicon and 0.03 wt.% oxygen. The temperature = 1800 K. Ignore all interaction coefficients. Solution The absolute maximum size would be obtained only if a very long time is allowed and the system attains equilibrium. For SiO2(s) = [Si]wt.% + 2[O]wt.% and a SiO2 = 1, K Si = [ W Si ] [ W O ] = 4.7 × 10 2

–6

(E6.1)

From reaction stoichiometry, 28 o o W Si – W Si = ------ ( W O – W O ) 32 o

(E6.2)

o

where W Si = 0.15 wt.% and W O = 0.03 wt.% or, WSi = 0.875 WO + 0.124

(E6.3)

Substituting WSi from Eq. (E6.1) into Eq. (E6.3) and solving by iteration, WO = 0.007 wt.% (Ans.). Material Balance for Oxygen Oxygen in 106 nuclei + residual oxygen in 1 cm3 of melt = initial oxygen in 1 cm3 of melt, i.e., 32 6 –2 –2 10 × V × ------ + 7.16 × 0.007 × 10 = 7.16 × 0.03 × 10 25

(E6.4)

where V is the volume of one particle of SiO2, 25 is the molar volume of SiO2 in cm3, and 7.16 is the density of liquid iron in gm cm–3. Solving Eq. (E6.4) for V and assuming the particle to be spherical, the radius of the particle is equal to 6.7 × 10–4 cm, i.e., 6.7 microns. Since growth by diffusion takes places for a short time, the actual radius will be less than this. ©2001 CRC Press LLC

5.2.2

KINETICS

OF

DEOXIDATION PRODUCT REMOVAL

FROM

MOLTEN STEEL

As Figure 5.9 shows, the total oxygen content of molten steel decreases more slowly than the dissolved oxygen content. This has been well established by several researchers. It may take 10 to 15 minutes, even in laboratory melts, to remove the total oxygen adequately. This behavior pattern demonstrates that the removal of deoxidation products from the melt is a slow process and is really the most important kinetic step controlling steel cleanliness. Growth by diffusion is expected to be complete essentially in seconds. Sample calculations31 demonstrate that the deoxidation products can assume a size of 1 to 2 microns at best. This is because there are too many nuclei in the melt and, hence, each one has limited growth. In contrast, microscopic observations of solidified steel reveal presence of a large number of inclusions of a size even above 50 microns. Therefore, other mechanisms of growth play an important role. The kinetics of removing deoxidation products from molten steel consists of the following steps: 1. growth 2. movement through molten steel to the surface or crucible wall 3. floating out to the surface or adhesion to the crucible wall Sano et al.29 and Lindberg and Torsell30 carried out fundamental investigations with laboratory melts. Out of these, the latter have received wide acceptance because their theoretical analyses were supported by inclusion counting and size analysis. In addition to diffusion, they considered the following additional mechanisms of growth. Ostwald Ripening (i.e., Diffusion-Coalescence) According to this mechanism, larger particles of deoxidation product grow at the cost of smaller ones. However, this mechanism does not make any significant contribution to the growth of deoxidation product. Stokes Collision In a quiet fluid and at low Reynold’s number (i.e., laminar flow), a spherical particle of solid, at steady state, moves according to the Stokes’ Law of Settling, and its terminal velocity (v) is given as gd ( ρ s – ρ f ) u t = ----------------------------18µ 2

(5.45)

where g is acceleration due to gravity, d is diameter of the particle, µ is viscosity of the fluid, ρs and ρf are densities of solid and fluid, respectively. This equation may be applied even to the motion of gas bubbles and liquid droplets, provided that these are small in size (less than a millimeter or so). Since deoxidation products are lighter than molten steel, they move upward. Equation (5.45) 2 shows that u t ∝ d , other factors remaining constant. Therefore, particles of different sizes would move at different speeds. During this process, many of them are likely to collide with one another. Lindberg and Torsell30 assumed that they would coalesce and form one particle as soon as they collide. This is the mechanism of growth by Stokes collision. Gradient Collision Suppose the melt is not quiet, and there is some stirring and consequent turbulent flow, and random motion of eddies. The the minimum size of such an eddy under conditions of Torsell’s experiments was estimated as 300 microns.30 Since inclusion sizes were much smaller than this, it was assumed ©2001 CRC Press LLC

that any oxide particle would move along with the eddy in which it is contained. Since different eddies have different velocities both in magnitude and direction, they would continuously collide, enhancing the chances of the collision of deoxidation products and leading to their coalescence and growth. Example 5.7 Liquid steel is being deoxidized by the addition of ferrosilicon at 1600°C. The deoxidation product is globular silica. Calculate the time required for particles of 5 and 50 microns diameter to float up through a depth of 10 cm and 2 m. Given Densities of liquid steel and silica are 7.16 × 103 and 2.2 × 103 kg m–3, respectively. The viscosity of liquid steel = 6.1 × 10–3 kg m–1 s–1, g = 9.81 m s–2. Solution Assume that the particles have a steady, terminal velocity given by Stokes law from the beginning. Then, 18µ H Depth of steel ( H ) Time to float up ( t ) = ----------------------------------------------- = ----------------------- × -----2 g ( ρl – ρs ) d ut

(E7.1)

If d is expressed in microns (10–6 m) and time in minutes, then 18 × 6.1 × 10 × 10 H H 4 - × -----2 = 3.6 × 10 × -----2 min t = -------------------------------------------------------------------------3 9.81 × ( 7.16 – 2.22 ) × 10 × 60 d d –3

12

(E7.2)

Calculations yield: H (in m) = 0.1 d (in microns) = 5 t (in min)

0.1 50

2 5

= 150 1.5 3 × 10

2 50 3

30

These theoretical expectations were confirmed by laboratory experiments30 as shown by Figure 5.12 for silicon deoxidation. Very little oxygen (total) is removed in the first stage, because the particles are small, and they do not float out rapidly. Then, particles grow rapidly due to collisions and start floating out, giving rise to rapid oxygen removal in the second stage. Since most large particles float out at this stage, further flotation and removal in the third stage is slow. Based on their work, Lindborg and Torsell30 proposed a schematic diagram of average particle radius vs. time for deoxidation by silicon (Figure 5.13). The decrease in average size after the peak is due to preferential removal of larger particles by flotation. It has been well established that stirring helps in the removal of deoxidation products.19,29,31 Stirring contributes to faster growth of oxide particles and hence helps in the removal of deoxidation product. Some researchers are of the view that a recirculatory motion of bath, such as in induction stirring, actually makes the floating out of inclusions difficult,25 but Miyashita et al.,31 on the other hand, have shown that, even in induction stirring, the rate of the floating out of inclusions is much higher than predicted by Stokes’ law ( Figure 5.14). Slow flotation of smaller inclusions (0

Hcr,v Vθ,I = 0 Hcr,d d

d drainage nozzle

drainage nozzle

primary liquid

primary liquid nozzle ouflow (a)

nozzle outflow (b)

FIGURE 5.22 (a) Vortexing funnel and (b) draining funnel during the pouring of liquid through a bottom nozzle.45 Reprinted by permission of Iron & Steel Society, Warrendale, PA.

The critical limiting height (HCr), below which such a funnel would reach down to the drainage opening, is of technological importance. In industrial processes, rotational (i.e., tangential) flow is very likely to be present and, hence, HCr for a vortexing funnel (HCr,v) is of importance. The findings may be stated as H Cr ,v Q n ----------- =  -------------------5 1⁄2  D ( gD ) 

(5.55)

where D = vessel diameter Q = discharge flow rate g = acceleration due to gravity The scatter of the data did not allow further quantification, but n was approximately 2. HCr,v /D ranged between 1 and 50 in their experiments. For a draining funnel, data of various investigators showed wide discrepancies. However, HCr,d /D was found to be in a range from 0.1 to 2. Hence, HCr,d constitutes the lower limit and HCr,v the upper limit of drainage height. In further studies, Shankarnarayan and Guthrie47 refined the correlation of Eq. (5.55) further. Since vortexing funnels are undesirable in terms of the cleanliness of steel, they designed “vortex busters” both for a water model and plant studies and carried out experiments. The object was to prevent vortexing. They have reported a significant decrease of HCr,V in the water model as well as improved steel cleanliness in plant studies by use of these vortex busters. Steffen48 has reported studies on flow phenomena related to slag carryover in water models as well as in 300t basic oxygen furnace (BOF) vessel. In ladles, drain sink occurred if the volume flow of the open channel at the ladle bottom was less than the corresponding out let capacity. Critical liquid height was not significantly dependent on the location of the bottom nozzle (centric or eccentric). Also, HCr was found to be proportional to the nozzle diameter (dn). Water model investigations by Mazumdar et al.49 demonstrated a significant decrease in entrainment of the upper liquid phase if its viscosity was larger. Increased waiting time after the blow ©2001 CRC Press LLC

from 15 to 180 s also lowered the drainage of the upper liquid phase by a factor of 4 to 5. This can be explained as due to the decay of vortexing flow. Dubke and Schwerdtfeger50 dropped balls of various sizes and densities into the draining liquid. Entrainment of the lighter liquid was found to decrease by a factor of 4 to 5 due to the presence of the ball. A ball density that allows partial immersion of the ball into the heavier liquid was found to be very effective. The ball settles into the mouth of the vortexing funnel and thus prevents the lighter liquid from flowing out. The studies on entrainment phenomena have enabled industries to control slag carryover during tapping with varying degrees of success. The devices are plugs of various shapes, e.g., as nail shaped. Fruehan51 has briefly discussed about devices for slag-free tapping. In electric arc furnaces, eccentric bottom tapping allows considerable reduction in the quantity of carryover slag. The use of slide gates decreases vortexing as compared to stoppers. As far as the BOF is concerned, the plugs are composites of refractory materials and iron, so the density is in between that of slag and of metal. If everything works right, the plug falls into the tap hole and blocks it before the slag flows through it. A teapot-type spout for tapping should be quite effective, too. Another device that is useful in reducing the amount of slag carryover is an electromagnetic sensor, which is placed around the tap hole. When slag starts coming out through the hole, the nature of the signal changes significantly due to differences in electromagnetic induction for slag and metal. The device is superior to visual detection. Poferl and Eysn52 have reported on plant trials with 130/140t converters at Linz and obtained the following average slag rates (kg/tonne steel): Without slag stopper

