the mathematical modeling revolution in extractive...

The 1987 Extractive Metallurgy Lecture The Metallurgical Society

The Mathematical Modeling Revolution in Extractive Metallurgy

JULIAN SZEKELY

A brief review is presented of the current state of extractive metallurgy, and it is shown that it is still a significant part of the national economy. Then a definition is given of mathematical models, and the general philosophy of modeling is discussed, together with the cost of models, hardware, and software options. Several illustrative examples are given, drawn from aluminum electrolysis, flash smelting, tundish operations, and plasma systems. The paper is concluded with the future modeling tasks facing us; these include the more widespread applications of models to represent both existing and new processing operations. It is stressed that models can play a major role in developing a holistic approach to metals and materials processing, where the primary extraction and refining operations are combined with the final processing steps.

The Extractive Metallurgy Lecture was authorized in 1959 to provide an outstanding man in the field of nonferrous metallurgy as a lecturer at the annual AIME meeting.

JULIAN SZEKELY is Professor of Materials Engineering at the Massachusetts Institute of Technology, a position he has held since 1975. A native of Hungary, he received the B.Sc., Ph.D., and D.Sc. degrees from Imperial College, London, England. His main research interests are in the mathematical modeling of metals and materials processing operations, with emphasis on new technologies, and he has written numerous books and over three hundred journal articles on this subject. His most recent and forthcoming books include The Mathematical and Physical Modeling of Primary Metals Processing Operations, with J. W. Evans and J. K. Brimacombe (Wiley, 1988); Ladle Metallurgy: Theory and Practice, with G. Carlsson and L. Helle (Springer, 1988); and Mathematical Modelling Strategies in Materials Processing, with W. Wahnsiedler (Wiley, to appear in 1989). His work has been recognized by many national and international awards, including the Mathewson Gold Medal (1973), the Extractive Metallurgy Science Award (1973), and the Howe Memorial Lectureship (1979) of the AIME. He received the Sir George Beilby Gold Medal of the British Institution of Chemical Engineers in 1974. In 1982 he was elected to the United States National Academy of Engineering.

I. INTRODUCTION

T O be chosen as the 1987 Extractive Metallurgy Lecturer is a great honor, which is amplified by the many distin- guished colleagues and friends of mine who have been so recognized in the past. These include mentors like the late Denys Richardson E~1 and colleagues including John Elliott, t2J Herb Kellogg, E31 Nick Themelis, t4J Noel Jarrett, f51 and Milt Wadsworth. [6]

Noel Jarrett's lecture, given last year, may serve as an excellent model, because it has dealt with both the economic and societal, as well as the technological, aspects of extractive metallurgy, that of aluminum. Noel has interpreted the term "extractive metallurgy" broadly to include recycling, and has indeed shown that recycling from scrap, rather than the recovery of metal values from their ores, ought to be the main emphasis of our activities-- at least in many instances.

The purpose of this lecture is to discuss the very major advances that have been made in the mathematical modeling of processing operations, and to explore the accom-

METALLURGICAL TRANSACTIONS B VOLUME 19B, AUGUST 1988--525

plishments and the potential for the application of these techniques for both existing and novel technologies in the extractive metallurgy field. It is unrealistic to consider the technological issues in isolation, and for this reason we shall begin by painting a broad-brush picture of the extractive metallurgy field worldwide and in the United States. This will be followed by a brief discussion of mathematical models in a generic sense, with a review of the con- temporary tools that have become available to the modeler. Then we shall present a selection of successful modeling examples and will then conclude with a brief review of some potentially exciting modeling applications.

II. THE STATUS OF EXTRACTIVE METALLURGY

For the purpose of argument, let us define extractive metallurgy as the extraction of metals from their ores, their refining and transformation into saleable semi-finished solid shapes, such as ingots, billets, sheet, plates, bars or slabs. Scrap recycling and the recovery of valuables from solid wastes (e.g., flue dust, electric furnace, dust, etc.) is explicitly part of these considerations.

In recent years it has become quite fashionable to "bury" extractive metallurgy; in the United States, the major losses and attendant restructuring by many of the major metals companies, the corresponding loss of jobs, and the virtual disappearance of R & D functions in many organizations represent a very gloomy picture indeed. The situation has not really been any better for chemical/extractive metallurgy in the United States academic field, with major cutbacks in research support and the markedly diminishing activities in many university departments. Indeed, at present there are very few funding constituencies for extractive metallurgy research. This is in marked contrast to new (exotic) materials development and characterization, which has been very we l l - - a t times lavishly--supported by both Federal agencies and by corporations.

Conventional wisdom has it that both industrial and academic operations should drastically curtail activities in the extractive metallurgy field, and concentrate instead on the more "high value added" materials, exemplified by mono- dispersed ceramic powders, semiconductors, and more recently, superconductors.

While no one can argue with a general trend that seeks to emphasize new materials and new technologies, a plea should be entered for a rational balance between the pursuit of entirely new materials concepts and the more traditional fields, either as a research or as an academic discipline.

This balance should be struck on both intellectual and pragmatic, economic grounds. Intellectually, established fields tend to be rather more rigorous, having undergone the usual weeding-out processes; this rigor and methodology could be helpful, both in research and education. In the economic sphere, as will be shown subsequently, traditional metals processing still has a very significant impact, both in terms of contribution to the GNP and as a source of employment.

Common sense would suggest that a reasonable strategy for the survival, indeed the prosperity, of metals companies would have to include a judicious mix of significantly improving existing operations, from the standpoint of cost and

product quality; and the aggressive development of new markets for the company's products, coupled with a diversification into contiguous areas where use can be made of existing competence.

For any aluminum company, such a strategy may involve effort to capture a larger part of the automobile market to include some engines and body components, as well as the implementation of new processing technologies such as spray forming, ET,SJ rheocasting [9,j~ and compocasting, [H'121 the development of new alloys, and metal matrix com- posites for aircraft applications, as well as a more aggressive stance toward recycling. The marked fluctuations in aluminum price may lead one to question the wisdom of abandoning reduction cell research altogether. In terms of new materials markets, ceramics (alumina, silicon carbide, titanium diboride, etc.), and semiconductors closely related to aluminum, such as silicon and gallium products, may be logical choices.

