Seetharaman S. Fundamentals of metallurgy

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thus the overall rate of reaction is controlled by the C±CO

reaction taking place

at the observed p

and p

Sohn and Szekely (1973) developed a theoretical basis for ana1yzing the

general case in which both reactions aff ect the overall rate and for establishing a

criterion for the controlling step. Let us consider a uniform mixture of solids B

and D. They may be in the form of small particles or pellets that undergo

reactions of the types discussed in Sections 7.3.1 and 7.3.2. We shall here

consider isothermal systems and the case of uniform gas concentrations between

the particles or pellets of solids B and D. This latter condition is applicable when

the overall reaction, and thus the overall gas production, is reasonably fast. This

condition is also valid if the mixture is placed in a container with only a small

opening that allows the gas product to exit but does not allow the back diffusion

of any gas from the surroundings. From the mass balances of the gaseous

species, the rates of net generation of the gaseous species are given as follows:

 ÿv

 av

7:110a

 cv

 v

7:110b

where v

and v

are the net forward rates of reactions 7.109a and 7.109b,

respectively, per unit volume of the solid mixture. These terms can be expressed,

7.14 Observed value s of p

ratios compared with the comp uted

equilibrium values for the component reactions in the carbothermic reduction

of stannic oxide.

306 Fundamentals of metallurgy

using equations 7.85 and 7.85b, as follows:







 

7:11a







 

7:11b

The reaction rate terms, dX

=dt, depend on the reaction mechanism of each solid.

It can be given by any of the solutions given earlier for the reaction of a single

solid such as the differential form of equations 7.71, 7.83, 7.91 or 7.94b. It must

be noted that the bulk concentration terms in these equations are now the

interstitial concentrations, which vary with time as the reaction progresses.

If the total pressure of the system is maintained constant, the following

relationships hold:



7:112a



7:112b

where dV =dt is the rate of volume generation of the gas mixture, and V

is the

volume of the mixture. We have made a pseudo-steady-state assumption that

the gas-phase concentrations at any time are at the steady-state values

corresponding to the amounts and sizes of the solids at that time, namely,

CdV=dt  VdC=dt.

We also have the condition that

 C

 C

7:113

Equations 7.110 to 7.113 may be solved for C

and C

. Using these values, the

rates of reaction of the solids may then be obtained from the appropriate

expressions used for dX

=dt and dX

=dt. Integration of these rates over dt will

yield incremental values of X

and X

. Using the new values of X

and X

equations 7.110 to 7.113 are solved again to obtain C

and C

at the next time

step. The procedure is repeated to give conversions as functions of time. In

certain simple cases, analytical solutions are possible (Sohn and Szekely,

1973).

Solid±solid reactions proceeding through gaseous intermediates with no net

production gas

This situation arises when ac  1 in reactions represented by equations 7.109a

and 7.109b. An example is the oxidation of metal sulfides with lime in the

presence of water vapor to produce the corresponding oxides (Sohn, 1983; Sohn

and Kim, 1984b, 1987, 1988; Soepriyanto et al., 1989). This reaction can in

general be expressed by the following:

The kinetics of metallurgical reactions 307

S  H

O  Me

O  H

S (7.114a)

CaO  H

S  CaS  H

O (7.114b)

Overall: Me

S  CaO  Me

O  CaS (7.114c)

Using this scheme, selected metal sulfides such as molybdenum disulfide and

zinc sulfide can be transformed into the corresponding oxides without producing

a sulfur-containing gas. The role of lime in this process is twofold: (1) it improves

the thermodynamics of the reaction between sulfide and steam, which has a very

low equilibrium constant due to the positive Gibbs free energy change, by

eliminating hydrogen sulfide, and (2) the reaction between hydrogen sulfide and

lime fixes sulfur as cal cium sulfide which can be further treated to recover sulfur.

Reactions 7.114a, b, c involve no net consumption or generation of the

gaseous intermediates, and the gaseo us species do not appear in the overall

stoichiometry. The gaseous species simply act as carriers of oxygen and sulfur

atoms. In this respect, the mechanism for this type of reaction might be termed

`catalysis by gases' (Sohn, 1991).