10–15

With slag stopper

4.45

With slag stopper and slag indication system

3.5

Besides the use of a slag stopper in a BOF, further deslagging by slag raking seems to be practiced in some modern plants. During this even electromagnetic stirring is being employed to push the BOF slag towards the front of the ladle for quick and efficient removal. In secondary refining, a slag of CaO-Al2O3-SiO2-CaF2 with a high content of CaO and substantial percentage of Al2O3 is aimed at for better deoxidation, desulfurization, etc., as well as for controlling dissolved Al in molten steel. This requires the addition of CaO, CaF2, etc., and deoxidizers to modify the carryover slag during tapping of steel as well as during the subsequent stage. The average slag composition in a ladle furnace after arc reheating is 50 to 56% CaO, 7 to 9% MgO, 6 to 12% SiO2, 20 to 25% Al2O3, 1 to 2%(FeO+MnO), and 0.3% TiO2, with small amounts of S and P.45 An already prepared synthetic slag speeds up refining. The concept is not new. In the Perrin process of early 1930s, steel was tapped onto a molten calcium (magnesium) aluminosilicate slag placed at the ladle bottom. Lime-alumina–based premelted synthetic slags are on the market now. Turkdogan44 has reported the findings of some plant experiments in which, during some tappings, only ferromanganese was added to the ladle, whereas for some others ferromanganese and calcium aluminate slag were added. Figure 5.23 presents the findings. Si and Al were less than 0.003% each. The figure demonstrates significant improvement in deoxidation upon use of slag. Of course, the extent of improvement depends on the steel grade. For Si-Mn killed or Al-killed steels, the difference would be less. Section 5.1.2 has already provided the necessary thermodynamic background for this. Turkdogan has also tried to show that reactions approached thermodynamic equilibria approximately. This was attributed to the strong stirring and mixing action of the tapping stream. These agree with the findings by Choudhary and Ghosh8 in EAF, reported earlier in this section. Section 5.2.1 has already presented a brief discussion on melting-cum-dissolution of ferroalloys, aluminum, and other alloying additions. In industrial processing, it is desirable that the solid additions remain submerged in liquid steel long enough for completion of melting-cum-dissolution. ©2001 CRC Press LLC

FIGURE 5.23 Deoxidation by ferromanganese with and without calcium aluminate ladle slag in furnace taping of low-Si, low-Al steel.44

Heavy ferroalloys, such as ferrotungsten, shall remain submerged. However, additions of lower density, such as Al, ferromanganese, etc., would tend to float up and react with slag and atmospheric oxygen. This problem is especially serious with aluminum. Guthrie22 has reviewed the subject and reported findings based on mathematical modeling, cold modeling, and actual plant trials. There have been subsequent mathematical exercises on the same subject, as well as water model experiments.53 Since additions are made primarily during the tapping of steel, the same situation was simulated. Attention was paid to the subsurface trajectories of buoyant (i.e., lighter than liquid bath) additions, and total immersion times were found to range between 0.1 and 40 s. Some guidelines for optimum particle size, location, and timing of the addition have been proposed. However, plant trials are a must since, besides the flow hydrodynamics caused by the tapping stream, there are several variables affecting alloy recovery, as discussed earlier. Moreover, significant improvements in the utilization of deoxidizers during tapping would be possible only with major investments. Baldzicki et al.54 have reported significant improvement in Al recovery by wire injection in Al-killed grades. Rapid mixing and homogenization of the bath, as well as metal-slag mass transfer, are very desirable for success. During tapping, stirring is effected by the tapping stream. Some limited control of the process is possible. During argon purging, the flow rate of argon is a key variable. Chapter 3 presented details of fundamental studies on fluid flow in gas-stirred ladles. Chapter 4 discussed mixing and mass transfer in gas-stirred ladles. These are being utilized for industrial process control, design, and optimization to some extent. Figure 4.4 illustrates that, for a ladle with a single porous plug, minimum mixing time is obtained if the plug is located at mid-radius. Accordingly, in smaller ladles with single plugs, the plug location is at mid-radius. For dual plugs, the two plugs at mid-radius are located in diametric opposition. Zhu et al.55 have reported their findings from water model and mathematical model work for mixing in a gas-purged ladle with multiple tuyeres at the bottom, and they arrived at the same conclusion. Nippon Steel Corporation of Japan has developed the CAS (composition adjustments by sealed argon bubbling) process. With an additional facility for oxygen blowing, it is known as CAS-OB and seems to be catching up as alternative to ladle furnaces. The features of CAS-OB have been presented by S. Audebert et al.56 Figure 5.24 illustrates the features of the process. Since it does not involve elaborate top and arc heating arrangements as does LF, capital cost is low. The high©2001 CRC Press LLC

5. O2 blowing into snorkel 1. Simple equipment & operation

Fume extraction O2 blowing Alloying C, Si, Mn, Al etc.

2. Alloy addition Ar atmosphere

á áá

Raising temperature of steel Low running cost and investment Quick treating and low temperature drop

áá á

Low reoxidation High yield Precise composition adjustment

áá á

Effective removal of oxides No disturbance of slag layer Quick homogeneity

3. Slag off for prevention of slagmetal reaction

Slag Molten steel

Snorkel 4. Ar stirring into snorkel

Ladle Porous plug Ar N2

} gas

FIGURE 5.24 Features of the CAS-OB process.56

alumina refractory snorkel had a life of about 60 heats. Deoxidizers and other alloying elements are added inside the snorkel during argon purging. This significantly prevents their oxidation loss due to reaction with atmosphere and slag. As mentioned in Section 4.2, stirring is most intense in the plume region, leading to faster melting and dissolution of additions. Temperature adjustment, as required, is done by feeding aluminum wire and oxygen blowing inside the snorkel. The oxidation of Al provides the necessary heat input. This way, heating is more rapid (7 C/min or so) and cheaper as well. However, the control of residual dissolved aluminum calls for good process control measures. Nilsson et al.57 have reported the performance of CAS-OB, commissioned at the SSAB Tunnplat AB, Lulea Works in 1993. They have claimed a reduction in total production cost as well as improvement of product quality. Section 5.1 has briefly discussed the principles and importance of an immersion oxygen sensor, which measures dissolved oxygen content of molten steel. The use of this to monitor bath oxygen levels in steelmaking furnaces and ladles is a must for scientific purposes and improved deoxidation control. Many investigators have reported extensive trials and the results achieved. For example, Anderson and Zimmerman58 have reported findings of trials at Republic Steel’s Warren BOF shop. The [Wc] · [Wo] product, before tapping of the BOF, was determined to be 0.0029, in comparison to 0.0020, the equilibrium deoxidation constant at 1593°C. Maximum vessel oxygen was found to provide a superior control criterion of the bath oxygen level rather than the traditional one, i.e., minimum tap carbon.58 This vessel oxygen level was also employed for BOF charge calculation, deoxidizer additions in ladles, and decisions about the deoxidation schedule. Significant improvements were found, including improved rimming action, improved yield of semi-killed grades, and an improved aluminum addition schedule. However, a lack of reproducibility in aluminum recovery was still a major problem.

REFERENCES 1. Steelmaking Data Sourcebook: The Japan Soc. for The Promotion of Science, The 19th Committee on Steelmaking (revised ed.), Gordon and Breach Science Publishers, New York, 1988. 2. Subbarao, E.C., ed. Solid Electrolytes and Their Applications, Plenum Press, New York, 1980. 3. Fruehan, R.J., Martonik, L.J., and Turkdogan, E.T., Trans. AIME, 245, 1969, p. 1501. ©2001 CRC Press LLC

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

Saeki, T., Nisugi, T., Ishikura, K., Igaki, Y., and Hiromoto, T., Trans. ISIJ, 18, 1978, p. 501. Plushkell, W., Stahl ûnd Eisen, 96, 1976, p. 657. Nilles, P., Defays, J., Cure, O., and Surinx, H., CRM Report, No. 51, Oct. 1977, p. 31. Jacquemot, A., Gatellier, C., and Olette, M., IRSIDE RE 289, 1975, also ref.19, p. 50. Ghosh, A. and Chaudhary, P.N., Trans. IIM, 38, 1985, p. 31. Ghosh, A. and Murty, G.V.R., Trans. ISIJ, 26, 1986, p. 629. Qiyong Han, Proc. 6th Int. Iron and Steel Cong., Nagoya, Vol. 1, 1990, p. 166. Kimura, T. and Suito, H., Metall. Trans. B., 25B, 1994, p. 33. Verein Deutscher Eisenhuttenleute, Slag Atlas, Verlag Sthalisen mbH, 1981. Turkdogan, E.T., in Chemical Metallurgy of Iron and Steel, Iron and Steel Inst., London, 1973, p. 153. Bagaria, A.K., Brahma, Deo, and Ghosh, A., Proc. Int. Symp. Modern Developments in Steelmaking, Chatterjee, Amit and Singh, B.N., ed., Jamshedpur, 1981, 8.1.1. Ghosh, A. and Naik, V., Tool and Alloy Steels, 17, 1983, p. 239. Wilson, W.G., Kay, D.A.R., and Vahed, A., J. Metals, 26, 1974, p. 14. Faulring, G.M., and Ramalingam, S., Metall. Trans. B, 11B, 1980, p. 125. .Plockinger, E. and Wahlster, M., Stahl Eisen, 80, 1960, p. 659. Olette, M. and Gatellier, C., in Information Symposium on Casting and Solidification of Steel, IPC Science and Technology Press Ltd., Guildford, U.K., 1977, p. 8. Kundu, A.L., Gupt, K.M., and Krishna Rao, P., Ironmaking and Steelmaking, 13, 1986, p. 9. Patil, B.V. and Pal, U.B., Metall. Trans. B, 18B, 1987, p. 583. Guthrie, R.I.L., Electric Furnace Proceedings, AIME, 1977, p. 30. Engh, T.A., Principles of Metal Refining, Oxford University Press, Oxford, 1992, Ch. 8. Argyropoulos, S.A. and Guthrie, R.I.L., in Heat and Mass Transfer in Metallurgical Systems, ed. Spalding, D.B. and Afgan, N.H., Hemisphere Publishing Corp., London, 1981, p. 159. Grethen, E. and Phillippe, L., in Production and Application of Clean Steels, Iron and Steel Inst., London, 1972. Von Bogdandy, L., et al., Arch. Eisenhuttenleute, 32, 1961, p. 451. Turpin, M.L. and Elliott, J.F., JISI, 204, 1966, p. 217. Turkdogan, E.T., JISI, 204, 1966, p. 14. Sano, N., Shiomi, S., and Matsushita, Y., Trans. ISIJ, 7, 1967, p. 244. Lindborg, U. and Torsell, K., Trans. AIME, 242, 1968, p. 94. Miyashita, Y. et al., in Proceedings 2nd Japan-USSR Joint Symposium on Physical Chemistry of Metallurgical Processes, Iron and Steel Inst., Tokyo, 1969, p. 101. Hirasawa, M., Okumura, K., Sano, M., and Mori, K., in Ref. 10, Vol. 3, p. 568. Mori, K., Hirasawa, M., Shinkai, M., and Hatamaka, A., Tetsu-to-Hagane, 71, 1985, p. 1110. Nogi, K. and Ogino, K., Canad Met. Qtly., 22, No. 1, 1983, p. 19. Kozakevitch, P. and Olette, M., in Production and Application of Clean Steels, Iron and Steel Inst., London, 1972, p. 42. Singh, S.N., Metall. Trans., 2, 1971, p. 3248. Lindborg, U., in Ref. 19, Vol. 2, 85. Bziva, K.P. and Averin, V.V., in Proceedings 2nd Japan-USSR Joint Symposium on Physical Chemistry of Metallurgical Processes, Iron and Steel Inst., Tokyo, 1969, p. 113. Nakanishi, K. et al., in Proceedings 2nd Japan-USSR Joint Symposium on Physical Chemistry of Metallurgical Processes, Iron and Steel Inst., Tokyo, 1969, p. 50. Suzuki, K., Kitamura, K., Takenouchi, T., Funazaki, M., and Iwanami, Y., Ironmaking and Steelmaking, 1982, p. 33. Lehner, T., Canad Met. Qtly., 20, No. 1, 1981, p. 163. Kim, S.H., Fruehan, R.J. and Guthrie, R.I.L., Steelmaking Proceedings, Iron and Steel Soc., USA, 1987, p. 107. Kikuchi, Y., et al., in Ref.10, Vol. 3, p. 532. Turkdogan, E.T., Ironmaking and Steelmaking, 15, 1988, p. 311. Shankarnarayanan, R. and Guthrie, R.I.L., Steelmaking Proceedings, Iron and Steel Soc., USA, 1992, p. 655. Steffen, R., in Int. Conf. Secondary Metallurgy (English preprints), Verein Deutscher Eisenhuttenleute, Verlag Stahleisen mBH, Dusseldorf, 1987, p. 97.