In the steel area, many Japanese companies have mounted a major effort at diversification into superalloys, titanium products, high-tech ceramics, semiconductors, and the like, drawing heavily on their existing processing expertise. In copper research, Mitsui Mining has taken a similar tack of diversifying into the more high-technology applications of copper products and their derivatives. There are rather fewer similar efforts in the United States.

The statistical information to be provided in the following may provide a sense of proportion regarding these matters.

Tables I, II, and III contain data on the worldwide and United States production of some key metallic materials, both in terms of tonnage and value.

In comparison, Table IV shows corresponding figures for electronic materials. The appropriate figures for "high-tech ceramics," which do include some of the packaging, would be comparable.

We should note, furthermore, that the comparison between the metals and the "high-tech" materials is not totally on the same basis. The metals are clearly of the semi- finished products, while values of both the semiconductor materials and the ceramic components may be significantly inflated over and above the purely material values, since they do include a significant component due to fabrication. Figure 1 shows the gross output by various selected in-

Table I. Annual Production (1986) and Value of Some Base Metals ~t3'~4~

U.S. Production World Production Metal Value Value

Carbon steel 82 M tons 781 M tons $16 to 25 billion $152 to 238 billion

Aluminum 3.5 M tons 17 M tons $3.7 to 5.7 billion $18 to 27 billion

Copper 1.35 M tons 9.1 M tons $0.6 to 1.78 billion $11 to 12 billion

Lead 0.39 M tons 3.8 M tons $0.14 to 0.22 billion $1.4 to 2.1 billion

Zinc 0.24 M tons 7.5 M tons $0.15 to 0.22 billion $4.6 to 4.9 billion

526- -VOLUME 19B, AUGUST 1988 METALLURGICAL TRANSACTIONS B

Table II. Annual Production and Values (1986) of Some Special Metals (Ni, Co, Ti, Si) [13'14!


Nickel 1.1 K tons 799 K tons $3.9 to 7.0 million $2.9 to 5.1 billion

Cobalt - - 35 K tons $0.60 to 0.79 billion

Titanium 17.5 K tons 87 K tons $0.14 billion $0.7 billion

Silicon 0.36 K tons 3.0 M tons $0.45 billion $3.7 billion

Table III. Annual Production and Values (1986) of Some Precious M e t a l s [13'14]


Gold 123 tons 1.7 K tons $1.2 to 1.6 billion $17 to 22.5 billion

Silver 1.2 K tons 13.9 K tons $0.18 to 0.21 billion $2.0 to 2.4 billion

Platinum Palladium Iridium Osmium Rhodium Rutherium

17 tons 278 tons $55 to 268 million $0.89 to 4.3 billion

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Table IV. Electronic Materials

A. World-Wide Ceramics Sales (1986) t~Sj

1986 1991 1995

$5.4 billion $12 billion $24 billion

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1986 1989

U.S. $8.5 billion $12.66 billion World-wide $23.36 billion $39.06 billion

be some important lessons here for the metals community as a whole.

Clearly, there are many key ingredients to a successful metals processing operation; these include management, marketing, and technology. In the area of technology, we have a new key tool, mathematical modeling, which has al- ready produced impressive results and promises a great deal more. In the remainder of this article I shall be concerned with a discussion of mathematical modeling, the techniques, hardware, software, successes, and future possibilities.

dustries in the United States. The very important point that must emerge from these data is the following:

While metals production has been declining in recent years, both in terms of absolute quantities and as a relative proportion of the economy as a whole, primary metals production and processing is still a major component of the national economy and is likely to remain so for the fore- seeable future. The recently experienced currency value re- alignments are likely to strengthen this argument.

One final point ought to be raised here. The late 1980's and early 1990's may be regarded as the era of the niche. Organizations have become successful by being original, and by carving out new markets and territories for them- selves. Following, rather than creating, new trends could become very hazardous, rather than a "safe" procedure. The well-documented success of many mini-mills in steel, particularly Nucor, Chaparral , and North Star S t e e l - - contrasted with the quite hard times facing the integrated producers - -may serve as excellent illustrations. There may

I l l . MATHEMATICAL MODELS: HARDWARE, SOFTWARE, AND M E T H O D O L O G Y

What is a mathematical model? A mathematical model is a set of equations, algebraic or differential, which may be used to represent and predict certain phenomena. The term model as opposed to law implies that the relationships employed may not be quite exact and that the predictions derived from them may only be approximate.

Mathematical modeling has been used for a long time, but in recent years major breakthroughs have been possible because of dramatic developments both in computing hardware and software, which allows for the far more effective manipulation of the modeling equations.

Major successes from everyday life include the very accurate prediction of election results from data based on an extremely small number of votes taken from representative precincts. In contrast, weather forecasting sounds a sobering note that many key problems still remain.


A. The Classification of Mathematical Models t''j

Within the more restrictive framework of extractive metallurgy, we may consider several types of mathematical models, namely:

(1) Fundamental, or mechanistic, models; (2) Empirical, or black box models; (3) Population balance models; and (4) Input-output, systems models.

Fundamental or mechanistic models, which will be the focal point of the discussion here, are based on basic physical or chemical laws such as thermodynamic equilibria, chemical kinetics, heat flow, fluid flow, mass transfer, deformation processing, and the like. These models tend to have a broad general validity and with some ingenuity the scope of application can often be expanded.

Empirical or "black box" models are based on obser- vations on a particular system, and not on fundamentals. At times there is no alternative to their use, but great care must be taken if these relationships are to be extrapolated or generalized.

Population balance models deal with distributed parameters, such as particle size distribution in a grinding circuit, the coalescence of inclusions in steel processing, or the vapor phase synthesis of fine ceramic particles. These tend to be a special class of mechanistic models because they are based on fundamental concepts.

Input-output or systems models satisfy overall conser- vation relationships and find a variety of uses, including cost analysis, scheduling, de-bottlenecking, and the like. At times such models may be effectively combined with a mechanistic modeling approach.

