Scandinavian Journal of Metallurgy 2003; 32: 33–36 Copyright C© Blackwell Munksgaard 2003Printed in Denmark. All rights reserved SCANDINAVIAN

JOURNAL OF METALLURGYISSN 0371-0459

Scaling laws and instabilities in electric field-enhancedsmelting and refining of steel

Adam Powell1 and David Dussault2

1MIT, Cambridge, MA, USA; 2Aerodyne Research, Billerica, MA, USA

In electric field-enhanced smelting of iron from slag, liquidiron formation at the cathode exhibits a Mullins–Sekerka in-stability, leading to the growth of fingers and enhancing thereaction rate. However, the liquid state of the growing ironphase leads to the expression of this instability that is qual-itatively different from the classical example of growth ofdendrites in solidification. As the molten iron fingers growinto the slag, there is a surface tension-induced instabil-ity that leads to breakup into droplets, and this breakup isaccelerated by the higher inward Lorentz force in the nar-rower regions due to higher current density. And if growth

is directed downward, there is a buoyancy instability dueto the higher density of iron than slag. These instabilitiesand their scaling behavior are treated analytically, and com-pared qualitatively with experimental measurements of fingersize.

Key words: electrochemistry, Mullins–Sekerka, steelmaking.

C© Blackwell Munksgaard, 2003

Accepted for publication 6 May 2002

Inherent to the steelmaking process is a tradeoff betweencarbon content in the hot metal, iron oxide content inthe slag, and time. That is, to achieve low carbon con-tent in the metal in a short time, one must tolerate largeamounts of iron oxide in the slag and the resulting loss ofyield, and vice versa. Or one can let the iron oxide reactwith carbon to increase the iron yield and lowerthe car-bon content, but this reaction is very slow, and limitedby ferrous ion transport to the slag–metal interface.

To improve the reaction rate between iron oxide inslag and carbon in hot metal, Uday Pal invented theelectric field-enhanced smelting and refining process,shown in Fig. 1. In this process, electrodes are made tocontact the slag and metal, allowing the reaction:

FeO + C −→ Fe + CO

to be split into two half-cell reactions at the cathode inthe slag and anodic slag–metal interface, respectively:

Fe2+ + 2e−−→Fe

C + O2 −→ CO + 2e−.

This enhances the reaction rate by increasing the sur-face area to which ferrous ions must be transported. Itis hoped that industrial use of this process will allowone to achieve very low carbon content in the metal as

well as low iron content in the slag, resulting in slightlyincreased iron yield, significantly less waste slag, lessfoaming, and slower refractory wear due to lower ironcontent in the slag, and possibly, less need for subse-quent decarburization processing. Put differently, thishas the potential to achieve low carbon content in thehot metal and low iron content in the slag in a shorttime, breaking the tradeoff of the traditional steelmak-ing process.

The process can operate in two modes. In passive op-eration, one short-circuits the process by simply insert-ing a conductor that contacts the slag and the hot metal,improving the kinetics for the existing thermodynamicdriving force. Alternatively, one may use separate elec-trodes with a potential applied between them to increasethe driving force and further accelerate the process (aspictured in Fig. 1). Both modes of operation have beendemonstrated experimentally.

Because this process is still limited by the mass trans-fer of ferrous ions to the cathode, its performance pro-cess depends critically on the complex behavior there,including several unstable phenomena and the inter-actions between them. This paper therefore focuses oncathode instabilities, giving both a qualitative descrip-tion of the instabilities and some quantitative anal-ysis to estimate the size of their features. To better

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Fig. 1. The electric field enhanced smelting and refining process.

understand the effect of these phenomena on overallreaction rate, a computational model of a two-fluidsystem will be necessary; such a model is outlinedhere.

Process instabilities

The electric field enhancement introduces several insta-bilities. With increasing reaction rate comes increasingproduction of carbon monoxide bubbles at the anodicslag–metal interface, which stirs the slag and increasesthe mass transfer coefficient, so that increasing the sur-face area for iron reduction gives a nonlinear increase inthe overall reaction rate of the process.

At the cathode, diffusion-limited growth of the liquidiron phase results in a Mullins–Sekerka instability, lead-ing to protrusion of liquid iron dendritic fingers into theslag, and in turn provides more surface area for ferrousion reduction and further accelerates the reaction. Thisis illustrated schematically in Fig. 2.