47. Mazumdar, S., Pradhan, N., Bhor, P.K., and Jagannathan, K.P., ISIJ Int., 35, 1995, p.92. 48. Dubke, M. and Schwerdtfeger, K., Ironmaking and Steelmaking, 15, 1990, p. 184. 49. Fruehan, R.J., Ladle Metallurgy Principle and Practices, Iron and Steel Soc., Warrendale, PA, USA, 1985, p. 655. 50. Poferl, G. and Eysn, M., in Ref. 46, p. 137. 51. Tanaka, M., Mazumdar, D. and Guthrie, R.I.L., Metall. Trans. B, 24B, 1993, p. 639. 52. Baldzicki, E.J., Tomazin, C.E. and Turacy, D.L., in Ref. 48, p. 129. 53. Zhu, M.Y., Inomoto, T., Sawada, I., and Hsiao, T.C., ISIJ Int., 35, 1995, p. 472. 54. Audebert, S., Gugliarmina, P., Reboul, J.P., and Sauermann, M., MPT, 1989, p. 26. 55. Anderson, E.D. and Zimmerman, E., in Ref. 48, p. 79.


6

Degassing and Decarburization of Liquid Steel

6.1 INTRODUCTION As stated in Chapter 1, the degassing of steel melts by subjecting them to vacuum treatment was introduced in the decade of the 1950s. The primary objective was to lower the hydrogen content in forging quality steels. The gases—hydrogen, nitrogen, and oxygen—dissolve as atomic H, N, and O, respectively, in molten steel. However, their solubilities in solid steel are very low. Chapter 5 has already dealt with oxygen. Solubilities of H and N in pure iron at 1 atm pressure of the respective gases are shown in Figure 6.1 to demonstrate this point. When liquid steel is solidified, excess nitrogen may form stable nitrides such as nitrides of aluminum, silicon, and chromium. However, hydrides are thermodynamically unstable. Therefore, the excess hydrogen in solid steel tends to form H2 gas in pores and also diffuses out to the atmosphere. H has a very high diffusivity even in solid steel due to its low atomic mass. In relatively thin sections, such as those manufactured by rolling, diffusion is fairly rapid. Hence, excess hydrogen is less, reducing the tendency toward development of high gas pressure in pinholes. Also, the bulk of the rolled products in the ingot route belong to rimming and semi-killed grades. Here, liquid steel tends to contain less hydrogen due to a flushing action by the evolution of CO gas. However, diffusion is not that efficient in forgings, due to their large sizes. Moreover, liquid steel in forging grades contains more hydrogen too, since these are killed steels. As a consequence

FIGURE 6.1 Effect of temperature on the solubilities of nitrogen and hydrogen in iron at 1 atm pressure for each gas.


of this, H rejected by solid steel accumulates in blowholes and pinholes. The gas pressure developed inside the latter tends to be high. During forging, the combination of hot working stresses and high gas pressure in pinholes near the surface tends to cause fine cracks in the surface region. Efforts to avoid these cracks led to commercial development of vacuum degassing processes. Hydrogen also causes a loss of ductility of steel. Hence, low H is a necessity for superior grades of steel with high strength and impact resistance. These considerations have led to hydrogen consciousness in rolled products as well for several grades of steel. The need to control the oxygen content of steel melt and deoxidation has been discussed in Chapter 5. The use of deoxidizers leads to the formation of deoxidation products affecting the cleanliness of steel. Vacuum treatment of liquid steel promotes a carbon-oxygen reaction and removal of oxygen as CO. This is clean deoxidation. Recognizing this, steelmakers also made deoxidation a target of vacuum treatment. This simultaneously lowers carbon as well and constitutes the basis for vacuum decarburization. Nitrogen affects toughness and aging characteristics of steel as well as enhancing the tendency toward stress corrosion cracking. Nitrogen is by and large considered to be harmful for properties of steel. Its strain hardening effect does not allow extensive cold working without intermittent annealing. Low nitrogen is essential for deep drawing quality steel. Very low nitrogen levels have become extremely important for ultra-low carbon, cold rolled steels with high formability for the automotive industry, subjected to continuous annealing. However, it is worth mentioning that there are applications where nitrogen has beneficial effects on the properties of steel.1 The grain refinement action of fine precipitates of aluminum nitride (AlN), and consequent beneficial effects on properties, have been known for a long time. Solid solution hardening and precipitation strengthening effects are utilized in high-strength steels. Nitrogen additions are also particularly beneficial for stability and pitting resistance of austenitic stainless steel grades. Precipitates of nitrides or carbonitrides of several alloying elements, such as aluminum, boron, chromium, niobium, etc., have been reported.2 Hydrogen is picked up by the steel melt during primary steelmaking from moisture and water associated with raw materials and atmosphere. Nitrogen, of course, is picked up from the air. Steelmakers endeavor to lower the extent of such pickup as well as to flush out these gases from the melt using various strategies. All of this is beyond the scope of discussion here, since we are concerned with secondary steelmaking. However, in this connection, it may be mentioned that nitrogen is to be largely controlled in the primary steelmaking and tapping stage. In Chapter 8, there is a discussion of this topic in connection with gas absorption from the atmosphere during tapping and teeming. As discussed in Section 5.1, total oxygen in steel is determined by inert gas fusion apparatus. A similar method is employed for determining the nitrogen and hydrogen content of steel. The sample is melted in graphite crucible under a flow of pure argon. N and H evolve as N2 and H2 in the gas stream, whose total quantity is determined spectroscopically. More recently, emission spectrometers have come on the market and are in wide use for the analysis of nitrogen and other alloying elements, as for oxygen in steel. Peerless and Clay2 have reported development of the “Nitris” system by Heraus Electro-Nite. It has been derived directly from the Hydris technique for the determination of hydrogen in liquid steel. It employs a disposable immersion lance. Through some pumping arrangement, the gas in equilibrium with the melt is collected. The partial pressures of N2 or H2 directly give values of the nitrogen or hydrogen contents of molten steel. This method does not require the collection of solid sample and handling of the same for subsequent analysis, and hence it is more rapid.

6.1.1

VACUUM DEGASSING PROCESSES

Review articles and monographs have been published on general features.3,4,5 Vacuum degassing processes are traditionally classified into the following categories: ©2001 CRC Press LLC

1. ladle degassing processes 2. stream degassing processes 3. circulation degassing processes (DH and RH) As stated in Chapter 1, an additional temperature drop of 20 to 40°C occurs during secondary processing of liquid steel. Temperature control is very important for proper casting, especially continuous casting. Therefore, provisions for heating and temperature adjustment during secondary steelmaking are very desirable. In vacuum processing, a successful commercial development in the decade of the 1960s was vacuum arc degassing (VAD), where arc heating is undertaken. Provision for heating is provided in an RH degasser as well. Stainless steels contain a high percentage of chromium. A cheap source of Cr is high-carbon ferrochrome. However, its addition raises the carbon content of the melt to about 1%, which is to be lowered to less than 0.03% in subsequent processing. Oxygen lancing has already been found to promote C–O reaction in preference to Cr–O reaction, and it has been practiced commercially. The use of a vacuum is of further help and led to the development of vacuum-oxygen decarburization (VOD) process for stainless steels in the decade of the 1960s. Some oxygen blowing is nowadays resorted to in vacuum degassers for the production of ultra-low carbon steels as well. The RH-OB process is an example. In vacuum degassing, the total pressure in the chamber is lowered, whereas, in degassing by argon purging, the total pressure above the melt is essentially atmospheric. Even then, degassing is effected. This is because partial pressures of H2, N2, and CO are essentially zero in the incoming argon gas. Therefore, degassing by bubbling argon through the melt without vacuum is possible in principle. But consumption and cost of argon would be high, and the processing time would be lengthy. Hence, it is not practiced for ordinary steels. However, decarburization of stainless steel melts by the argon-oxygen decarburization (AOD) process is still popular. Besides degassing, modern vacuum degassers are used to carry out various other functions such as desulfurization, decarburization, heating, alloying, and homogenization, thereby achieving more cleanliness as well as inclusion modification. Adaptation of vacuum processes to produce ultra-low carbon steels is an important development direction. It is to be recognized that not all of these functions are equally important. A plant’s management has to fix its targets and accordingly has to decide priorities. These in turn dictate the choice of process, facilities required, and operating practices. Some broad guidelines are noted below3. 1. The treatment time in vacuum degassing should be short enough to logistically match with the converter steelmaking on the one hand and continuous casting on the other. This is one of the challenges. Higher pumping capacity for the vacuum systems is a prerequisite. For a modern 200t VD unit, a capacity higher than 500 kg of air/hr at 1 torr is common (1 torr = 1 mm Hg). 2. Injection of argon below the melt is a must for good homogenization, mass transfer, and inclusion removal. Design and location of tuyeres for such injection play an important role toward achievement of the targeted goals. Some plants have even adopted powder blowing with the gas for desulfurization, as in injection metallurgy. 3. In early vacuum degassers, deoxidation by carbon was one of the objectives. Nowadays, it is carried out principally by deoxidizers such as ferrosilicon, aluminum, and calcium silicide, either in the ladle prior to degassing or in the VD unit itself during degassing. The carbon-oxygen reaction is promoted in vacuum degassing either for deep decarburization in ultra-low carbon steels, for enhancing rates of removal of nitrogen and hydrogen, or for both. 4. The carryover slag from a steelmaking converter poses problems during secondary steelmaking and has to be considered. Its modification by additions such as deoxidizers and CaO is to be included in the refining program for achieving defined objectives. ©2001 CRC Press LLC

Figure 6.2 shows the various degassing processes schematically. Sales trends for the period 1981–91 are presented in Figure 6.3, showing the dominant processes in the international market during that period. The situation has not changed. One dominant process is RH (Ruhrstahl Heraus) and its variants, such as RH-OB. These come under the category of circulation degassing processes. Another dominant process is vacuum degassing in the ladle (VD), and its variants, VAD, VOD, etc.