B. The Role of Mathematical Models [18l

Figure 2 illustrates the role of mathematical models in materials processing. It is seen that models, through their representation of processes or operations in a quantitative, mathematical form, may play a key role in process control and process optimization, as well as in the planning and interpretation of measurements. Furthermore, mathematical models may provide a clear link to artificial intelligence-- perhaps better termed knowledge-based engineering.

Briefly commenting on these, both process optimization and process control require a quantitative representation of the process. In optimization we seek to maximize the profit

Fig. 2 - - T h e role of mathematical models in materials processing.

Plant and Physical Modeling Pilot Plant Data

Fig. 3 - - T h e balance between mathematical modeling, physical modeling, and plant-scale experimentation.

or minimize the production cost; in process control* we

*Process control represents a special problem, in that the calculations have to be carried out very rapidly so that a meaningful response is possible in real time. For this reason, control algorithms often require a drastic simplification of a mechanistic model.

need a quantitative relationship for the control algorithm. The planning and interpretation of experimental mea-

surements is perhaps one of the most important and often most misunderstood role of mathematical models. Very often, the measurements are made first and then some form of modeling is undertaken as an afterthought. It is important to realize that unless the measurements are properly interpreted, and this interpretation leads to appropriate conclu- sions, the whole effort may turn out to be useless. For this reason, it is almost always desirable to do some calculations (i.e., modeling) first, and then use this as a basis both for planning the measurements and interpreting the data being obtained.

It is important to stress that calculations and measurements are not alternatives, but most often must be pursued in a complementary fashion as illustrated in Figure 3, indicating that mathematical modeling, physical modeling, and actual (plant-scale) measurements may all be ingredients of a successful program. Indeed, in many instances several iterations may be required between mathematical modeling and physical measurements before the desired level of understanding finally emerges.

C. How Are Mathematical Models Built?

Figure 4 shows a schematic flow chart of mathematical model development. It is seen that the first (and usually the most difficult) step is the identification of the problem; processing problems are hardly ever presented to the in- vestigator in a neatly defined form. Most often one has to "dig deep," form a mental picture of the key parameters affecting the system, and then express this physico-chemical picture in mathematical form.

Once the problem is formulated, the next step is scaling, scoping, and order-of-magnitude analysis. This technique, which has frequently been used by applied mathematicians, is a very powerful one, providing useful insight into the behavior of the system. Through order-of-magnitude analy-

MATHEMATICAL MODELING

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528--VOLUME 19B, AUGUST 1988 METALLURGICAL TRANSACTIONS B

~ o b l e m ~ " ' ~

Scoping, Scaling Asymptotic Solutions

Numerical ~).,~_ -1~( Experiments Solutions .,

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Fig. 4--The general methodology of mathematical model development.

ses, we can define the time scale of the problem, that is the time required to approach thermal or chemical equilibrium; and the velocity scale, that is the typical values of the fluid velocity and the like. Furthermore, this preliminary scoping analysis is extremely useful in defining the range of parameters that must be covered in the subsequent computational work.

As seen in the figure, the next two parallel steps are machine computation and experimental work. Machine computation is often necessary because purely analytical results or order-of-magnitude estimates will not provide adequate detail. Experimental work will be needed, both for

testing the theoretical predictions and also for providing property values (viscosity, thermal and electrical conduc- tivity, reaction rate constant, and the like), should these not be available from prior work.

The following key step is the synthesis, that is the development of a quantitative understanding of the system, upon which the subsequent implementation may be based. This implementation may involve a decision whether to proceed with a development project, the major modification of an existing operation, the installation of a new control system, and the like.

The key point that one must re-emphasize regarding this flow chart is that once a problem has been identified, any investigation should start with (simplified) calculations first, and then use this as a framework for planning experimental and computational programs to be pursued in an interactive manner.

D. The Building Blocks of Mathematical Models

How are mathematical models put together? In the early days of modeling, most of the models were "custom- crafted," often starting from first principles and the consideration of elementary control volumes. The present-day modeler is in a much stronger position, because he or she can rely on:

(a) many precedents and analogous situations; (b) readily available and inexpensive hardware; and (c) a wide range of software packages.

At present, the effective path in the development of mathematical models relies on two key components:

(i) The actual building blocks of models, which are the basic physical (including mechanical) and chemical laws; some of these are summarized in Table V. (ii) Drawing on the analogy to previous work, through the consideration of some basic physical situations. Some standard, relatively well-studied physical situations are summarized in Figure 5, while some less well understood, but still very important systems are shown in Figure 6.

Commenting on these briefly, the fairly well-studied systems include stirring of liquid metals with a vertically injected gas stream, with applications in ladle metallurgy, the BOF, combined blowing, gas phase deoxidation of molten

Table V. The Building Blocks of Mathematical M o d e l s tl3a41

Component Application Remarks

Navier-Stokes equations Fourier's Law

Fick's Law

Convective transport Maxwell's equations

Thermodynamics Kinetics Constitutive relationships

fluid flow heat conduction

diffusion

heat and mass transfer electrodynamics MHD

equilibria phase diagrams rate predictions deformation processing

complex vector-tensor manipulation relatively simple, more complex if phase change is involved simple for one component, very complex for multi-component systems combines fluid flow, heat and/or mass transfer very complex problems, i.e., plasmas, induction furnaces, etc. often routine calculations needs experimental input very complex, needs realistic input


Fig. 5 - Some well-studied basic physical situations in extractive-process metallurgy. (a) Vertical gas injection into melts. (b) Horizontal gas injection into melts. (c) Gas jet impinging onto a melt. (d) Induction melting, stirring. (e) Submerged resistance, arc melting, smelting. ( f ) Gas flow through packed beds.

E. Hardware

In recent years there have been major developments regarding computing hardware. As shown in Figure 7, the cost of doing a given calculation has dramatically decreased, so that we can now routinely tackle complex three-dimensional transient heat and fluid flow problems. This figure also shows that in terms of instructions/second �9 dollar one tends to get rather better value from the smaller machines than from the mainframes.