The kinetics of this reaction can be described by equation 7.110. The

difference between this case and the reactions with net generation of gases is

that, since no gases are formed or consumed,

 av

7:115

Equations 7.111 and 7.113 are also valid in this case. The solution of these

equations gives the conversion as a function of time. Using this approach, the

detailed kinetics of reactions of various sulfide minerals with lime has been

described (Sohn and Kim, 1987, 1988; Soepriyanto et al., 1989; Sohn, 1991).

Successive gas±solid reactions in which the reactant gas reacts with the first

solid, producing an intermediate gas which in turn reacts with the second solid

Two notable examples, in which successive gas±solid reactions play a major

role, utilize lime as a scavenger of sulfur-containing gases . The first is the

hydrogen reduction of metal sulfides in the presence of lime (Kay, 1968;

Habashi and Yos tos, 1977; Sohn and Rajamani, 1977; Shah and Ruzzi, 1978;

Rajamani and Sohn, 1983; Sohn and Won, 1985; Won and Sohn, 1985), which

can be represented by the following general scheme:

 Me

S  H

S  xMe (7.116a)

S  CaO  H

O  CaS (7.116b)

 Me

S  CaO  H

O  xMe  CaS (7.116c)

In this reaction scheme, lime serves two purposes: (1) it fixes sulfur in the solid,

thereby reducing the emission of hydrogen sulfide into the atmosphere, and (2)

308 Fundamentals of metallurgy

the presence of lime improves the otherwise unfavorable thermodynamics of

reaction 7.116a by removing hydrogen sulfide from the gas phase. Reaction

7.116b is highly favorable thermodynamically. The second example is the

roasting of sulfide minerals in the presence of lime (Haver and Wong, 1972;

Bartlett and Haung, 1973; Haung and Bartlett, 1976).

Sohn and co-workers (Sohn and Rajamani, 1977; Rajamani and Sohn, 1983;

Sohn and Won, 1985) developed a model for successive gas±solid reactions in a

porous pellet. The model takes into consider ation the effects of the relative

amounts of the solids, grain sizes, the pellet size and porosity, and the diffusion

of gaseous species, as well as the effect of improved thermodynamics in systems

represented by the first example. The model describes not only the rates of

reaction of the solid reacta nts but also the fraction of the intermediate gas (the

sulfur-containing gases in the above examples) that is captured by the second

solid (lime in the above examples).

Staged reaction of a solid with a gas in which the solid forms a series of

thermodynamically stable intermediate phases

A notab le example of such a system is the gaseous reduction of hematite (Fe

)

to iron through the successive formation of magnetite (Fe

) and wustite (FeO)

(Sohn and Chauba1, 1984; Sohn, 1981).

Simultaneous reactions between solid reactants and gases

This type of system can be describ ed by writing the expressions for the

consumption of the gaseous reactants by the different solids as well as the

conversion of different solids by reac tion with different gases in equations 7.82

and 7.83 together with equat ions 7.2 and 7.3 and solving the resulting equation

by a numerical method. The reader is referred to the literature for further detail

(Sohn and Braun, 1980, 1984; Lin and Sohn, 1987; Paul et al., 1992).

Multiparticle systems

In the above sections, the discussion mainly involved the analysis of reactions

taking place in a single particle or pellet of solids. The eventual objective for

studying single particle systems is, of course, to apply the results to the analysis

and design of multiparticle systems of industrial importance. Examples of

multiparticle gas±solid contacting equipment include packed beds, moving beds,

fluidized beds, and rotary kilns. The extension of single particle studies to

multiparticle systems will depend on the nature of the particulate asse mblages,

the mode of gas±solid contacting, and the spatial variation of the gas properties

within the system. Therefore, general analyses of these processes will not be

presented here, and the reader is referred to the literature for developments in

this area (Zhou and Sohn, 1996; Paul et al., 1992; Sohn, 1991; Rhee and Sohn,

The kinetics of metallurgical reactions 309

1990; Sohn and Lin, 1990; Gao et al., 1983; Herbst, 1979; Szekely et al., 1976;

Evans and Song, 1974).

