Seetharaman S. Fundamentals of metallurgy

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in a plant situation to know if the gas flow is constant during injection. Very

often gas leaks out through cracks in the refractory, etc. It should also be pointed

out that the phenomenon model employed in this approach for the slag±metal

mixing

should only be considered very preliminary. It requires further

improvement on the basis of rigorous theoretical consideration and experimental

verification. However, the comparison shows that by buildi ng the process model

of a gas-stirred ladle using the micro-models as building blocks, it is possible to

predict desulphurization with acceptable accuracy.

9.7 Conclusions

In this chapter, the principal factors in process design have been briefly

discussed. The focus has been given to the theoretical consideration. However,

this consideration is far from sufficient. Very often, a lot of phenomena in the

process are poorly understood. It is also very common that the ther modynamic

data are missing or subjected to larger uncertainties. Even the physi cal

properties an d kinetic data are not available in many occa si ons. The

unavailability (or/and the uncertainty) of the basic data would hinder us from

finding out the optimum solutions. When, inevitably, we hit the wall of our

understanding, experimental investigations must be carried out. These

investigations should include (i) laboratory experiments, (ii) physical model

experiments, (iii) pilot plant experiments, and finally (iv) industrial experiments.

Interested readers should read the corresponding chapters in the book and other

literature to prepare their knowledge for a promising process design.

9.8 References

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1962, 224, 878±887.

2. Masson CR, `Thermodynamics and constitution of silicate slags', J. Iron Steel Inst.,

1972, 210, 89±96.

3. Masson CR, Smith IB and Wihiteway SG, `Activities and ionic distributions in

liquid silicates: application of polymer theory', Can. J. Chem., 1970, 48, 1456±1464.

4. Lin PL and Pelton AD, `A Structural Model for Binary Silicate Systems', Metall

Trans B, 1979, 10, 667±675.

5. Jeffes JHE, `The relationship of structure and thermodynamic properties of

polyanionic compounds', Canadian Met. Quart., 1981, 20, 37±50.

6. Kapoor ML and Frohberg MG, Int. Metallurgical Chemistry ± Applications in

Ferrous Metallurgy, Sheffield, July 1971, Iron Steel Inst., 17±23.

7. Gaye H, Riboud BA and Welfringer J, `Slag Modelling: a Tool for Evaluating

Metallurgical Treatments', Process Technology Proceedings. Vol. 6; Washington,

DC, USA, 6±9 April, 1986, 631±639.

8. Hillert M, Jansson B, Sundman B and A

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Solutions With Different Tendency for Ionization', Metall. Trans. A, 1985, 16A,

261±266.

396 Fundamentals of metallurgy

9. Selleby M, `An assessment of the Ca-Fe-O-Si system', Metall. Trans. B, 1997, 28B,

577±596.

10. Pelton AD and Blander M, `Thermodynamic Analysis of Ordered Liquid Solutions

by a Modified Quasichemical Approach ± Application to Silicate Slags', Metall

Trans B, 1986, 17B, 805±815.

11. Temkin M, `Mixtures of fused salts and ionic solutions', Acta Phys. Chim. URSS,

1945, 20, 411±420.

12. Lumsden J, Physical Chemistry of Process Metallurgy, Part 1, New York,

Interscience, 1961, 165±205.

13. Sommerville ID, Ivanchev I, Bell HB, Int. Metallurgical Chemistry ± Applications in

Ferrous Metallurgy, Sheffield, July 1971, Iron Steel Inst., 23±25.

14. Ban-Ya S and Shim J-D, `Application of the Regular Solution Model for the

Equilibrium of Distribution of Oxygen Between Liquid Iron and Steelmaking Slags',

Canadian Metall. Quarterly, 1982, 21, 319±332.

15. BjoÈrkvall J, Sichen Du and Seetharaman S, `Thermodynamic model calculations in

multicomponent liquid silicate systems', Ironmaking and Steelmaking, 2001, 28,

250±257.

16. BjoÈrkvall J, Doctoral Thesis, Departmenta of Materials Science and Engineering,

Royal Institute of Technology, Stockholm, ISSN 1403-493X, ISBN 91-7170-523-6,

24 March, 2000.

17. Kalyanram MR, Macfarlane TG and Bell HB, `The activities of calcium oxide in

slags in the systems CaO-MgO-SiO

, CaO-Al

-SiO

and CaO-MgO-Al

-SiO

at 1500ëC', J. Iron Steel Inst., 1960, 195, 58±64.