Table VI shows a listing of the principal hardware possibilities, together with brief notes about costs and principal applications. This is a very rapidly changing field, so this table will have to be updated periodically.

The following general remarks may be applicable here. The distinction between the "high end" of personal computers (e.g., 386-based machines or the Mackintosh II) and the low end of engineering workstations is becoming quite blurred.

At the present cost level, it is hardly appropriate to use personal computers for serious engineering calculations. The bulk of this work is now being done on engineering workstations, of which the MicroVAX II and III, the Sun III and IV,

Fig. 6 - - Some less well-understood basic physical situations in extractive- process metallurgy. (a) Atomization of a liquid metal stream with a gas jet. (b) Mechanical stirring of liquid metals. (c) Mold filling. (d) Emulsifi- cation at a slag-metal interface. (e) Surface waves with liquid metals.

copper, and the like (a). Horizontally injected gas streams into liquid metals (mattes) include the AOD process copper converting and the Noranda Process (b). Gas jets impinging on liquids include basic oxygen steelmaking (c), while induction stirring and melting is extensively used in both the steel and the aluminum industries (d). Submerged resistance and arc furnaces are extensively used in the extractive metallurgy of ferroaUoys and in scrap melting (arcs) (e). Finally, packed beds are used in blast furnace operations, while fluidized beds are used in the roasting of copper sulfides and in the flash drying of alumina (f).

Among the less well-understood systems, the atomization of liquid metal streams with gas jets is a key step in powder metallurgy (a). Mechanical stirring (quite well understood in the contexts of chemical engineering practice) is being used in aluminum and lead processing (b). Mold filling (c) is common to most nonferrous operations. The emulsifica- tion of slag-metal systems is a key desired step in ladle metallurgy and highly undesirable in seeking to avoid copper losses in converting (d). Finally, surface waves are important (and not well understood) in the context of aluminum smelting and steel tundish operations (e).

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Table VI. Principal Hardware Possibilities and Applications

Price Range

I. Workstation class: $20 to 100 K MicroVAX II DEC 8250, 8350 Sun 3, 4

ti. Minicomputers: $30 to 200 K MicroVAX III DEC 8500, 86800 MASSCOMP 5000 Intel iPSC Sugarcube

III. Mini-supercomputers: $300 to 800 K DEC VAX 8700, 8800 Convex Intel iPSC/2-VX/d3

IV. Mainframe class: $500 to $2,000 K IBM 3808, 3090 Cyber 875 FPS Array Processors Intel iPSC/2-VX/d4

V. Supercomputers: several millions Cray -1 and X-MP of dollars Cyber 200 Fujitsu VP400 Intel iPSC/2-VX/d5

VI. Massively parallel processors: several millions Intel iPCS/2-VX/d6 of dollars

and the Apollo DN 3000 series are good, but certainly not all-inclusive examples.

Mainframe machines are, of course, being used quite extensively for computing by major organizations, but the purchase of new standard mainframes for doing engineering/ modeling work is becoming a rather less attractive proposi- tion in many cases. Of course, these machines are still domi- nant in accounting, banking, inventory-control applications and the like, although networked workstations may represent a useful alternative in certain cases.

The new and emerging superstars regarding hardware are the vector array and parallel processing machines, offering tremendous computational speed for suitably posed problems. The vector processing mainframes (e.g., those made by Cray Research, Floating Point Systems, or IBM) are very expensive. However, there is a new class of machines with the potential of offering vector array or parallel processing capability at a quite modest cost. The principal impediment at this point is the adaptation of the existing programs to these new machines, and the lack of a suitable preprocessing capability to do this task automatically.

E Software

One of the key developments in recent times is the ready availability of software to perform numerous computational tasks that would be part of a modeling operation. As shown in Table VII, software packages are available to do thermodynamic equilibrium calculations, solve heat flow and fluid flow problems, solve heat conduction and thermal stress problems, and even tackle problems with free boundaries. The intelligent use of these available software packages, with suitable modification, has to be part of most rational modeling strategies.

Table VII. Partial List of Available Software

Name Application Notes

A N S Y S structural, thermal, finite element electrical, magnetic

ABAQUS structural, thermal, finite element nonlinear analyses

NASTRAN structural finite element FIDAP fluid dynamics finite element PHOENICS fluid dynamics, heat finite volume

transfer, chemical reaction

FLOW 3-D fluid dynamics finite difference FLUENT fluid dynamics, finite difference

heat transfer NEKTON fluid dynamics, finite element

heat transfer (spectral) NISA fluid dynamics, finite difference

heat transfer IMSL math/statistical library LINPACK math library MINPACK minimization/optimization

library EISPACK eigenvalue library ASPEN flowsheet simulation sequential

modular FLOWTRAN flowsheet simulation SOLGASMIX chemical equilibrium ECES aqueous chemistry F*A*C*T phase diagrams,

chemical equilibrium ROMULUS solids modeling

PATRAN solids modeling, finite element pre- and post- processing

SLAM II event simulation MAGNUM magnetics finite element

G. What do Mathematical Models Cost?

The cost of mathematical models may range from a low of about $20,000 to $30,000 for a simple exercise, to many millions of dollars for a really complex model. A mean value in the range of about $120,000 might provide a first estimate of a reasonably complex model that might take about one man-year to develop. The principal cost item is the time of the programmer and the systems analysts (in the region of 80 pct or more of the total cost); thus the cost of the hardware and the software tends to be a relatively minor part. This is a fact often not appreciated by technical management, who too often indulge in false economies by not investing in appropriate hardware and software, thus making the analysts' work unduly protracted. In deciding whether to undertake a modeling exercise, the cost of the model will, of course, have to be weighed against the benefits that may be derived; by the same token, one must determine the proportion of the effort that has to be devoted to the complementary physical modeling, pilot plant work, and plant- scale experimentation.


H. What Constitutes an Optimal Modeling Strategy?

In tackling a practical problem, a decision has to be made regarding the extent to which one is to use mathematical modeling, physical modeling, pilot plant and plant-scale work. In the majority of cases, as was illustrated in Figure 3, we may need a combination of these approaches. We should note that pilot-plant scale work and plant-scale experimentation both tend to be very expensive, so if possible one should emphasize mathematical and physical modeling work.