7.5 Liquid±liquid reactions

There are instances during treatment of molten metals where reactions between

immiscible molten phases need to be considered. Most often this involves the

partition of a solute or impurity element between the two phases. As a general

picture, imagine the interface between two phases as shown in Fig. 7.15 and the

partition of an element i.

From a kinetic point of view, the rate controlling step is one or several of the

following:

(i) Reactant species has to move between the bulk of phase 1 to the interface.

(ii) Reaction has to take place at the interface between phases 1 and 2.

(iii) Reactant has to move between the interface and the bulk of phase 2.

At high temperatures, in the absence of surface active elements that might

sterically hinder the reaction, the chemical reaction is often rapid enough to

establish local equilibrium and avoid concentration build up at the interface. The

flux of the solute can then be written,

 k

C

ÿ C

  k

C

ÿ C

 7:117

Here the ks are mass transfer coefficients. It should be noted that if the species i

could be in different chemical forms depending on the phase, i.e. Ca and Ca

The partition coefficient is defined as: m  C

. The interfacial concentra-

tions can then be eliminated from the equation 7.117 to yield:



 

 mC

ÿ mC

7:118

For the case where the interface is not in equilibrium, a similar expression can be

derived, assuming that forward and back reactions are simple:

7.15 Schematic of a liquid±liquid interface.

310 Fundamentals of metallurgy



rxn



 

 mC

ÿ mC

7:119

Here an additional resistance term containing the chemical reaction rate constant

has been added.

Reactions between molten metals and slags are an example of liquid±liquid

reactions. During metal extraction and liquid state processing slags are often

used to achieve a multitude of purposes such as thermal insulation, contami-

nation prevention, lubrication, and heat transfer control. Generally, industrial

slags and fluxes contain SiO

, Me

O (metal oxides) and, depending on the slag,

additional compounds like Al

, CaF

and P

. The ratio SiO

/Me

O is an

indication of the degree of polymerization. In this section some of the

fundamental aspects of chemical reactions between slags and metals are

discussed with the case of FeO/Fe as an example. The activation energy for the

interfacial reaction is 120 kcal per mole (Richardson, 1974) for a reaction

between molten iron and slag:

FeO  Fe 

O (7.120)

Due to the relatively low activation energy this reaction is expected to be mass

transport cont rolled. Slags have in general higher viscosities and due to their

ionic/polymeric nature the transport of Fe

±O

2ÿ

in the slag phase is likely the

rate determining step. In general, the activation energy for the chemical reaction

step is expected to be related to the enthalpy change. When transferring species

from a slag to a metal, more bonds have to be broken than formed, since the

solutes share ionic bonds in the slag phase. Thus, the activation energies would

be expected to be high for those elements that share many bonds in the slag

structure such as Si. The activation energy for the reaction,

SiO

 Si  2O (7.121)

is 270 kcal per mole.

One of the noteworthy aspects of melt±melt reactions is the ability to achieve

large interfacial areas to enhance reaction rates. Kozakevitch (Kozakevitch et

al., 1955 and Kozakevitch, 1969) followed the desulfurization of liquid steel by

molten slag by using X-rays to observe the change in the shape of a sessile drop

of the steel in liquid slag (held in a crucible). It was noted that there was a

marked change in the shape of the drop from non-wetting to wetting and then

back to non-wetting conditions. These changes are equivalent to the following

changes in the metal±slag interfacial tension (

(i) a high initial 

;

(ii) a reduction to a very low 

value while there is rapid desulfurization;

(iii) an increase in 

to its initial value when desulfurization is complete.

Studies on the dephosphorization of steel have yielded similar behavior

(Jakobsson et al., 1998).

The kinetics of metallurgical reactions 311

These dramatic changes in 

associated with rapi d mass transfer are usually

referred to as dynamic interfaci al tension. When 

is very low, any disturbance

or turbulence can cause droplets of one phase to move into the other phase. This

is usually referred to as emulsification and the formation of a metal emulsion in

the slag phase (or vice versa) leads to fast kinetics for the refining reactions

because of the huge surface area/mass ratio. Thus from a refining process

viewpoint a low interfac ial tens ion is advantageous and mo der n metal

production processes make use of this.