18. Wagner C, Thermodynamics of Alloys , Reading MA, Addison-Wesley, 1952.

19. Lupis CHP and Elliott JF, `Generalized interaction coefficients', Acta Metallurgica

sinica, 1966, 14, 529±538.

20. Pelton AD and Bale CW, `A modified interaction parameter formalism for non-

dilute solution', Metallurgical and Materials Transcription, 1986, 17A, 1211±1215.

21. Pelton AD, `The polynomial representation of thermodynamic properties in dilute

solution', Metallurgical and Materials Transcription B, 1997, 28B, 869±876.

22. Harja JP, Reddy SR and Frohberg MG, `Deviation and consistency of partial

function of the ternary system involving interaction coefficients', Metallurgical and

Materials Transcription B, 1995, 26B, 495±502.

23. Ma ZT, Ohser J and Janke D, `Thermodynamic treatment of multicomponent system

using interaction parameters', Acta metallurgica sinica, 1997, 10(2), 375±385.

24. Darken LS, `Thermodynamics of binary metallic solutions', TMS±AIME, 1967, 239,

80±89.

25. Reddy RG and Reddy SR, `Thermodynamic properties of quarternary systems',

Minerals, Metals and Materials Society/AIME, 1999, 243±253.

26. Tao DP, `Prediction of thermodynamic properties of multicomponent liquid alloy by

binary infinite dilute activity coefficients', Metallurgica l and Materials

Transcription B, 2001, 32B, 1205±1211.

27. Guochang J and Kuagdi X, `A discussion on thermodynamics of pyrometallurgy for

ferroalloys', China Science and Technology Press, 8th International Ferroalloys

Congress Proceedings, 1998, 290±293.

28. Richardson FD, Physical Chemistry of Melts in Metallurgy, vol.2, Academic Press,

London, 1974, 291±296.

29. Engh TA, Principles of Metal Refining, Oxford University Press, Oxford, 1992, 51.

Improving process design in steelmaking 397

30. Nilsson R, Sichen Du and Seetharaman S, `Estimation of sulphide capacities of

multi-component silicate melts' Scand. J. Metal., 1996, 25, 128±134.

31. Young RW, Hassall GJ, Duffy JA and Xu Z, `Use of Optical Basicity Concept for

Determining Phosphorus and Sulhur Slag±Metal Partitions', Ironmaking and

Steelmaking, 1992, 19, 201±219.

32. Reddy RG and Blander M, `Modeling of Sulfide Capacities of Silicate Melts',

Metall. Trans. B, 1987, 18B, 591±596.

33. Han Q, Rare earth, alkaline earth and other elements in metallurgy, 1998, Japan

Technical Information Service, Tokyo, Japan, 42±55.

34. Martinez GP, `Gas-Fired Belt Furnace Cuts Sintering Costs', Meta Powder Report,

1992, 47, 31±33.

35. Saxena N, Dwyer J, Haring N, Kirschner M and Nayar H, `Predicting part

temperatures during sintering under production conditions using a simple PC-based

model' Advances in Powder Metallurgy and Particulate Materials, 1996, vol. 3,

Washington DC, USA, June 16±21, 1996, 12.43±12.55.

36. Brandberg J, Johansson E, MagneloÈv M, Rutqvist E and SjoÈstroÈm U, Steel Grips, in

press, 2005.

37. Cramb AW, Chung Y, Harman J, Sharan A and Jimbo I, `The slag/metal interface

and associated phenomena. A history of pneumatic steelmaking', Iron and

Steelmaker (USA), 1997, 24(3), 77±83.

38. Dayal P and Sichen Du, Unpublished work, Department of Materials Science and

Engineering, Royal Institute of Technology, SE-100 44 Stockholm, 2004.

39. Beskow K, Dayal P, BjoÈrkvall J and Sichen Du, Ironmaking and Steelmaking, in

press, 2005.

40. Jakobsson A, Viswanathan NN, Sichen Du and Seetharaman S, `Interfacial

phenomena in some slag±metal reactions', Metall & Materials Trans. B, 2000,

31B, 973±980.