FIGURE 6.2 Some degassing processes.

FIGURE 6.3 Share of various degassing processes in the world market, 1981–1991. Courtesy of Messo Metallurgie.


Figure 6.4 shows the RH process schematically. Molten steel is contained in the ladle. The two legs of the vacuum chamber (known as snorkels) are immersed into the melt. Argon is injected into the upleg. Rising argon bubbles have a pumping action and lift the liquid into the vacuum chamber, where it is degassed and comes down through the downleg snorkel. The entire vacuum chamber is refractory lined. There is provision for argon injection from the bottom, heating, alloy addition, sampling, and sighting of the interior of the vacuum chamber. Figure 6.5 shows the VAD process schematically. Heating is by arc with graphite electrodes, as in an electric arc furnace (EAF). Heating, degassing, slag treatment, and alloy adjustment are done without interruption of the vacuum. Even in late 19th century, vacuum treatment of steel melt was advocated. A major constraint was the availability of large-capacity industrial vacuum generating systems. Comprehensive discussions on this subject are available in the early publications on vacuum metallurgy.6 Figure 6.6 shows a typical system. Mechanical booster pumps remove the bulk of the air and gas. However, they are not capable of lowering vacuum chamber pressure to as low as approximately 1 torr (1 mm Hg). This is achieved by the use of steam ejector pumps in conjunction with mechanical pumps.

1 2 3 4

5 6

7

8 9 10

11

12 13

14

FIGURE 6.4 RH degasser. Courtesy of Messo Metallurgie.


1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Vacuum connection Television camera Sightglass with rotor Alloying hopper Alloying feeder Heating transformer Graphite rod vacuum vessel Upper part Middle part Lower part Lifting gas connection Upleg snorkel Downleg snorkel Teeming ladle

1 2 3 4 5 6 7

1. Temperature and sampling lance 2. Telescopic tubes for vacuum-tight electrode sealing 3. Bus tube supporting arms 4. Secondary bus 5. Water-cooled flexible highcurrent cable 6. Electrode tensioning device 7. Vacuum hopper for alloying agents 8. Guide column for electrode control 9. Sight glass with rotor 10. Sampling valve and hopper 11. Heat shield 12. Vacuum connection 13. Vacuum treatment vessel 14. Teeming ladle with steel 15. Porous inert gas bubbler 16. Diaphragm for steel outlet at ladle breakout

8

9

10 11 12

13 14 15

16

FIGURE 6.5 VAD unit. Courtesy of Messo Metallurgie.

Ejector pumps work on the principle of the diffusion pump. A jet of steam issues through a nozzle at high velocity and drags surrounding gas along with it (known as entrainment). Dusts coming out of the vacuum chamber, including condensed particles of volatile matters, settle down with condensed steam and are removed as sludge from time to time.

6.2 THERMODYNAMICS OF REACTIONS IN VACUUM DEGASSING 6.2.1

PRINCIPAL REACTIONS

Chapter 2 reviewed the basics of metallurgical thermodynamics relevant to secondary refining of liquid steel. Appendix 2.1 through 2.4 presented tables for important thermochemical data. Table 6.1, therefore, contains equilibrium data pertaining to the principal degassing reactions only, viz., ©2001 CRC Press LLC

FIGURE 6.6 Vacuum generating system.

1 [ H ] = --- H 2 ( g ) 2

(6.1)

1 [ N ] = --- N 2 ( g ) 2

(6.2)

[ C ] + [ O ] = CO ( g )

(6.3)

TABLE 6.1 Equilibrium Relations of Degassing Reactions7 SL. No.

Reaction

Equilibrium relation

1.

1 [ H ] = --- H 2 ( g ) 2

[ h H ] = K H ⋅ p H2

2.

1 [ N ] = --- N 2 ( g ) 2

3.

[ C ] + [ O ] = CO ( g )

Unit of h

K vs. T relation

ppm

1905 logK H = – ------------ + 2.409 T

0.77

[ h N ] = K N ⋅ p H2

ppm

518 logK N = – --------- + 2.937 T

14.1

[ h C ] [ h O ] = K CO ⋅ p CO

wt.pct

1160 logK CO =  – ------------ – 2.00  T 

4.7 × 10−4∗

ppm

1160 logK CO =  – ------------ – 6.00  T 

0.47∗

1⁄2

1⁄2

At hC – 0.05 wt.%, i.e. 500 ppm. Note: 1 mm Hg = 1 torr = 1.315 × 10–3 atm.

*


Values of h at 1600°C and 1 mm Hg

In Table 6.1, T is temperature in Kelvins, h denotes activity of solute dissolved in molten steel, and p is partial pressure of the concerned gas in atmosphere. In the binary iron alloys Fe-H, Fe-N, h may be taken as equal to concentration of H and N, respectively, in parts per million. This is because concentrations of H and N are small and lie in the Henry’s law region. Hence, the activity coefficient (fi) may be taken as 1 (Section 2.6). This approximation is quite valid for ordinary lowcarbon and even microalloyed steels. However, the influence of alloying elements on hH and hN would be significant for high-carbon and high-alloy steels, and solute–solute interactions are to be taken into account. Calculations have already been illustrated in the solved Example 2.4 in Chapter 2. The above comments are applicable for carbon-oxygen reaction also. However, in this case, some departures from Henrian behavior are possible even in a simple Fe-C-O ternary melt, depending on the concentrations of carbon and oxygen. Dissolved oxygen cannot be removed from the melt as gaseous O2. A sample calculation on –3 the basis of Eqs. (5.2) and (5.3), and assuming that hO = WO, shows that, at p O2 = 10 atmosphere and 1600°C, WO is 26 wt.%. This demonstrates the impossibility of the removal of dissolved oxygen as O2. Table 6.1 also shows that it is possible to obtain very low and completely satisfactory levels of H, N, and O in the melt from a thermodynamic point of view. The C–O reaction constitutes the basis for vacuum decarburization as well. For example, at 1600°C and 1 torr* pressure of CO, if the oxygen content in liquid steel (= hO) is 25 ppm, then the carbon content (= hC) would be equal to 7.1 ppm only, which is indeed very low. However, such low values are not obtained in practice. This is due to kinetic limitations.

6.2.2

SIDE REACTIONS

In addition to the principal degassing reactions discussed above, several other reactions occur during vacuum degassing to a minor extent. A brief discussion of some of these is presented below. Decomposition of Inclusions Suppose that the inclusion is a nitride (such as AlN). Its decomposition is given by AlN (s) = Al + N

(6.4)

Under vacuum, N decreases, thus favoring decomposition of AlN. Oxide inclusions can be decomposed, in principle, by reduction with carbon. For example, SiO2 (s) + 2C = Si + 2CO(g)

(6.5)

A lowering of the CO pressure helps this reaction to proceed in the forward direction. Thermodynamic predictions about inclusion decomposition can be made only through calculations under specific conditions. It would depend on the stability of the oxide. For example, Al2O3 would be more difficult to decompose than SiO2. There have been a number of investigations of the breakdown of nonmetallic inclusions upon vacuum treatment of steel, and decreases have been found.8 Reaction of Liquid Steel with the Refractory Lining Besides the above reactions, which are encouraged by lowering the chamber pressure, some more are illustrated with examples below. * 1 torr = 1.315 ×10–3 atm.


SiO2 (s) + C = SiO (g) + CO (g)

(6.6)

MgO (s) = Mg (g) + O

(6.7)

CaO (s) = Ca (g) + O

(6.8)

MgO (s) + C = Mg (g) + CO (g)

(6.9)

In all these reactions, one or more gaseous species are generated. Therefore, the lowering of pressure tends to lead them to the forward direction as shown. SiO is a gas. Mg and Ca are stable gases at steelmaking temperatures. They also have negligible solubility in liquid steel. Some data indicate that melt-refractory reactions do occur in industrial vacuum degassing. Example 6.1 In a vacuum degassing process, MgO is being used as ladle lining and the temperature is 1850 K. Make a thermodynamic assessment of reaction of MgO with molten steel containing 0.2 wt.% C and 0.001 wt.% O. Ignore interaction coefficients. Given 1 MgO(s) = Mg(g) + --- O2 (g) 2 ∆G

0 1

= 6.085 × 105 + 1.005 T log T – 112.84T

(E1.1)

J/mole

Solution 1. Consider Reaction (6.7). This reaction is a combination of Reaction (E1.1) and Eq. (5.1). 1 3 o --- O 2 ( g ) = O; ∆G O = 117.3 × 10 + 2.889 T, 2

J/mole

(5.1)

So, ∆G 7 = ∆G 1 + ∆G O o

o

o

(E1.2)

Carrying out the calculations, ρ Mg × [ h o ] o ∆G 7 at 1850 K = 2.94 × 105 J mole–1 = –RT ln K7 = –RT ln -----------------------a MgO = – RT ln ( p Mg × x [ W O ] ) since aMgO = 1 and hO = WO, because interaction coefficients are ignored. Hence, KMgO = pMg × WO = 0.5 × 10–8. Since WO = 10–3, pMg = 0.5 × 10–8/10–3 = 0.5 × 10–5 atm. Thermodynamically, Reaction (6.7) would occur only if pressure on the ladle wall is less than 10–5 atm. This is not achievable in vacuum degassing. 2. Consider the alternate reaction, i.e., Eq. (6.9). This reaction is the sum of Reactions (6.7) and (6.3). So, K9 = K7 × K3 ©2001 CRC Press LLC