IV. SOME ILLUSTRATIVE EXAMPLES

During the past decade we have seen a very marked evolution in modeling efforts. Far from being a scientific curi- osity, successful modeling examples now fill the materials and metallurgical literature. Indeed, one is faced with an embarrassment of riches, and any selection has to be somewhat arbitrary. We shall confine ourselves to four specific examples: two from the nonferrous metallurgy field, one from steel processing, and one from a new technology.

A. Aluminum Electrolysis

As was discussed earlier, over five million tons of aluminum are being produced annually, and all the primary processing involves the use of a process whose principles were established over a hundred years ago: the electrolytic decomposition of alumina, dissolved in a molten salt, using a graphite anode and a graphite cathode, and collecting the aluminum product in the molten state. During the past hundred years, major advances have been made by vastly in- creasing the scale of the operation, the electric efficiency, and also the cell productivity. Mathematical modeling has played an important role in these accomplishments.

Figure 8 shows a schematic sketch of a typical section of a Hall cell. It is seen that carbon anodes are being immersed into a molten bath that consists of alumina dissolved in cryolite. The electrolytic decomposition of the alumina yields the desired aluminum product. The key issues in Hall cell design and operation are to increase the cell productivity and electric efficiency, and to reduce the specific power con- sumption. One very important objective is to reduce the anode-cathode distance and hence the cell voltage. Two major impediments must be overcome in order to accom- plish this.

One of these is that, unlike the idealized sketch shown in Figure 8, the bath-metal interface is not flat and tends to be unsteady. It follows that short-circuiting may occur if the anode and the cathode are brought too close together without effectively controlling the surface deformation and the surface waves.

The other point is that any Hall cell must be operated so that a solidified shell of the bath is formed on the side walls; this "freeze" plays an important role in protecting the walls. In order to maintain this "freeze," the resistive (Joule) heating due to the passage of the current between the electrodes has to be matched with the heat loss through the walls.

From the standpoint of transport fundamentals, Hall cell operations represent a very complex, three-dimensional, unsteady state heat flow, fluid flow, and mass transfer problem in a multi-phase (bath, metal, and gas bubbles) system. The situation is further complicated by the presence of strong electromagnetic forces and very complex geometries.

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It follows that the optimization of the system will require a good detailed, quantitative understanding of the complex interrelationships between the process parameters. A great deal of useful work has been done in this area by Evans []9J and Moreau, [2~ and there must be much unpublished material in the archives of aluminum companies. Here we shall cite the very recent, excellent work of Wahnsiedler, t2'l with Figure 9 showing a computed bath-metal interface, and Figure 10 depicting the computed velocity fields. As noted earlier, these calculations require the solution of a set of simultaneous three-dimensional electromagnetic force field and fluid-flow field equations. Finally, Figure 11 shows the transient response of the interface height to an interruption in the anode current.

Work of this type has been very helpful in both the optimization of existing cell operations and in the development of new cell technologies.

B. Modeling of the Flash Smelting Process

The classical technology for smelting sulfide copper ores involved the use of large, unwieldy reverberatory furnaces with an extremely low surface area per unit volume and attendant very slow processing rate in terms of tons/unit reactor volume-time. One potentially attractive technology, pioneered by the Outokompu Company of Finland, is the flash smelting process sketched in Figure 12, where fine copper sulfide particles (together with a flux) are injected into a furnace with oxygen-enriched air, forming a dust-laden turbulent jet.

The inherent attractiveness of this system is due to the fact that full advantage is being taken of the very high surface- to-volume ratio of the fine ore particles; thus, a fast specific processing rate may be achieved. At the same time, for the proper design and scale-up of these units it is essential to have information on the temperature and concentration profiles and of the local reaction rates.

The accurate modeling of this system is quite a complex task, because it involves a reacting, two-phase gas-solid mixture with several components. Convective and radiative heat exchange is another complicating factor.

Useful modeling efforts in this field have been done by Themelis and co-workers [22! and by Sohn and his group. [23l

Here we shall cite the work of Hahn and Sohn, which is the most comprehensive to date. Figure 13 shows an outline of the concentration isopleths, while Figure 14 demonstrates excellent agreement between the theoretical predictions and


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Fig. 9 - - T h e computed bath-metal interface in a Hall cell, after Wahnsiedler. tLgt Reproduced by permission.

the experimental measurements. Work of this type is in- valuable in scale-up, and is an excellent illustration of how mathematical modeling can aid in the development of new technology.

C. Tundish Design Calculations

The third example is taken from steel processing, and may serve to illustrate how modeling can help to improve existing technologies, and also lead to the development of new concepts.

Figure 15 shows a schematic sketch of a tundish serving as a buffer vessel between the ladle and a continuous casting machine. Such tundishes are also used in aluminum, copper, and other metals processing operations, albeit on a different scale. A common feature of most conventional tundishes employed in steel technology is that they are essentially a horizontal trough, say a few meters long, about a meter wide and a meter deep. Steel is being poured in from a ladle at one end or at the center, while one or more steel streams are being discharged into the mold of a continuous casting machine through exit nozzles located at the bottom.

The originally envisioned function of tundishes was to act as a buffer between the ladle and the continuous casting machine, evening out the flow changes brought about by variations in the metal head in the ladle. Subsequently it

was discovered that tundishes can play a very important role in promoting the flotation of inclusion particles, and also in affecting the surface quality of the continuously cast products.

In recent years, a great deal of work has been done in mathematically modeling the three-dimensional fluid-flow field and associated temperature field, tracer dispersion, and inclusion behavior in tundishes, t24-27j Such mathematical modeling has usefully complemented physical modeling t28J and a very limited number of plant-scale investigations. Most tundishes employed today use dams, weirs, and baffles to regulate the flow. But there is still a surprising number of very poor tundish designs in current operation, much to the detriment of product quality. Mathematical modeling can play a key role in determining the optimal tundish design and the optimal location of the weirs and baffles.