41. GoÈrnerup M and Jacobsson H, `Foaming Sl ag-Practice in Electric Stainless

Steelmaking', Iron and Steel Society/AIME, 55th Electric Furnace Conference

Proceedings, Chicago, USA, 1998 57±69.

42. Pehlke RD and Elliott JF, `Solubility of nitrogen in liquid iron alloys II. Kinteica',

Trans. Metallurgical Society of AIME, 1963, 227, 844±855

43. Elliott JF, `The role of interfaces in pyrometallurgical processes', Extractuve

Metallurgy 85 Symposium, London, September 1985, c171±c178.

44. Kumar J, Tripathi N and Sichen Du, `Importance of interfacial phenomena in the

removal of inclusions by gas bubbles from steel ± An experimental study using cold

model', Steel Grips, 2003, 1, 133±139.

45. Jonsson L, Sichen Du and JoÈnsson P, `A new approach to model sulphur refining in a

gas-stirred ladle ± a coupled CFD and thermodynamic model', ISIJ International,

1998, 38, 260±267.

46. Ji F-Z, Sichen Du and Seetharaman S, `Viscosities of multi-component silicate melts

at high temperatures', International J. of Thermophysics, 1999, 20, 309±322.

47. Launder BE and Spalding DB, `The numerical computations of turbulent flows',

Comp. Meth. In Appl. Mech. & Eng., 1974, 3, 269.

398 Fundamentals of metallurgy

10.1 Introduction

The formation of solid from a liquid is an interesting phenomenon as the

following issues are experimentally observed:

The initiation of solidification requires either undercooling or

supersaturation.

Most common examples of the initiation of solidification are heterogeneous

where the first solid forms on a pre-existing solid surface.

· If the pre-existing heterogeneous sites are removed from the liquid, the

initiation of solidification appears in the body of the liquid and the number of

observed growing crystals is a function of either undercooling or

supersaturation.

· At high cooling rates in multicomponent or structurally complex liquids,

glass formation occurs when the liquid is supercooled to its glass transition

temperature before the onset of crystal formation. In this manner amorphous

solids can be formed.

· During solidification at high cooling rates it is possible to produce metastable

crystalline phases where the first phase to precipitate is not the thermo-

dynamically mos t stable phase. Transition through metastable states to the

stable state can often occur during the solidification process.

· After the initiation of solidification, it is common to observe the formation of

cells and dendrites during the growth phase of solidification.

· Three types of solidification structure are commonly seen in castings:

equiaxed, columnar and eutectic.

· Solidification structures are commonly observed to coarsen during the

solidification process.

· Partitioning of elements (segregation) between the solid and the liquid is

common.

· The dissipation of latent heat th rough the growing shell or into an

undercooled liquid often controls the solidification rate in metals and leads

to recalescence of the liquid and solid in undercooled growth.

Solidification and steel casting

A W C R A M B , Carnegie Mellon University, USA

Growth rate can be controlled by atom attachment at the interface, mass

transfer in the liquid, heat transfer or a mixture of these processes.

Generally there is a volume change during solidification that leads to either

porosity or swelling in castings depending upon the sign of the change.

The process of solidification commonly causes fluid flow due to Marangoni

and natural convection related to thermal and composition gradients induced

during the process.

From the above observations it is clear that in order to understand solidi fication

and casting there is a very complex interplay between many factors. There are a

number of excellent texts on solidification that have been writte n over the last

twenty years

1±10

and this subject continues to develop. In this chapter it is the

intention to give an introduction to our current understanding of the phenomena

that are related to the initiation of solidification and to the factors that cont rol

growth rates. These issues will then be discussed with respect to issues in the

casting of steels.

10.2 Solidification fundamentals

10.2.1 The conditions necessary for solidification

Solidification is the formation of a solid from a liquid or a gas. Solidification

generally occurs during cooling but can also occur at constant temperature by

chemical change of the system. For example, if a pure liquid metal, which is

initially at a temperature above its melting point, is cooled continuously, at some

temperatur e below its equilibrium melti ng point, sol id will form. The

temperature at which solidification actually occurs (T

) when subt racted from

the equi librium melting temperature (T

), is the undercooling (T) necessary

for the initiation of solidification. This phenomenon was first recognized by

Fahrenheit in 1724.