(E1.3)

At 1850 K, K3 = 500 and K7 = 0.5 × 10–8 Hence, p Mg × P CO K9 = 250 × 10–8 = ----------------------WC

(E1.4)

Noting that WC = 0.2, pMg × pCO = 0.5 × 10–6, toward the top of the melt, pCO may be taken as 10–3 atm. Then, pMg = 0.5 × 10–3 atm and hence the extent of this reaction would be appreciable. But at a depth, pCO is close to 1 atm. So pMg would be very low, and as such this reaction should be negligible. Volatilization Many elements have high vapor pressures and therefore are expected to be distilled off to some extent during vacuum treatment. An idea can be obtained if some calculations of vapor pressures (pi) are carried out for Fe-i binary solution. Wi -----Mi ---------------------pi = p ⋅ ai = p ⋅ γ i X i = p ⋅ γ i ⋅ W i W Fe ------ + --------M i M Fe o i

where

o

pi = ai = Xi = γi = Mi =

o i

o i

(6.10)

vapor pressure of pure element i at temperature under consideration activity of element i dissolved in liquid iron mole fraction of i in liquid iron activity coefficient (Raoultian) of i in liquid iron molecular mass of element i

For a dilute binary solution in iron, wFe ≈ 100, and γi = constant = γ i (Henry’s Law constant). Hence Eq. (6.10) may be simplified as o

o o W i M Fe p i = p i γ i ⋅ ---------------100M i

(6.11)

Table 6.2 shows some sample calculations based on Eq. (6.11). Again, the temperature dependence o of p i is given by the Clausius–Clapeyron equation, viz., ∆H v ∆S v o log p i (atm) = – ---------------------- + -----------------2.303 RT 2.303 R where ∆Hv = enthalpy of vaporization of the element per mol ∆Sv = entropy of vaporization of the element per mol R = universal gas constant = 8.314 J/mol/K ©2001 CRC Press LLC

(6.12)

TABLE 6.2 Equilibrium Vapor Pressures of Some Elements Dissolved in Molten Iron at 1600°C o

Element (i)

Mi

at 1600°C, milliatmosphere (approx.) (Ref. 9)

Al

27.0

2.66

Cu

63.5

1.2

Mn

54.9

Si

28.1

0.027

Sn

118.7

pi

γi

o

p I , milliatmosphere (calculated) at 1600°C (Ref. 10)

@ Wi = 0.05

@ Wi = 1

0.029

8.0 × 10–5

1.6 × 10–3

8.6

4.5 × 10–3

0.09

1.3

0.44

8.8

0.0013

0.35 × 10–6

6.9 × 10–6

2.66

2.8

1.7 × 10–3

0.035

665

Fe

55.85

0.76

–

–

–

S*

32.1

–

–

10–5

–

P*

31.0

–

–

10–9

–

*

Method of calculation discussed in text.

Table 6.2 indicates that volatilization of Mn should be the most predominant, followed by that of Fe. This agrees with observations. Significant loss of Mn occurs during vacuum treatment. The dust collected at the exit of the vacuum chamber shows that it consists primarily of Fe and Mn. Tix11 reported that the flue dust in a ladle degassing installation consisted of (in weight percent) the following: FeO, 17.9; MnO, 47.0; Zn, 1.4; Cu, 2.6; Sn, 0.2; and Pb, 1.0. Olette12 reported the results of extensive investigation carried out at IRSID, France, on vacuum distillation of minor elements from liquid iron alloys. Experiments had been carried out in a laboratory induction furnace. They found phosphorus to remain constant, and As, S, Sn, Cu, Mn, and Pb exhibited increasing elimination in the order given. Mn evaporated at such a velocity that its elimination could be considered as a degassing process. The above observations are qualitatively in line with Table 6.2. S and P are gaseous at steelmaking temperatures. Several gaseous compounds of these elements have been identified, e.g., S, S2, S4, S6, and S8 for sulfur, and P, P2, and P4 for phosphorus. Therefore, their total vapor pressures are really the sum of all the gaseous components. However, the calculations in Table 6.2 have been performed assuming S2 and P 2 . According to Table 6.2, the vaporization of Al and Si should be negligible. However, Olette12 reported much higher vaporization rates for these elements. These were explained by the formation of volatile suboxides, SiO and Al2O. This is in line with studies on the characterization of hightemperature vapors where various other volatile suboxides such as SnO, AlO, and ZrO have been identified. Sehgal13 and others found that there was an appreciable loss of silicon from liquid steel under vacuum only if the latter contained sulfur. The results were interpreted by the formation of a volatile species, SiS. Ohno14 has reviewed the kinetics of evaporation in detail. He has shown how the formation of volatile compounds like SiS, CS, CS2, SO, SO2, etc. enhances the rate of elimination of S under vacuum. Deoxidation by Si was also helped thermodynamically and kinetically under vacuum or argon atmosphere due to the formation and removal of volatile SiO. The above findings are based on laboratory/bench-scale studies employing shallow melts, and somewhat leisurely experiments. In principle, they are applicable to industrial degassing processes, too. However, the author has little information on their quantitative significance. ©2001 CRC Press LLC

6.3 FLUID FLOW AND MIXING IN VACUUM DEGASSING The nature of fluid motion and the turbulence intensity during vacuum treatment of liquid steel are of considerable importance due to their significant influence on mixing, mass transfer, inclusion removal, refractory lining wear, entrapment of slag, and reaction with the atmosphere. Rising gas bubbles are either the only source or the principal source of stirring. The bubbles are gases evolved due to degassing (CO, N2, and H2) as well as injected argon gas. As stated earlier, argon injection through submerged tuyeres or porous plugs is a must for a successful process. The basics of fluid flow and flow in a gas-stirred liquid bath were discussed in Chapter 3. In vacuum degassing, the chamber pressure is very small as compared to the ferrostatic head of the liquid in a vessel. As a consequence, the bubbles expand enormously when they rise to a free surface. This causes the phenomenon known as bubble bursting, as a result of which liquid metal droplets are ejected into the vacuum chamber in large numbers. For the purpose of understanding, the situations prevalent in vacuum degassing may be simplified into two categories. The first category basically is a ladle stirred by argon gas from bottom. Chapter 3, Section 3.2 presented discussions on fluid flow in steel melts in gas-stirred ladles. Brief presentations have been made on the following: • • • •

Growth and motion of single bubbles Bubble size and shapes Gas holdup and dynamics in bubble swarms Characteristics of the rising plume, viz., gas holdup, bubble size, and bubble frequency distributions • Flow field in the liquid bath outside the plume-isopleths of velocity, turbulent kinetic energy, etc. • Rate of energy input per unit mass (ε), importance of buoyancy and εb

6.3.1

FLUID FLOW

IN

LADLE DEGASSING

Generally speaking, the above are basically applicable to flow in the melt during ladle degassing, since here also argon is introduced through purging plugs located at the ladle bottom. However, the situation would differ from that in Chapter 3 in the following ways: 1. The gas pressure above the melt is close to zero. 2. The argon bubbles pick up CO, N2, and H2 as they rise through the melt. Both these factors are expected to lead to massive volumetric expansions of the bubbles, especially when they approach the top surface of the melt. Let us consider the issue of reduced gas pressure above the melt. From Boyles law, Pn Vb,n = PVb

(6.13)

where P, Vb are pressure on the bubble and the volume of the bubble, respectively, at any depth below the free surface. The subscript n denotes the condition existing at nozzle exit at bottom. Let atmospheric pressure = 0. Then, P = ρlgz, where ρl is liquid density and z is the depth below the free surface. Taking z = 2 m at the nozzle exit, and considering the size of the bubble at z = 0.1 m, Vb/Vb,n = 20, indicating a 20× expansion in volume. At z = 0.02 m, Vb/Vb,n = 100. Szekely and Martins15 argued that rapid radial expansion of the gas bubble in vacuum processing would impart a radial velocity to the surrounding fluid. The corresponding radial acceleration requires a radial pressure gradient. For this pressure gradient to be maintained, the pressure in the bubble must be higher than the pressure in the bulk of the liquid at the level of the bubble. Rapid ©2001 CRC Press LLC

expansion also would not allow instantaneous attainment of terminal velocity at a location, and it calls for modification of the drag coefficient relation. Quantitative predictions of bubble radius as a function of z agreed reasonably well with experimental data on growth of an n-pentane bubble in n-tetradecane at room temperature for a freeboard pressure of 1 mm Hg, as shown in Figure 6.7.16 Such bubble expansion recently has been observed in a silicon oil room-temperature model.17 The bubble shape also changed from oval to spherical-cap. Mixing time tended to be constant, independent of the stirring power of gas per unit bath volume in low vessel pressure, presumably due to the dissipation of most of the expansion energy. In a buoyant plume of rising gas-liquid mixture, bubbles may be emerging as single ones from the nozzle. But, at a short distance above, they coalesce and disintegrate, exhibiting a spectrum of sizes. Such a phenomenon will occur here also, thus rendering theoretical predictions of the situation extremely difficult. It is possible to state with fair confidence that the plume characteristics, as well as the flow field in the liquid outside the plume, can be assumed to be the same as in an ordinary gas-stirred ladle situation toward the bottom part of the liquid. However, toward the top, some departure is expected. Splashing of liquid droplets above the bath by rising gas bubbles is a common experience. The extent of such splashing increases with an increase in the gas purge rate. It also increases with increasing bubble size. This phenomenon occurs in the traditional open-hearth steelmaking process and is held to be responsible for fast transfer of oxygen from the gas phase to the bath. Rimming phenomena during solidification of steel ingot provide another example. Visual observations during degassing of liquid steel also show droplet ejection in the vacuum chamber on an extensive scale. Richardson18 has reviewed fundamental studies on bubble bursting, carried out on water and mercury at room temperature. A film of liquid tends to stick to the bubble due to surface tension effects while the bubble tries to emerge from the bath. The rupture of this film is responsible for ejection of droplets. If a layer of slag is present at the top of the melt, such droplets cause the formation of a slag-metal emulsion. Chapter 4, Section 4.4.2, reviewed this.