Figure 16 shows the computed velocity fields in a central vertical slice of a typical steel tundish, the key parameters of which are listed in Table VIII. Here, (a) depicts the flow pattern in the absence of flow-control devices, and a strong recirculating flow field occupying a significant part of the tundish is clearly seen. (b) shows the effect of a dam and a weir in controlling the flow, and the improvement is readily apparent. Finally, (c) illustrates the effect of placing a strong magnetic field on the tundish, which is seen to effectively streamline the flow.

METALLURGICAL TRANSACTIONS B VOLUME 19B, AUGUST 1988 - - 533

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(a)

Fig. 1 0 - - T h e computed velocity field in the metal pad of a Hall cell, after Wahnsiedler. u91 Reproduced by permission.

This behavior is perhaps more readily discerned using the concept of tracer dispersion or the "C" curves given in Figure 17. These represent the concentration of a tracer which has been added to the inlet stream in a stepwise manner at the outlet, as a function of the dimensionless residence time; the latter is defined as the ratio: tundish volume/volumetric flow rate of the metal.

It is seen that in the absence of flow control devices there is very marked short-circuiting and by-passing, as evi- denced by a sharp tracer peak at 0 ~ 1. This is somewhat ameliorated by the use of inserts or baffles, but plug flow would be approached only through the use of an externally imposed magnetic field.

D. Plasma Systems

Figure 18 shows a sketch of two types of plasma torches used in materials processing applications.[291 Here, (a) represents a nontransferred arc, where an arc is being struck between two electrodes and the resultant Joule heating and electromagnetic forces produce a stream of hot gas with temperatures up to 15 kK, which is then used to heat and melt metallic substances. Such torches are also used for spraying and for the synthesis of fine ceramic particles.

The sketch in (b) represents a transferred arc plasma, where the arc is struck both between the electrodes, and also between the central electrode and a metal surface or bath upon which the plasma impinges. Plasmas offer very inter- esting opportunities in extractive metallurgy, through providing an electrically derived heat source of very high temperature and intensity. Initially, there were many ideas regarding the use of plasmas that ranged from iron oxide reduction and the smelting of sulfide ores, and the recovery of zinc from electric furnace wastes, to the melting of titanium scrap, stainless steel scrap, and the like.

The consensus that is emerging, and which has been well articulated in a recent NMAB report, t3~ is that the ideal use of plasmas in an extractive metallurgy context will be for the melting of refractory metals of high specific value (e.g., titanium, stainless steel scrap, etc.) where emphasis is on product cleanliness; where there is an absolute need for a high-temperature (say above 1800 to 2000 K) heat source; and where a very high thermal efficiency is not mandatory.

The use of plasmas for the more conventional smelting applications (e.g., iron oxide reduction, lead smelting, etc.) seems to be rather less promising; however, there may be some niche-type applications, especially in connection with recycling.


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Fig. 10 Cont . - -The computed velocity field in the metal pad of a Hall cell, afierWahnsiedler, t~9~ Reproduced by permission.

In recent years there have been major advances in the modeling of plasma systems, including both transferred and nontransferred arcs . 131-351 These modeling exercises have

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Fig. l l - - T h e computed transient response of the bath-metal interface in a Hall cell after a current interruption, after Wahnsiedler.t'gJ Reproduced by permission.

been successful in that the theoretical predictions were found to be in good agreement with experimental measurements. As an example, Figure 19 shows a comparison between the theoretically predicted and the experimentally measured temperature fields in a nontransferred a r c . [34]

Figure 20 shows a comparison between measurements and predictions in a transferred arc. [35] The agreement in both cases appears to be quite good, indicating significant progress in this field.

Our ability to model these plasma systems provides an excellent base upon which to build new materials- processing technology.

V. DISCUSSION

In this paper, we have sought to make the point that mathematical modeling is being established as an extremely valuable and cost-effective tool for both process optimization and new process development. The deployment of this technique is very timely because of the ready availability and marked decrease in the cost of both hardware and software.

During the past few years, major successes have been reported with modeling, not only in the mechanistic model-


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Fig. 12--A schematic sketch of the flash smelting process, after Sohn. (23] Reproduced by permission.

ing area--the topic of this discussion--but also with systems modeling and the representation of distributed pa- rameter systems. Indeed, in a recent plenary lecture, Herbst ~36j has shown that the modeling and optimization of com- minution circuits has provided some 15 to 25 pet increase in capacity, without any additional capital investment in the majority of cases.

What Are the Tasks Ahead?

The tasks ahead fall in two categories:

A. The implementation of modeling schemes for the optimization of existing processing operations.

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AXIAL DISTANCE (m)

Fig. 13--The predicted contours of the SO2, 02 concentration and of the temperatures for a flash smelter, after Sohn.t23J Reproduced by permission.

B. The application of modeling techniques in new process development.

A. Implementation of Existing Processes

During the past decade a very impressive number of models has been developed, representing a broad range of basic physical situations. As noted earlier, these include induction furnaces, submerged resistance furnaces, gas bubble-stirred systems, submerged jets, packed beds, fluidized beds, particles entrained in gas jets, and laminar and turbulent flow- through vessels, troughs, and the like.

There are many situations where these existing models could be readily adapted in a most cost-effective manner from one system to another. Thus the knowledge of turbulent flow phenomena in ladles holding molten steel, a well-explored field, could be easily used in improving analogous processing operations in aluminum, lead, zinc, and copper processing. The knowledge gained in the induction melting of one kind of metal could be readily applied to the treatment of many others. By the same token, the very extensive experience gained in blast furnace technology could and should be readily adapted to counter current moving bed-type operations in nonferrous metals processing.

An excellent recent example that may be cited is the CREM Process invented by Charles Viv~s , [37] that is the


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electromagnetic stirring of continuously cast aluminum slabs and billets. While electromagnetic stirring has been used in the steel industry for some time, elegant calculations coupled with physical modeling have led to the application of these concepts in a rather new form, in the production of cast aluminum products of very high surface quality.

INLET STREAM

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Fig. 16--The computed velocity field in the central vertical field of the tundish: (a) in the absence of flow control arrangements, (b) with a dam and a weir insert, and (c) when a 3-kilogauss magnetic field is imposed on the system in the horizontal direction, perpendicular to the main direction of the flow.