Similarly, if a liquid binary solution is held at constant

temperature and its chemistry is changed by adding or removing one of the

constituents, at some point, when the equilibrium solubility for the constituent is

exceeded, solid will be precipitated. In this case, we will refer to solidification as

being from a supersaturated liquid or gas. From these two examples, it can be

seen that the initiation of solidification normally occurs by undercooling below

an e quilibrium temperature or by su per saturation above an equilib rium

composition.

We can view this process schematically for solutions in Fig. 10.1 where a

partial phase diagram for a binary solution is drawn. If the starting position is A,

then solidification can be initiated by (1) reducing temperature below the

position of the equilibrium liquidus (from A to B) or (2) increasing the

composition at constant temperature past the equilibrium liquidus (from A to C).

While path 1 is well recognized, path 2 is not as obvious; however, if one

considers a binary alloy where the solute readily evaporates at processing

400 Fundamentals of metallurgy

temperatures, the alloy, if held at constant temperature, will naturally follow

path 2. In this diagram the liquidus and solidus curves are, by definition,

calculated using bulk thermodynamic properties or measured during melting

experiments.

Solidification can also occur by chemical reaction. For example, in liquid

metals there is significant solubility of oxygen and sulfur that increases with

temperature. As most casting processes initiate at temperatures significantly

above the liquidus, oxides and sulfides will precipitate during cooling. These

precipitation reactions are also driven by chemical and/or thermal changes.

Figure 10.2 is a schematic of an equilibrium diagram indicating the

conditions for precipitation of C from a solution containing A and B. The solid

and dashed lines are the equilibrium position for A and B in solution and in

equilibrium with pure solid C at temperatures T

and T

(where T

> T

). In this

diagram decreasing temperature stabilizes solid C. If one begins with a solution

of A and B at temperature T

and then one adds B at some concentration above

that indicated by the dashed line, solid C will precipitate and grow until the

concentration of A and B in solution is that given by the diagram, taking into

account the stoichiometry of the reaction. The supersaturation necessary to

initiate precipitation of solid C is related to the position of the equilibrium line

and can be manipulated by decreasing temperature. Thus increasing composition

and decreasing temperature can both lead to a critical supersaturation and the

formation of solid C.

10.2.2 Issues of thermodynamics

In order to understand solidification, we must first understand phase diagrams of

the type schematically shown in Figs 10.1 and 10.2. These diagrams are

10.1 Schematic of a solidification process in a binary solution.

Solidification and steel casting 401

constructed using bulk thermodynamic parameters and denote regions of

stability where bulk parameters are important; however, during solidification,

the first solid to form occurs by atom clustering either within the bulk

(homogeneously ) or on surfaces (hete rogeneously ). In addition, during

solidification, there is formation of a solid that has both volume and a surface.

Thus, if we look at the energy change during the solidification process, we can

immediately see that for homogeneous nucleation of solid within a liquid mass,

then the total change in free energy (G) must be related to the volume free

energy change plus the energy necessary to create the surface:

G  G

 V

 

s=l

 A

10:1

where G

is the free energy change per unit volume created and 

s=l

is the

interfacial energy per unit area between the solid and the liquid. As the volume

free energy can be calculated in two different ways, depending upon whether

temperature or chemical change is driving precipitation, we can write G

follows:

G

 H

v;l!s

ÿ TS

v;l!s



H

v;l!s

T

 ÿ

H

v;s!l

T  S

v;l!s

T 10:2

10.2 Equilibrium diagram for the precipitation of C from a solution containing A

and B.

402 Fundamentals of metallurgy

G

 G

 RT ln Q  RT ln

 ÿRT ln S 10:3

where H

and S

are enthalpy and entropy change per unit volume

respectively of the phase transformation; T

is the equilibrium temperature for

the phase transformation; T  T

ÿ T and is the undercooling; G

is the

standard Gibbs free energy for the reaction; K

is the equilibrium constant for

the reaction; Q is the reaction product calculated using the initial activities of the

components and S is the supersaturation ratio (K

=Q). Thus one can relate G

to the undercooling (T), if we assume that for a small range of undercoolings

that the thermal contribution due to the phase transformation (S  H=T

)

accounts for more than 99% of total change of entropy, or to the natual

logarithm of the supersaturation ratio (S). Equation 10.2 is Volmer's equation

from 1939 and equation 10.3 is that of Guggenheim from 1957.