FIGURE 6.7 Plot of bubble radius against height for the growth of n-pentane bubble in n-tetradecane at a freeboard pressure of 1 mm Hg.16


6.3.2

FLUID FLOW

AND

CIRCULATION RATE

IN

RH DEGASSING

In modern RH degassers, argon is also bubbled through bottom purging plugs in the ladle for better mixing. However, theoretical computations of the flow field in the vessel have been carried out on a traditional RH degasser without gas purging from bottom. Figure 6.8 shows the flow pattern of the melt schematically. One of the latest studies is by Kato et al.,19 who also carried out water model and plant studies. The computed flow pattern (i.e., velocity distribution) is shown in Figure 6.9. It agreed reasonably with experimental observations in water model. Figure 6.10 shows a comparison of calculated and measured liquid velocity at a location in the water model. The speed of degassing in an RH unit increases with an increased rate of circulation (R) of liquid steel through the vacuum chamber. R ranges between 10 and 100 tonnes/min and has been a subject of study for some time. Circulation velocity increases with an increasing argon flow rate in the upleg of the degasser. Recently, Kuwabara et al.20 have reported extensive measurements of circulation rates in several RH degassers in Japan. They also computed R from the energy balance by considering buoyant force on bubbles and frictional dissipation in uplegs and downlegs of the vacuum chamber. This yielded the equation

where

R = AX

(6.14)

X = Q1/3 d4/3 {ln (P1/P2)}

(6.15)

A = a constant

and

where

Q d P1 P2 R

= = = = =

argon injection rate, Nm3/s I.D. of leg, m pressure at base of downleg pressure in vacuum chamber circulation rate, kg/s

A plot of R vs. X (Figure 6.11) yielded a straight line, confirming Eqs. (6.14) and (6.15). The value of A was obtained as 7.42 × 103.

FIGURE 6.8 Schematic flow pattern in the melt in an RH degasser.


0.3 m s

—1

FIGURE 6.9 Computed velocity field for a water model of an RH degasser.19

FIGURE 6.10 Comparison of computed and experimentally measured velocities in liquid for a water model of an RH degasser.19

Circulation rates had been estimated from radio tracer data in a 150t RH degasser.21 Data of Kuwabara et al. have been collected in a water model. Recently, Hanna et al.22 also reported an extensive water model investigation on circulation rate. They have also discussed the limitations of water models, since these cannot properly simulate bubble expansion due to temperature and pressure changes. Hence, they are approximate guides only. However, they have proven to be quite effective toward evolving more efficient degassers. ©2001 CRC Press LLC

FIGURE 6.11 Variation of the circulation rate (R) with the gas flow rate parameter (X) in RH degasser.

Hanna et al.22 also investigated the influence of other variables, such as the location and number of argon injection ports in the upleg and the depth of immersion of the legs into the bath liquid, on circulation rate. These have minor influences (at most 25% or so) but are important for more efficient design.

6.3.3

MIXING

IN

DEGASSER VESSELS

In both ladle and RH degassing, the vessel containing molten steel is a ladle, and the following discussions pertain to it. Chapter 3 presented a brief review about the rate of stirring energy input (ε). Chapter 4 reviewed the fundamentals of mixing phenomena and the relationship of mixing time (tmix) with ε for gas-purged ladles. Hence, the discussions here will be very brief and restricted to ladle and RH degassers only. In ladle degassing, mixing is due to agitation by rising gas bubbles, both argon as well as CO, N2, and H2. As already stated in Section 3.2.3, ε due to buoyancy, i.e., εb, has been accepted as a measure of the rate of energy input into the bath due to gas flow. εb per unit mass of liquid, i.e., εm, as defined by Eq. (3.64), is the popular parameter employed. Hence, in ladle degassing also, we can employ the tmix vs. εm correlations recommended in Section 4.2.2. This can constitute the basis of design and process control. It is difficult to quantitatively take into account the influence of other gases on εm and tmix. Nor is it justified in view of the uncertainty. So far as an RH degasser is concerned, very little data are available on tmix. Nakanishi et al.21 injected a radio tracer at the base of the upleg in a 150t RH unit. Tracer intensity was monitored at the bottom of the downleg. Theoretical computations with the aid of a two-dimensional turbulent Navier–Stokes equation were performed. Experimental data approximately matched the assumptions that the melt inside the vacuum chamber was perfectly mixed, with eddy diffusivity ranging between 100 and 500 × 10–4 m2/s, depending on the argon injection rate. Tracer additions made at the bottom of the ladle were found to be uniformly dispersed in 8 to 10 min, whereas additions in vicinity of upleg were dispersed in 4 to 5 min. In another paper,23 Nakanishi et al. suggested that the tmix vs. εm correlation proposed by them [i.e., (Eq. 4.4)] may be employed for the RH ladle with 1 2 ε m = --- U R/M 2 ©2001 CRC Press LLC

(6.16)

where U is the linear velocity of metal in the downleg in meters per second, M is the total mass of steel in kilograms, and εm is in watts per kilogram. Kato et al.19 carried out a numerical analysis of fluid flow in an RH vessel to calculate the rate of carbon removal. They also collected samples from several depths of 240t and 300t degasser ladles to determine the average carbon removal rate. A comparison of the two approaches showed that the experimental rate was somewhat lower than the perfectly mixed assumption and somewhat higher than the plug flow assumption, demonstrating that the actual flow was nonideal. Carbon concentration in the vertical direction was found to be rather uniform. According to them, no dead zone existed.

6.4 RATES OF VACUUM DEGASSING AND DECARBURIZATION 6.4.1

BEHAVIOR

OF

GASES

IN INDUSTRIAL

VACUUM DEGASSING

The hydrogen content can be lowered to levels of 1 to 2.5 ppm by nearly every method, independent of the time of treatment and quality of steel.24 It has also been found that the results obtained on killed steels agree well with the theoretical equilibria derived from Sieverts’ law for the hydrogen content of steels if the results are related to the total pressures employed in the vacuum-treatment process. In degassing of semikilled or rimming steels, lower final hydrogen levels than found in killed steels usually may be obtained by most processes. The reason for this phenomenon is that the hydrogen partial pressure of semikilled or rimming steels is lower at the same total pressure due to carbon monoxide given off by these steels. Figure 6.12 presents some data.

FIGURE 6.12 Influence of total pressure on hydrogen removal from molten steel in vacuum degassing processes.24


In vacuum arc remelting, the nitrogen content can be lowered to 5 to 10 ppm. Sometimes this level can be achieved in vacuum induction melting as well. However, typically, the nitrogen content is brought down to 25 to 30 ppm at the most in all vacuum degassing processes. This is somewhat insensitive to processing details. A sample calculation based on Table 6.1 shows that, at p N 2 = 0.1, 1, and 4 milliatmospheres (matm), the equilibrium nitrogen content of steel would be 4.3 ppm, 14 ppm, and 28 ppm, respectively, at 1600°C. Since the total pressure in the vacuum chamber during degassing lies in the range of 1 to 24 10 matm, and the exit gas contains anywhere between 10% to 50% N 2 , p N ranges between 0.1 2 and 5 matm. Therefore, the nitrogen content of molten steel either attains equilibrium with exit gas or may be somewhat higher during vacuum degassing. Suzuki et al.25 have put extent of nitrogen removal as 10 to 35%. The slowness of nitrogen removal may be ascribed to a lower value of diffusion coefficient as compared to that of hydrogen, and additional retardation by dissolved oxygen and sulfur as will be discussed later. It is also possible that the somewhat higher value of N may be due to stable nitride inclusions, such as AlN. Figure 6.1324 shows the C vs. O relationship after vacuum degassing and compares it with C – O equilibrium in molten steel at various values of pCO. At low carbon, the oxygen content corresponds to pCO = 100 torr (131.5 matm). At high carbon, value of O is lower. However, it corresponds to pCO close to 1 atm. Therefore, the C – O relationship is far off from equilibrium in vacuum degassing. Such a behavior pattern may be ascribed to the following causes. 1. The oxygen content indicated in Figure 6.13 is total oxygen content. The dissolved O is somewhat lower. This reduces the difference with the equilibrium curve somewhat.

FIGURE 6.13 Influence of carbon content on ultimate total oxygen content of the steel melt in vacuum degassing processes.24


2. The diffusivity of oxygen in liquid steel is an order of magnitude lower as compared to that of hydrogen. 3. The equilibrium value of dissolved oxygen is really negligible. This also has a magnifying effect on the discrepancy between the actual value and the equilibrium value of O. 4. The reaction between melt and oxide refractory, as discussed in Section 6.2, also has been attributed to this behavior pattern.

6.4.2

RATES

OF

REVERSIBLE DEGASSING PROCESSES

The degassing processes are unit processes. The ladle or the degassing chamber is a reactor in accordance with the terminology adopted in chemical engineering. In metallurgical engineering, we can profit greatly by exploiting some of the concepts and mathematical techniques that have already been developed by chemical engineers and subsequently extended to metallurgical engineering. Considerable progress has been made in the last two decades in this direction. Degassing processes, such as ladle degassing, cycling and circulation degassing (DH and RH), and argon purging in a ladle may be classified as semi-batch processes. The liquid metal is taken out only after the batch processing is over. But the gases are introduced/removed continuously. In stream degassing, however, the metal is introduced into the vacuum chamber continuously, and gases are withdrawn continuously as well. Therefore, it may be classified as a continuous stirred tank reactor (CSTR). The rates of these processes theoretically may be estimated by performing calculations based on • materials and heat balance • reaction equilibria • reaction kinetics and mass transfer In our present state of knowledge, this can be accomplished by making some simplifying assumptions. However, a lot more experimental data are required to do a better job. Major uncertainties relate to fluid flow, mixing, and phase dispersions. We shall present a brief discussion of these later. Under the circumstances, it is often advisable to carry out a considerably simplified mathematical analysis for evaluating the process rates. For this, the process may be treated as isothermal and isobaric. Moreover, it may be assumed that mass transfer and reaction kinetics are extremely fast. This is not a bad assumption for many steelmaking reactions because of the high temperature and intense agitation. This allows rapid attainment of equilibrium. Therefore, for problem solving, we assume the process to be thermodynamically reversible, i.e., equilibrium is established rapidly at every stage instantaneously. Also, the melt is well mixed. In other words, at any instant of time, equilibrium is assumed to exist among reactants and products. This is illustrated through Example 6.2 on RH degassing. Example 6.2 Calculate the rate of circulation of molten steel through the vacuum chamber in the RH degassing process to lower the hydrogen content of steel from 4 to 1.5 ppm in 15 min. Assume that the molten steel attains equilibrium with respect to hydrogen inside the vacuum chamber. Given 1. Temperature = 1577°C, weight of steel in the ladle = 50 tonnes, pressure inside the vacuum chamber = 0.1 milliatmosphere 2. Composition of molten steel: C, 0.05%,; Cr, 5%; Ti, 0.5%; Ni, 2%, remainder Fe C