Fig. 2. Mullins–Sekerka instability: The diffusion boundary layer before asmall protrusion is thinner, so diffusion there is faster, and the perturbationgrows.

Fig. 3. Capillary instability: A cylindrical fluid–fluid interface breaks intodroplets.

To complicate matters further, linear perturbationanalysis of a cylindrical fluid–fluid interface indicatesthat non-axisymmetric transverse modes decay, but ax-isymmetric longitudinal modes longer than the circum-ference grow. For this reason, the liquid iron fingers inthe slag described above will become axisymmetric andbreak into droplets whose centers are a distance apartand equal to the circumference of the original cylinder.When a droplet separates from a finger, the electricalconduction path to the cathode is lost and there is nonet reaction on the droplet surface (though there canbe iron reduction on one side of the droplet and oxida-tion on the other if a current passes through it). This isshown in Fig. 3. The instability is complicated by Lorentzforces, which act in the inward radial direction, and arelarger in narrower parts of a cylindrical finger, possiblyaccelerating a breakup into droplets.

Also, the density difference between liquid iron andslag will result in iron gathering at the bottom of thecathode and dripping downward to the hot metal below.If the reaction is very fast, this dripping becomes a steadystream that connects the cathode to the anode and short-circuits the process, reducing the reduction efficiencyconsiderably.

All of these instabilities come together to produce verycomplex liquid metal shapes near the cathode, a cartoonof which is shown in Fig. 4. The resulting overall reactionkinetics is likely to be similarly complex, and stronglydependent on the shape of the cathode and fluid flowpattern in the slag. In order to design the cathode shapeintelligently, it is helpful to fully understand the detailsof fluid flow and reaction rate density at the metal–slaginterface. Toward this end, a numerical modeling effortis under way, and presented below are a few helpfulanalytical results.

Quantitative analysis

The smallest wavelength at which a sinusoidal pertur-bation to a flat surface will grow into dendritic fingersis given by the root mean square of the capillary anddiffusion lengthscales dCa and dD. The diffusion length-scale is the approximate thickness of the Fe2+-depletedboundary layer, approximated by the ratio of ferrous

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Scaling and instabilities in smelting and refining of steel

Fig. 4. A picture of what the cathode might look like.

ion diffusivity DFe2+ to the velocity of the slag into theinterface uslag. The mass-weighted velocity relative tothe interface is in turn given by the ratio of ferrous ionflux times molar mass to slag density J 2+

Fe >MFe

ρslag, and molar

flux is the ratio of current density J to charge requiredto reduce a mole of atoms nF . We therefore obtain:

dD ∼ DFe2+

uslag= DFe2+ρslagnF

J MFe.

Using DFe2+ = 10−6 cm2

s , n = 2 for ferrous ion reduction,Faraday’s constant 96540 coul

mol , approximate current den-sity of 0.1 − 1 A

cm2 , and slag density of 3.5 gcm3 we arrive

at a rough approximation for the diffusion lengthscaleof ∼ 0.1–1 mm.

The capillary lengthscale is the ratio of interfacial en-ergy per unit area γslag−metal to driving force per unitvolume of metal produced. Because this is an elec-trochemical reaction, we can use the Nernst equation�G = −nF E , and for volumetric driving force we candivide this by the molar volume Vm. This gives the ex-pression

dCa ∼ γslag−metalVm

nFE.

A slag–metal interfacial energy of 100 Jcm2 , molar volume

of 8 cm3

mol , and typical open-circuit voltage of 0.1–0.25Vgive a capillary length of 0.15−0.4 mm. Note that this isnot the lengthscale associated with the capillary insta-bility, but gives an approximate minimum radius of theliquid dendritic fingers necessary for their growth.

The Mullins–Sekerka wavelength for this process isthus on the order of a fraction of a millimeter.

The size of droplets of a dense fluid dripping througha less dense fluid is determined by the ratio of surfacetension to net gravitational force per unit volume. Theformer is inversely proportional to length and the latterto volume, so that the square root of this ratio gives theapproximate size of such droplets,

R ∼√

γ

g�ρ.

In this case, the density difference �ρ is about 2.5 gcm3 ,

and the surface tension approximately 100 Ncm , these and

the gravitational acceleration give R ∼ 2 mm.