If we assume that the solid preciptates as a sphere, then equation 10.1 must go

through a maximum at small radii, as the volume term must become smaller

than the area term, as r

< r

, when r < 1. A schematic of this situation is given

in Fig. 10.3 where the maximum free energy is G



and the radius

corresponding to the maximum is r



From equation 10.1, assuming solidification is driven by undercooling, the

critical radius, r



, and the critical free energy, G



can be written as follows:





2

s=l

G

v;s!l



2

s=l

H

s!l

T

10:4

G









G

v;s!l







H

v;s!l

T

10:5

Thus both the critical radius and the critical free energy decrease with increasing

undercooling.

For a reaction driven process, r



and G



can be written as follows:

10.3 Schematic of the total free energy as a function of radius for a sphere.

Solidification and steel casting 403





2

s=l

G

v;s!l



2

s=l

RT ln S

10:6

G









G

v;s!l







RT ln S

10:7

where r



and G



decrease with decreasing temperature and increasing

supersaturation.

The nucleation of a phase was viewed by Gibbs

and Volmer and Weber

as a heterophase fluctu ation in an undercooled or supersaturated phase where

numbers of atoms must cluster together to form the potential starting points for

the phase transformation. In Fig. 10.3, at small radii G is positive and reaches

a maximum as a function of r. Thus if we use Gibbs view of fluctuations , this

maximum represents an unstable condition as, if a fluctuation occurred in the

system that leads to the formation of a sphere of radius r



, then any change to the

size of the droplet due to evaporation or condensation, would lead to the droplet

either diminishing or growing, as both processes would lead to a decrease in the

free energy of the cluster. Thus r



is the smallest radius that has the potential to

increase in size. Equations 10.4 to 10.7 are related to one's assumption of the

geometry associated with the solidification process and would change if one

assumed a different geometric shape and a number of such relationships can be

developed, if one changes geometry.

The most obvious change in geometry is to assume that the solidification

process occurs on a pre-existing surface (heterogeneous nucleation). If one

assumes a spherical cap which forms an equilibrium shape on a pre-existing

surface, then Young's relation can be used to relate the various surface energies

via an equilibrium contact angle () where:



cÿl

 

cÿs

 

sÿl

cos  10:8 

and 

cÿg

is the surface energy between the pre-existing solid and the liquid, 

cÿs

is the interfacial energy between the pre-existing solid and the newly solidified

material and 

sÿl

is the interfacial energy between the newly formed solid and

the liquid. This leads to the following relationship for G



G



het



2  cos 1 ÿ cos 

G



hom

 SG



hom

10:9

The function S varies from 0 to 1 as  varies from 0 to 180. Thus wetting

conditions lead to very small values of G



. Of course, as r



is related to the

shape of the curve (where dG=dr  0), not its maximum value, heterogeneous

nucleation does not change the critical radius; however, it strongly affects the

volume of the critical nucleus and thus the number of atoms that must cluster

together in order to form a critical nucleus:



het



hom

 S 

G



het

G



hom

10:10

404 Fundamentals of metallurgy

In the above discussion, it was assumed that the pre-existing solid was flat;

however, if one assumes that surfaces are rough, the concave areas are more

effective in decreasing the necessary volume for nucleation than either flat or

convex areas, thus heterogeneous nucleation is favored in valleys rather than

hills.

Another interesting issue is to consider solid nucleation at a liquid±liquid

interface

as shown schematically in Fig. 10.4. In this figure the various

interfacial energies are related as shown.

In this case:

G

het

G

hom

 f 

 2 ÿ 3cos 

 cos





1 ÿ cos



1 ÿ cos



 

3=2

2 ÿ 3cos 

 cos





" #

10:11

10.2.3 Effects of size

Another way of looking at this thermodynamic problem is to understand the

effect of surface energy on the equilibrium position of the reaction. The first

attempts to understand this issue was in the formation of liquid droplets from the

vapor where surface energies and contact angles are apparently easy to measure.

It is well known that the effect of surface energy on a droplet is to increase the

10.4 Nucleation at a liquid±liquid interface.

Solidification and steel casting 405