Cr

Ti

3. e H = + 0.04,e H = 0.005,e H = – 0.22 ©2001 CRC Press LLC

Solution 1. Hydrogen balance: Rate of removal of hydrogen from steel (g/min) ( m˙ 1 ) = rate at which hydrogen is transferred to vacuum (g/min) ( m˙ 2 )

(E2.1)

d [ ppmH ] × 10 d [ ppmH ] 6 m˙ 1 = – W × 10 × --------------------------------------- = – W × ----------------------dt dt

(E2.2)

Now, –6

where W is the weight of steel in tonnes, t is time in minutes, and [ppmH] denotes concentration of H in parts per million at any instant in time. Again, m˙ 2 = R × 10 × ( [ ppmH ] – [ ppmH ] eq. ) × 10 6

–6

= R ( [ ppmH ] – [ ppmH ] eq. )

(E2.3)

where R is the circulation rate of liquid steel through the vacuum chamber in tonnes/min, and [ppmH]eq. denotes [ppmH] in equilibrium with p H2 in the vacuum chamber (as assumed in problem). Equating Eqs. (E2.2) and (E2.3), R d [ ppmH ] ------------------------------------------------- = ----- dt W [ ppmH ] – [ ppmH ] eq

(E2.4)

Integrating Eq. (E2.4) within limits t = 0, [ppmH]initial and t = tf, [ppmH] = [ppmH]final, W [ ppmH ] initial – [ ppmH ] eq 50 4 – [ ppmH ] eq R = ----- ln --------------------------------------------------------- = ------ ln -----------------------------------t f [ ppmH ] final – [ ppmH ] eq 15 1.5 – [ ppmH ] eq

(E2.5)

For calculation of [ppmH]eq, assume that the gas in the vacuum chamber is primarily H2 (not a bad assumption, since it is a killed steel). So p H 2 = 0.1 × 10–3 atm. C Cr Ti From Table 6.1, hH = 0.623. Again, log f H = e H × W C + e H × W Cr + e H × W Ti from Eq. (2.56). Substituting values, fH = 0.851. So, 0.623 [ ppmH ] eq = ------------- = 0.73 0.851 Or, from Eq. (E2.5), R = 4.82 tonnes/min

6.4.3

KINETICS

OF

DEGASSING

AND

(Ans.)

DECARBURIZATION GENERAL FEATURES

Degassing and decarburization reactions involve two phases: molten steel and gas. The overall reaction consists of the following kinetic steps: 1. Transfer of dissolved gas-forming elements H, N, C, and O from the interior (i.e., bulk) of the liquid to the gas/liquid interface 2. Chemical reactions [i.e., Reactions (6.1) to (6.3)] at the gas–liquid interface ©2001 CRC Press LLC

3. 4. 5. 6.

Transfer of gaseous species H2, N2, and CO from the interface to the bulk gas Nucleation, growth, and escape of gas bubbles Mixing in the bulk liquid Mixing in the bulk gas

From our knowledge of reaction kinetics here as well as in similar situations, it has been taken as established that steps 4 and 6, i.e., mass transfer and mixing in the gas phase, are very fast and are not rate controlling, even partially. Chapter 5, Section 5.2.1 has reviewed the fundamentals of nucleation in connection with the formation of deoxidation products. Equation (5.39) gives the relationship between critical radius of nucleus (r*) with other variables for homogeneous nucleation, i.e., 2σ r* = – ------------------( ∆G/V )

(5.39)

For the equilibrium of the gas bubble with the liquid at constant temperature, dG = 0 = VdP + (dG)chemical

(6.17)

∆G *  ∆G -------- =  -------- = – ∆P  V  chemical  V  in Eq.(5.39)

(6.18)

2σ r* = ----------∆P*

(6.19)

So,

and hence,

where ∆P* = excess pressure inside the gas bubble with radius r* The gas law for the critical nucleus may be written as n* * * p b ⋅ V b =  ------ ⋅ RT N

(6.20)

Here, ∆P* ≅ P b, *

*

4 *3 * V b = --- π r , 3

R = 82.06 × 10

–6

3

–1

m K mol

–1

*

with P b and V b in atm and m3 respectively, N is the Avogadro number, and n* is the number of molecules in the critical nucleus (assume 100). Combinations of Eqs. (6.19) and (6.20) yield 3 1⁄2

6 σ * P ex ( atm ) = 1.54 × 10  ----- T

(6.21)

where σ = 1.6 Nm–1 for molten steel. Taking T = 1850 K, Eq. (6.21) gives a value of ∆P* as 7.2 × 104 atm. ©2001 CRC Press LLC

It is impossible to generate such high excess pressure via the steelmaking reactions. In connection with the basic open hearth (BOH) process of steelmaking, this issue had been investigated, and it was concluded that nucleation of gas bubbles is not required. They grow on existing gascontaining cavities in the refractory lining of the hearth.26 Nucleation of gas bubbles is not at all a problem, as it appears from several other studies, including carefully conducted cold model studies.25,27 In the early days, ladles for degassing were not fitted with gas purging arrangements. It was found that, initially, there was vigorous evolution of gas bubbles. After a time, when gas evolution ceased, degassing rates were very slow. The need for some stirring was sensed. This was achieved by electromagnetic stirring in ASEA-SKF ladles. Here, stirring helped mass transfer. Now, ladles have argon purging facilities. As a result, nucleation and growth are not required, as H2, N2, and CO either would be picked up by argon bubbles or would escape into vacuum chamber directly at the stirred top surface. It has been established that the rates of nitrogen absorption and desorption by molten iron and steel are partially controlled by slow surface reaction. This will be discussed separately later. For the time being, surface reaction is assumed to be fast. Hence, kinetic steps 1 and 5, viz., mass transfer and mixing in the liquid, have been generally considered to be slow and rate controlling, either singly or jointly. Section 4.5, in Chapter 4, discussed the issue of mixing vs. mass transfer control in steelmaking. It presented an analysis by Ghosh28 that tried to show that the mixing times for 95% mixing are in the same overall range as the 95% conversion times for mass transfer controlled reactions for some steelmaking processes. Since both have rate expressions as for first-order reversible processes, it is often difficult to say whether a process is controlled by slow mixing or slow mass transfer. For mass transfer between a gas and a liquid, surface renewal theory is to be applied. Section 1⁄2 4.3.2 has reviewed it. It predicts that km,i should be proportional to D i , where km,i is the mass transfer coefficient, and Di is diffusivity of solute i. It was also pointed out in Section 4.4.2 that, in view of uncertainties in the surface area of the melt, experimental rate data generally allow determination of the lumped parameter ka with the help of Eq. (4.36) or (4.37). If it is desired to further generalize the empirically determined rate without attributing it to mass transfer or mixing, it is better to define the empirical (i.e., experimentally determined) rate constant as dW e – ----------i = k i,emp ( W i – W i ) dt

(6.22)

Takemura et al.29 determined values of ki,emp parameters for the removal of carbon and hydrogen in RH injection process. Figure 6.14 presents these as a function of argon gas flow rate. It may be noted that there is virtually little difference between them, i.e., ki,emp parameters were approximately the same for both carbon and hydrogen. DC = 7.2 × 10–9 m2 s–1 and DH = 10–7 m2 s–1 at 1600°C. Hence, if mass transfer were rate controlling, then from Eq.(4.25), –7 1⁄2 k H ,emp 10  ------------- =  ----------------------= 3.73 – 9  7.2 × 10  k C,emp

Since this is not the case, it may be concluded that mass transfer was not rate controlling. On the other hand, rate control by mixing theoretically predicts same value of ki,emp for both carbon and hydrogen. Bauer et al.30 measured rates of removal of hydrogen, oxygen, sulfur, and nitrogen in a 185t VOD unit. Quantitative comparison of ki,emp parameters was not possible on the basis of their data. ©2001 CRC Press LLC

FIGURE 6.14 ki,emp vs. argon flow rate for carbon and hydrogen in the RH-injection process.29

But they indicated behavior similar to that obtained by Takemura et al.29 Therefore, it does not seem to be correct to assume mass transfer control for degassing processes.

6.4.4

IMPORTANCE

OF THE

akm PARAMETER

Regardless of whether the mass transfer is rate controlling, the degassing rate can be speeded up only if the ak parameter is large. Assuming that k = km, k can be increased by enhanced stirring. But there is a limit to it. As discussed in Section 4.3.2, k m,i = (DiS)1/2 in turbulent flow. S is Danckwerts’ surface renewal factor. S has been found to range between 5 and 25 s–1 in mild turbulence, going up to 500 s–1 in high turbulence. This gives the maximum ratio of ( k m,i ) high turbulence 500 1 ⁄ 2 ------------------------------------ =  --------- = 10  5  ( k m,i ) low tubulence In contrast to this, the specific surface area (a) can be increased by a factor of even 104 by the creation of small gas bubbles and metal droplets ejected into the gas space.31 Hence, for fast degassing, a large value of specific surface area is a prerequisite. It can be achieved only if a large number of gas bubbles are present in the melt during processing or if the molten steel is dispersed into the gas phase as fine droplets. Let us examine the various degassing processes from this point of view. 1. In ladle degassing, as soon as the chamber is evacuated, rapid growth and evolution of bubbles occur due to the initially large thermodynamic supersaturation. Rapid evolution of bubbles also causes ejection of fine droplets of liquid steel into the empty space of the vacuum chamber, causing a further rate increase. However, after few minutes, bubble evolution ceases, and rate of degassing decreases drastically unless there is argon purge from the bottom. 2. In stream degassing, the steel is introduced continuously into the vacuum chamber as a stream. Rapid formation and bursting of gas bubbles in the stream cause the latter to disintegrate into fine droplets. Hence, degassing is fast throughout the processing. ©2001 CRC Press LLC

3. In the DH process, molten steel is drawn into the vacuum chamber as a shallow pool. Again, rapid gas evolution and droplet ejection lead to fast degassing of the melt inside the chamber. 4. In RH process, steel is continuously drawn into the vacuum chamber by both the vacuum action and the lifting effect of rising argon bubbles. These argon bubbles expand and burst out of the melt in a vacuum chamber, thus assisting in the creation of drops and bubbles and assuring a fast rate of degassing throughout the processing.