It is perhaps paradoxical, but mathematical modeling has been used to a much greater extent in the traditional metals processing fields than in the "high-tech" operations. Thus heat and fluid flow in steel tundishes is well understood, but the largely analogous problems, concerning flow in titanium-

Table VIII. Principal Input Parameters for the Tundish Calculations

Working fluid steel Tundish length 6.79 m Tundish width 0.65 m Melt height 0.75 m Inlet stream velocity 7.67 m/s Inlet nozzle diameter 54 mm Outlet nozzle diameter 54 mm Ratio of inclusion particles to fluid density 0.5


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melting hearths or superalloy-pouring systems supplying atomizing nozzles, have hardly received any attention.

The main barrier to the implementation of these concepts tends to be the nonavailability of suitably trained personnel (even the most user-friendly computational package requires experienced users in this field!). Another key problem is the lack of appreciation of these techniques by senior management, many of whom had their professional engineering careers in the pre-computer age. One should stress here that there are many outstanding opportunities here, with a very impressive potential for return on investment.

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Fig. 1 9 - - A comparison between the experimentally measured and the theoretically predicted temperature profiles in a nontransferred arc system.

B. The Use of Models in New Process Development

It is axiomatic that as the United States metals producers are in the commodity business, paying United States wages, they will not be able to survive unless they are the technology leaders, or at least among the technology leaders, in the world. This mandates the need for a much more rapid rate of developing new products and new processes than has been the case in the past.

There is a critical need to promote recycling (the aluminum companies have been quite successful here), the recovery of valuable by-products from waste streams and perhaps most important of all, the development of new markets and new product lines. Mathematical models can have a major effect during the whole sequence of the development process. Initially, models can help in assessing the potential feasibility of a concept, then will provide guidance regarding the measurements that have to be taken and their interpretation, and finally in the scale-up process.

Fig. 18--Typical examples of plasma torches: (a) nontransferred arc, (b) transferred arc.

Fig. 2 0 - - A comparison between the experimentally measured and the theoretically predicted temperature profiles in a transferred arc system.


An excellent example of successful modeling of new technologies has been the development of Allegheny Ludlum Steel's continuous strip casting process, where intelligent calculations have paved the way toward a radically new technology, t38j

Metallurgy and the materials field are unique in engineering, because ultimately we have to be concerned with the structure and the properties of the final solid product. It is this structure and property relationship that drives both the extraction and the refining processes, because of the key role played by specifications for purity, internal structure, and composition (as well as the end processing sequence) for the final product.

The interdependence of these processing operations is clearly illustrated in Figure 21. It is seen that the nature of the raw material, i.e., virgin ore, recycled scrap, or material recovered from waste materials and the actual refining will have a marked effect on the composition, structure, and properties of the finished products.

As an example, using present-day technology, many of the impurities, such as copper, lead, and zinc contained in "normal" steel scrap, will rule out the production of high- grade sheet products. Starting with virgin ore and limiting the use of "bought" scrap, with the proper technology, will enable one to produce very high quality end products. In- deed, as demonstrated by Emi, 1391 using sophisticated ladle metallurgy techniques it is possible to produce steel that contains less than 50 ppm of all other impurities. Such steels would have extremely good "superplastic" formability.

At the same time, we should note that certain secondary processing techniques, such as rapid solidification and spray forming, might allow one to "get away" with higher im- purity contents than the traditional processing and still yield high-quality products. While the recycling of aluminum cans is very attractive from the standpoint of energy conser- vation, the impurities contained in the cans may interfere with the formability of the products, so there is a need either to change the initial can composition, to impose an additional refining step in the melting sequence, or to modify the forming step.

In the past, artificial barriers have been erected between the scientists and engineers concerned with the structure and properties of the finished product, and those whose main

interest was in metals extraction and refining, that is the manufacture of the semi-finished products, slabs, billets, rods, or bars. This barrier is not only artificial and unneces- sary, but also positively harmful. The major challenges facing us need a holistic approach to these problems, fully integrating the "primary," "secondary," and the finishing operations, in order to arrive at the optimal strategy. Mathe- matical models can play a key role in bringing about such an approach.

ACKNOWLEDGMENTS

The author wishes to thank his numerous students, past and present, who played a key role in doing some of the work that has been cited, and who are now leading major new initiatives in the modeling fields. Thanks are also due to the various Federal agencies that have generously supported this work over the years; particularly noteworthy in the field has been the support of NASA (NAS3-24620, NAS3-24642, NAG3-594, NAS3-25074, NAG3-808) for our electromagnetically driven flow work, and the DOE (DE-FG02-85ER-13331) for our plasma program. Dr. W. Wahnsiedler and Professor H. Y. Sohn kindly supplied the original material used in Figures 8 through 11 and 12 through 14, respectively. Morris Cohen, John E Elliott, and Noel Jarrett kindly commented on the draft of the manu- script, but the responsibility for the opinions expressed here is solely the author's.

REFERENCES

1. F.D. Richardson: 1971 TMS-AIME Extractive Metallurgy Lecture, Metall. Trans., 1971, vol. 2, pp. 2747-56.

2. J.F. Elliott: 1976 TMS-AIME Extractive Metallurgy Lecture, Metall. Trans. B, 1976, vol. 7B, pp. 17-33.

3. H.H. Kellogg: 1966 TMS-AIME Extractive Metallurgy Lecture, Trans. TMS-A1ME, 1966, vol. 236, pp. 602-15.

4. N.J. Themelis: 1972 TMS-AIME Extractive Metallurgy Lecture, Metall. Trans., 1972, vol. 3, pp. 2021-29.

5. N. Jarrett: 1986 TMS-AIME Extractive Metallurgy Lecture, Metall. Trans. B, 1987, vol. 18B, pp. 289-313.

6. M.E. Wadsworth: 1969 TMS-AIME Extractive Metallurgy Lecture, Trans. TMS-A1ME; 1969, vol. 245, pp. 1381-94.

7. S. Safai and H. Herman: in Treatise on Materials Science and Tech- nology, Academic Press, New York, NY, 1987, vol. 20, pp. 183-214.