7.6 Solid±liquid reactions

7.6.1 Solidification

The transformation of an element of liquid into solid will follow a heat

balance:

c

 L

ÿ Q

7:122

where , c

and L are the density, specific heat capacity and latent heat of the

metal respectively. S and V are the surface area and volume of the element and f

is the fraction solid. Q is the net heat flux from the volume element. This

equation can be closed by specifying an f

-t-T relationship if Q and f

can be

identified. Q is dependent upon the thermal field caused by the stru cture around

the element, the mold characteristics and fluid flow in the melt. The fraction

solid, f

, depends on the solidification kineti cs, which is governed by the rates of

nucleation and growth which are also influenced by fluid flow. The strong

interdependence of melt fluid flow, heat transfer and crystal growth complicates

the theoretical treatment of solidification. Recently, studies have presented

numerical modeling techniqu es that can simulate solidification structures

ranging from dendrite morphology (as reviewed in Boettinger et al., 2000) to

microporosity (as reviewed in Lee et al., 2001) and particle pushing (Stefanescu

et al., 1998). The dendrite mor phology models range from phase field

simulations which accurately track the development of a dendritic grain in

simplified thermal conditions to cellular automata models which track only the

dendrite tip locations in complex geometries for the prediction of grain

structures (Gandi n et al., 1993). Microporosity models range from analytic

solutions to complex simulations of the interactions of pores and the developing

microstructure (Lee et al., 2001). Particle pushing models have also spanned a

range from analytic solutions (Stefanescu et al., 1998) to the use of phase field

(Ode et al., 2000). One common feature of many of these models is their

highlighting of the importance of the kinetics of solidification processes upon

the final structure. Information on high temperature kinetics is, however,

difficult to obtain, especially on nucleation rates.

312 Fundamentals of metallurgy

7.6.2 Metal±melt refractory reactions

Refractory vessels that contain molten metals and slags are subjected to

degradation due to chemical or mechanical attack by the molten phases.

Chemical attacks occur through the reduction of oxide compounds in the

refractory by more reactive elements in the melt.

For example, at high Al levels, there can be a significant reaction with the

MgO in the refractories according to:

3hMgOi  2

Al ) 3Mg (v)  hAl

i (7.123)

This reaction rate can be controlled by the mass transport of Al to the reaction

site or by the gas phase mass transport of Mg vapor away from the reaction site

since its partial pressure is equal to or less than 1 atmosphere. It is also possible

that the Mg (v) can react to produce undesired spin el inclusions in the metal

melt.

Erosion at the slag±gas or slag±metal interfaces is usually referred to as `slag

line attack'. It is caused through the establishment of Marangoni flows at the

interfaces. Mukai (1998) investigated this form of erosion for oxide refractories

and MgO/C refractories. For oxide materials (e.g. SiO

) and systems where the

dissolution of SiO

leads to an increase in surface tension, the region of the slag

film in longer contact with the refractory will have higher SiO

contents. These

regions will have higher surface tension and Marangoni flow will occur towards

these regions. Since these regions occur at the top of the slag film there will be a

flow of slag up the refractory until the Marangoni forces are balanced by

buoyancy forces. However, if SiO

dissolution causes a decrease in surface

tension of the slag the erosion pattern is different with rotational and cyclical `up

and down' motions of the slag.

The erosion of MgO /C and Al

/C refractories occurs by a two-stage

process. When MgO particles protrude from the surface the refractory will be

wetted by the slag and the slag will dissolve the MgO leaving C particles

protruding from the surface. Under these conditions the metal phase will wet the

refractory better than the slag. The metal thereupon dissolves the carbon leaving

MgO protruding from the surface and the whole process is then repeated.

Tsotridis and Hondros (1998) developed a mathematical model to predict

erosion wear in vessels. This accounted for interfacial wear due to both thermo-

and diffuso-capillary flows and erosion in the lower part of the vessel du e to

buoyancy flows.