6.4.5

KINETICS OF DESORPTION AND ABSORPTION OF NITROGEN BY LIQUID IRON

It has already been mentioned that degassing of nitrogen has to be considered separately, since its kinetics includes some special features. Pehlke and Elliott,32 in their pioneering study, measured the rate of absorption and desorption of nitrogen by a clean liquid iron surface in an inductively heated melt in a modified Sieverts apparatus. They derived the following important findings. 1. Absorption and desorption were first-order reversible processes with approximately the same rate constant. Rate (r) of desorption was given by r = AkN ([WN] – [WN]e) where

(6.23)

A = liquid-gas interfacial area [WN], [WN]e = weight percent of dissolved nitrogen in iron respectively at that instant and at equilibrium with the partial pressure of nitrogen above the melt kN = first-order rate constant

2. The increase in oxygen content of the liquid iron decreased the rate drastically. Figure 6.15 presents a typical behavior pattern, which shows that k N ∝ [ W O ] . Several investigators measured the surface tension (σ) of liquid iron with variable concentrations of oxygen dissolved in it. Figure 6.1633 shows the variation of σ with ln[WO] at 1550°C. From Gibbs Adsorption Isotherm, dσ dσ Γ O = – --------- = – ------------------------------dµ O RTd ( ln [ a o ] ) where

(6.24)

ΓO = excess oxygen at surface µ O = chemical potential of oxygen dissolved in liquid iron

Noting that aO ∝ WO, from Figure 6.16, it was inferred on the basis of the above equation that ΓO is positive. In other words, oxygen is surface active in liquid iron and prefers to stay at the surface. Darken and Turkdogan34 critically reviewed this inference. Theoretical calculations revealed that most of the surface would be covered by oxygen atoms at WO = 0.05 wt.% at 1550°C. Figure 6.17 shows the fraction of surface covered (θ) as a function of WO. On the basis of the above, Pehlke and Elliott32 quantitatively explained the dependence of kN on WO by assuming that the rate was controlled by slow surface reaction, and that adsorbed oxygen atoms acted as barriers to it, consequently retarding the rate. Sulfur is also surface active in liquid iron. It also has been found to retard nitrogen absorption/desorption rates. This phenomenon has been well established in laboratory as well as in industry through numerous subsequent studies. As an example, Figure 6.18 shows influence of sulfur on removal of nitrogen from molten steel ©2001 CRC Press LLC

FIGURE 6.15 Influence of oxygen content on the absorption rate of nitrogen by liquid iron at 1823 K.32

FIGURE 6.16 Variation in the surface tension of liquid iron with its oxygen content at 1823 K.34 Reprinted by permission from American Chemical Society.

during degassing in a 185t ladle.30 Hence, the melt should be well deoxidized and desulfurized before attempting to remove nitrogen. Although Pehlke and Elliott claimed the kinetics to be exclusively controlled by slow surface reaction, this view was not accepted by all. In view of its importance, there have been several fundamental laboratory investigations in the last two decades. The current view may be summarized as follows. • The rate is primarily controlled by mass transfer in liquid iron at low oxygen and sulfur levels. • At normal oxygen and sulfur levels in liquid iron, partial rate control by slow interfacial reaction is also exhibited. Some investigators even considered mass transfer of N2 in gas phase as well. • It has also been concluded by several investigators that the interfacial reaction is a secondorder process, i.e., the rate of interfacial chemical reaction (rC) is given by ©2001 CRC Press LLC

FIGURE 6.17 Fraction of surface covered (θ) vs. [WO] for liquid iron at 1823 K (estimated).34

FIGURE 6.18 Influence of sulfur content of liquid steel on the extent of nitrogen removal during ladle degassing.30

r C = Ak C ( [ W N ] – [ W N ] e ) 2

where

2

(6.25)

kC = chemical rate constant

A quantitative correlation of rate with oxygen and sulfur content of steel has been proposed by Fruehan and Martonik35 at constant temperature as k m,N k N = ----------------------------------------------1 + a[W O] + b[W S]

(6.26)

where kN is the actual first-order rate constant and is less than the mass transfer coefficient for nitrogen in liquid steel (km,N), and a and b are empirical constants. ©2001 CRC Press LLC

Evaluation of kC based on Eq. (6.25) requires the elimination of mass transfer effects from actual rates through theoretical analysis. It has been done by several investigators. Harada and Janke36 have summarized their own findings as well as those of some other recent studies. KC was correlated with variables at 1600°C by an equation of the following type: aφ ( f N ) k C = -----------------------------------------1 + b [ hO ] + c [ hS ]

(6.27)

where a, b, and c are empirical constants. Figure 6.19 presents the results of some investigations.36 It may be noted that data obtained at reduced pressures (curve 1 and data points of Ref. 36) corresponding to vacuum degassing conditions do not agree with those obtained at normal pressures (curves 3, 4, and 5). There was no satisfactory explanation for this. Formation and evolution of tiny gas bubbles at reduced pressures, causing surface flows in melt, may be responsible for such a discrepancy. It was stated in Section 4.3.2 that the presence of surface-active species on the surface of a liquid would retard the motion of fresh eddies coming from the bulk of the liquid (Figure 4.9a), resulting in a smaller value of km as compared to that for a clean surface. Richardson37 suggested that this may, in principle, explain the retardation of rate in the presence of oxygen and sulfur where mass transfer is occurring in a turbulent flow situation. This alternative mechanism has not been seriously considered yet. As discussed in Section 6.4.1, industrial vacuum degassing is not very efficient in the removal of nitrogen. Suzuki et al.25 suggested the extent of removal to be 10 to 35%. To a significant extent, the retarding influence of oxygen and sulfur is responsible for this, as well as for the irreproducible nature of removal. However, as stated earlier, it has been possible to achieve fairly low nitrogen levels in a well desulfurized and well deoxidized melt. Of course, even then, vacuum degassing alone is not enough. Nitrogen is picked up by molten steel at all stages of processing, viz., primary steelmaking, tapping, and teeming, due to contact with N2 in an air/gaseous environment. A lownitrogen steel can be produced (WN < 20 ppm) only if precaution is taken at all stages to prevent its absorption. Chapter 8 offers further discussion on this subject.

FIGURE 6.19 Variation of kC for nitrogen desorption with dissolved oxygen and sulfur content of molten steel at 1873 K.36


6.4.6

KINETIC CONSIDERATIONS DECARBURIZATION

FOR INDUSTRIAL

VACUUM DEGASSING

AND

In industrial vacuum degassing, as already mentioned in Section 6.1.1, the treatment time should be short enough to logistically match with converter steelmaking on the one hand and with continuous casting on the other hand. To achieve this, in addition to the proper choice and design of the process, the principal variables are: • pumping rate of vacuum equipment (also known as exhaust rate) • rate of injection of argon below the melt Increasing the Ar flow rate increases the rate of degassing and gas evolution. This tends to raise chamber pressure and requires a higher exhaust rate. The dynamic balance between the two determines the chamber pressure. This is illustrated for ladle degassing by Figure 6.20.38 In the initial stage, gas evolution is much faster, leading to higher chamber pressure. Predeoxidation is helpful, since it lowers the extent of CO evolution, thus allowing quicker attainment of a steady vacuum. The need for optimization has been illustrated by Soejima et al.39 (Figure 6.21) theoretically. The figure shows that, for an RH degasser, below a certain exhaust rate, the argon flow had no effect on rate constant k. Nor is there any advantage in having a high pumping rate if Ar flow rate is not adequate. Reaction sites are as follows: • • • •

argon bubble/melt interface free surface of the melt surfaces of ejected liquid metal droplets separately formed gas bubbles through growth inside the melt

Argon bubbling as such is ineffective without vacuum. This can be illustrated by analyzing the dehydrogenation of steel melt by argon purging, assuming it to be a thermodynamically reversible process. It is based on: 1. Hydrogen balance, viz., the rate of removal of hydrogen from molten steel = the rate at which hydrogen is going out with the exit gas 2. The assumption that the argon leaving the ladle is in equilibrium with the molten steel at that instant

FIGURE 6.20 Variation of chamber pressure with treatment time for ladle degassing.38


FIGURE 6.21 Influence of pumping rate and argon flow on decarburization rate constant in RH degasser.39

This analysis would predict the highest possible rate of hydrogen removal by simple argon purging from the bottom. Going through the derivation steps, the following equation was derived:31 2 2 3     KH KH 10 ---------V = M  ppmH + ------------------------+ ---------------------------[ ppmH ] –    o 2 2 11.2 f H [ ppmH ]  f H [ ppmH ] o   

(6.28)

where M is the mass of steel in tonnes and V is the volume of Ar (in Nm3) to be passed for lowering H from [ppmH]o to [ppmH]. Assuming [ppmH]o = 4, [ppmH] = 2, T = 1873 K, and fH = 1, calculations show that 1.67 Nm3 of argon would be required per tonne of steel. It is indeed a very high and uneconomical consumption of argon. As discussed in Section 6.3.1, argon bubbles would expand enormously as they approach the top surface of melt in the vacuum chamber. Hence, the dominant volume would be present only below the top surface, and this is the region where bubbles pick up large quantities of H2, CO, and N2. Bannenberg et al.40 have illustrated this through their mathematical modeling exercise for a ladle degasser. Yano et al.41 have developed a dynamic model in connection with improvement of the RH process for production of ultra-low-carbon and low-nitrogen steel. They have not considered ejected droplets separately but have taken them as part of the free surface. Higbie’s surface renewal theory (Section 4.3) was employed for calculation of rate constants at each site. Steel in the reaction vessel was assumed to be perfectly mixed in agreement with that by Kato et al.19 The chamber pressure was calculated by balancing the gas exhaust rate and the gas forming rate. Effective reaction surface area was estimated by fitting the calculated rate with measured values. Figure 6.22 shows some calculated results of Yano et al.41 The process was divided into two stages. Stage I was characterized by the rapid generation of gases (principally CO). Hence, the reaction inside the melt had a dominant share in decarburization. In stage II, this subsided due to a lowering of the contents of C, H, and N in the melt. Then, the free surface (including ejected droplets) was found to play the most dominant role, followed by the argon bubble/melt interface. We may assume that this pattern would be valid only qualitatively. Quantitatively, the ratios are expected to exhibit a range depending on the assumptions in model formulation and the nature of plant data. Yamaguchi et al.42 carried out kinetic studies in an RH degasser of Kawasaki Steel Corporation in connection with production of ultra-low carbon steel. The mechanism was assumed to be rate ©2001 CRC Press LLC

controlled jointly by the mass transfer of carbon and oxygen in the molten steel in the vacuum vessel. In the low carbon range (

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