Experimental evidence

In some of Pal’s experiments, the slag and metal aroundthe cathode were quenched and sectioned, and smallfingers were found that were approximately 1

4 mm indiameter and 1–4 mm long. The agreement of the fingerdiameter with the estimated capillary lengthscale is agood indication that the lengthscale of these fingers ina real process is governed by the competition betweenfree energy released by the iron reduction and surfaceenergy added by finger growth.

On the other hand, that the observed length of someof the fingers is several times their circumference mightseem to contradict the prediction of finger breakupdue to the capillary instability. However, breakup intodroplets is not instantaneous, the length of the fingers isgoverned by the ratio of a finger’s longitudinal growthrate to that of the perturbation leading to breakup, notunlike the length of the unbroken stream beneath aslowly running faucet. The time required to break updepends strongly on the nature of the perturbations thatlead thereto, and therefore cannot be understood inde-pendently of the larger fluid flow picture.

Future numerical modeling

A useful numerical model of the cathode will have to cal-culate the rate of ferrous ion reduction as a function ofcathode geometry and macroscopic flow conditions inthe vicinity of the cathode. To do this accurately, the de-tailed shape of the liquid metal–slag interface will haveto be calculated. One might be tempted to use a VOFmethod to predict coarse-grained and time-smoothedmetal fraction, but predicting reaction rate from this re-quires topological information about electrical connec-tivity which such a model would not have. Indeed, cal-culation of the length of connected liquid iron fingers isnot possible without a detailed model of the interactionbetween fluid flow and interface shape, as discussedabove.

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Powell & Dussault

A numerical model of the flow field and interfaceshape will also be able to predict the tendency for liquidmetal to bridge the gap between the cathode and hotmetal and “short-circuit” the process, which is a veryuseful result.

However, numerous factors could complicate sucha model. Local heating due to endothermic reactionsmay play a role, and the enthalpy released by eachof the half reactions must be assessed and incorpo-rated independently. Because of large curvature gra-dients along the interface, it is likely that the electricfield normal to the interface will vary considerablyas well, so electrocapillarity must be included in themodel. As discussed above, Lorentz forces may play arole in the capillary instability. The presence of currentin the slag could enable stirring using a DC magneticfield, in which case Lorentz forces would definitely beimportant.

It is worth pointing out here that a two-dimensionalmodel would not capture the capillary instability (a pla-nar liquid layer is stable, whereas a cylinder is not). Ax-ial symmetry would resolve this problem, but wouldnot correctly predict any side branching from the liq-uid iron fingers (analogous to secondary dendrites),nor cross-currents driving their breakup. It seems,therefore, that a fully three-dimensional model will benecessary.

Although it is certain that stirring produced by bub-bling will affect mass transfer at the cathode, it is notclear whether the bubbles will directly affect the liquidmetal shape and reaction kinetics there. Including themin a model would enormously increase its complexity.One could avoid this by using silicon in hot metal as areducing agent, and correlate the results of such a modelwith those from experiments, and this might help oneto gain insights into cathode design. However, it wouldnot represent the real process, and might not result in adesign optimized for the presence of bubbles.

Thus the required fields for the simplest cathodemodel would include

� Fluid velocity and pressure,� Temperature,� Concentrations and oxidation states of species in-

volved: at least Fe, Ca, and electrons in a simplifiedmodel without bubble interactions, and

� Electrostatic potential, and perhaps magnetic vec-tor potential.

One important issue that must be resolved is the rep-resentation of the interface. At the present time, Eulerianrepresentations look promising, such as phase field orlevel set methods, because of the difficulty involved infollowing the topological changes in interface using La-grangian methods that explicitly track the interface.

Because of the number of degrees of freedom, the com-plexity of the phenomena involved, and the necessity ofthree dimensions, this model presents many challenges.

Conclusion

Simple analytical models of lengthscales in cathode in-stabilities presented here have been shown to agree withexperimental observations. However, understanding ofcathode phenomena to a degree required for processoptimization will require an extraordinarily complexnumerical model. Such a model is under development,and will provide a useful tool in the engineering of thispotentially significant improvement to the traditionalsteelmaking process.

Address:A. PowellMIT Rm. 4-11777 Massachusetts Ave.Cambridge, MA 02139-4307USAe-mail: hazelsct@mit.edu

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