7.6.3 Electrochemical reactions

Electrochemical reactions are a type of chemical reaction where electric energy

is consumed or created. An electrochemical reaction could be e.g.:

A  ne

! A

nÿ

7:124

The kinetics of metallurgical reactions 313

This is a reduction reaction. The quantitative link between electrical energy and

chemical reactions was established by Faraday (see, for example, Uhlig and

Revie, 1985) who found that, for a reaction such as the one above, the loss of

product A (in moles = m) is related to a current at A according to: m  n  F  it.

The chemical free energy of the reaction is G

and a corresponding reversible

potential can be computed, using the Nernst equation, G

 ÿnFE

(where F

is Faraday's constant).

The reaction above cannot occur by itself since the electrons consumed need

to be supplied by a different reaction, called an oxidation reaction.

nÿ

! ne

 B 7:125

The two reactions have to occur at a location called electrodes, where electrons,

ions and reactants (A and B are usually solids or gases) have to be present. The

electrode where oxidation occurs is called the anode and the one where

reduction occur s is called the cathode. The electrons given off at the anode are

supplied to the cathode through an electronic conducting phase. To close the

circuit, an ionic current is passed between the electrodes through an ionic

pathway, called the electrolyte. The entire electrochemical system, consisting of

electrodes, electrolyte and electron lead, is called an electrochemical cell. The

overall cell reaction is the sum of anode and cathode reactions and the cell free

energy and reversible cell potential can be calculated accordingly. There are two

basic types of electrochemical cells. In galvanic cells (see Fig. 7.16a), the cell

free energy is negative and thus the reversible cell potential is positive. In these

types of cells, to which most corrosion cells, fuel cells and batteries belong,

electrical energy is released. The second type is electrolysis cells (Fig. 7.16b),

which have a negative cell potential and thus are not spontaneous. Electrolytic

metal extraction processes are examples of this latter type.

For an in-depth understanding of electrochemical kinetics there are several

excellent textbooks available (e.g. Bard and Faulkner, 2001). In this chapter, the

topic is introduced in two cases, corrosion in an aqueous solution and

electrolytic Al extraction.

Corrosion of Zn in a de-aerated environment

Consider Zn metal reacting to form ions in the presence of water according to:

Zn ! Zn

 2e

(7.126)

The standard half cell potential of this anodic oxidation reaction is: E



0.763 V (Uhlig and Revie, 1985) vs the standard hydrogen potential at room

temperature. The Zn ions dissolve in the water that serves as an electrolyte. The

electrons are supplied through the Zn metal itself and delivered to a cathodic

reduction reaction:

+ 2e

 H

(g) (7.127)

314 Fundamentals of metallurgy

The standard half cell potential vs the standard hydrogen electrode is 0 V for this

reaction. This reaction will have to occur at the Zn surface too and therefore the

exact location of the anode and cathod e cannot be specified. This cell can be

viewed as the case in Fig. 7.16a, with the Zn dissolution reaction occurring at the

anode and the hydrogen reduction reaction at the cathode. Applying Kirchoff's

law to the cell, one obtains:

ÿ j

j  E

ÿ j

j ÿ i 

ohm

 0 7:128

In this equation, E

and E

are the reversible cell potentials:

 E



 

7:129

and

 E

2

 

7:130

The terms 

and 

are called over-potentials and account for the kinetic

resistances at the electrodes. In general they include the effects of (i) charge

transfer reaction resistance, (ii) mass transport resistance and (iii) intermediate

chemical reaction and adsorption effects. As a result, they are non-linear

functions of current, which can be derived similarly to what was done for

reaction rate theory but adding the effect of electrical potential on stabilizing or

de-stabilizing the presence of electrons:

j

j 



2:3  log

0;H





0; t



 

7:131

Here,  is the transference number (between 0 and 1), usually close to 0.5. i

0,H

the exchange current of the charge transfer reaction and is a measure of the

electrocatalytic nature of the electrode. Roughly it corresponds to the chemical

reaction rate constant. The second term in the parenthesis is the ratio between

reactant concentration at the electrode and in the bulk that depends on the mass

transport of reactants to the reaction site. It should be mentioned that the mass

transport is carried out through migration, diffusion and fluid flow. The

combined flux equation being:

7.16 (a) Galvanic cell and (b) electrolysis cell.

The kinetics of metallurgical reactions 315