/ IDI = Dec is ion Points

Fig. 21 - -The metals and materials processing sequence, also showing key decision points.


8. E.J. Kubel, Jr.: in Advanced Materials and Processes incl. Metal Progress, 1987, vol. 132, pp. 69-80.

9. A.C. Fonseca de Arruda and M. Prates de Campos Filho: in Proc. Solidification Technology in the Foundry and Casthouse, Metals Society, London, 1980, p. 143.

10. M.C. Flemings, R. G. Riek, and K. P. Young: Materials Science and Engineering, 1976, vol. 25, p. 103.

11. A. Mortensen, J.A. Cornie, and M.C. Flemings: Proc. Conf. Mechanical Properties of Aluminum Castings, American Foundry- men's Society, May 27-28, 1987, Rosemont, IL, 1987.

12. K.K. Chawla: Composite Materials: Science and Engineering, Springer-Verlag, New York, NY, 1987.

13. Mineral Commodity summaries, U.S. Bureau of Mines, 1987. 14. American Metal Market, several issues in 1987 and 1988. 15. Ceramic Bulletin, 1987, vol. 66(12), p. 1698. 16. Semiconductor International, Oct. 1987, p. 100. 17. J. Szekely, J.W. Evans, and J.W. Brimacombe: The Mathematical

and Physical Modeling of Primary Metals Processing Operations, John Wiley, New York, NY, 1987.

18. J. Szekely and W. E. Wahnsiedler: Mathematical Modeling Strategies in Materials Processing, John Wiley, New York, NY, 1988, in press.

19. S.D. Lympany and J. W. Evans: Metall. Trans. B, 1983, vol. 14B, pp. 63-70.

20. R.J. Moreau and J.W. Evans: J. Electrochem. Soc., 1984, vol. 131(10), pp. 2252-59.

21. W.E. Wahnsiedler: in Proc. TMS-A1ME Extractive Metallurgy Symp. on Mathematical Modelling of Materials Processing Operations, Palm Springs, CA Nov. 29-Dec. 2, 1987, J. Szekely, L.B. Hales, H. Henein, N. Jarrett, K. Rajamani, and I. Samarasekera, eds., The Metallurgical Society, Warrendale, PA, 1987, pp. 643-70.

22. Q. Jiao, L. Wu, and N. J. Themelis: in Proc. TMS-A1ME Extractive Metallurgy Symp. on Mathematical Modelling of Materials Processing Operations, Palm Springs, CA, Nov. 29-Dec. 2, 1987, J. Szekely, L. B. Hales, H. Henein, N. Jarrett, K. Rajamani, and I. Samarasekera, eds., The Metallurgical Society, Warrendale, PA, 1987, pp. 835-58.

23. Y.B. Hahn and H.Y. Sohn: in Proc. TMS-A1ME Extractive Metal- lurgy Symp. on Mathematical Modelling of Materials Processing Operations, Palm Springs, CA, Nov. 29-Dec. 2, 1987, J. Szekely, L. B. Hales, H. Henein, N. Jarrett, K. Rajamani, and I. Samarasekera, eds., The Metallurgical Society, Warrendale, PA, 1987, pp. 799-834.

24. O.J. llegbusi and J. Szekely: unpublished research, 1987. 25. K-H. Tacke and J.C. Ludwig: Steel Research, 1987, vol. 58(6),

pp. 262-70. 26. Y. Sahai and R. Ahuja: Ironmaking and Steelmaking, 1986, vol. 13,

pp. 241-47. 27. J. Szekely and O.J. Ilegbusi: The Mathematical and Physical Mod-

elling of Tundish Operations, Springer-Verlag, New York, NY, 1988, in press.

28. E Kemeny, D.J. Harris, A. McLean, T.R. Meadowcroft, and J. D. Young: Proc. Second Process Technology Conference, Continuous Casting of Steel, Chicago, Feb. 23-25, 1981, pp. 232-45.

29. W.C. Roman: in Proc. MRS Symp. on Plasma Processing and Syn- thesis of Materials, J. Szekely and D. Apelian, eds., 1984, vol. 30, pp. 61-76.

30. National Materials Advisory Board, Report of the Committee on Plasma Processing of Materials, NMAB, Commission on Engineering and Technical Systems, National Research Council, Publication NMAB-415, Washington, DC, National Academy Press, 1985.

31. H.A. Dinulescu and E. Pfender: J. Appl. Phys., 1980, vol. 51, pp. 3149-57.

32. J. McKelliget, J. Szekely, P. Fauchais, and M. Vardelle: J. Plasma Chem. and Plasma Proc., 1982, vol. 2(3), p. 317.

33. A.H. Dilawari and J. Szekely: J. Plasma Chem. and Plasma Proc., 1987, vol. 7(3), pp. 317-39.

34. A.H. Dilawari, J. Szekely, J. Batdorf, and C. B. Shaw, Jr.: unpublished research, 1988.

35. J. McKelliget and J. Szekely: Metall. Trans. A, 1986, vol. 17A, pp. 1139-48.

36. J.A. Herbst: in Proc. TMS-A1ME Extractive Metallurgy Symp. on Mathematical Modelling of Materials Processing Operations, Palm Springs, CA, Nov. 29-Dec. 2, 1987, J. Szekely, L.B. Hales, H. Henein, N. Jarrett, K. Rajamani, and I. Samarasekera, eds., The Metallurgical Society, Warrendale, PA, 1987, pp. 23-47.

37. C. Viv~s and B. Forest: Light Metals, 1987, pp. 769-78 and 779-84. 38. R.K. Pitier: in Proc. Nicholas J. Grant Symposium on Processing and

Properties of Advanced High-Temperature Alloys, MIT, Cambridge, MA, June 16-18, 1985, S. M. Allen, R. M. Pelloux, and R. Widmer, eds., Columbus, OH, ASM, 1985, pp. 1-10.

39. T. Emi and Y. Iida: Proc. SCANINJECT II1, MEFOS, Lule~, Sweden, 1983.


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