of pneumatic steelmaking processes, including the converter must again be pointed out that the assumption of the rate-

Mathematical Modeling of the Argon-Oxygen and AOD processes, that there exist distinct regimes of rate controlling steps all being the liquid-phase mass transfer, in
Decarburization Refining Process of Stainless Steel: control for decarburization at high and low carbon levels.
However, the oxygen blown into the bath during the refining
the cases of high and low carbon levels, is not reasonable.
Additionally, the diluting role of the inert gas blown to
Part I. Mathematical Model of the Process process cannot usually be completely absorbed.[12] The the carbon monoxide formed and its effect on the rate of
decarburization were not fully reflected. Also, it is not likely
assumptions of complete utilization of oxygen, of a reaction
scheme, and of isothermal conditions are potential sources that the oxygen in the gas blown through the tuyeres from
JI-HE WEI and DE-PING ZHU of weakness in the model developed by Fruehan. the vessel bottom is completely consumed by the oxidation
Deb Rey and Robertson[5,6] considered that all the injected of the various elements in the steel. The estimation of a
Some available mathematical models for the argon-oxygen decarburization (AOD) stainless steel– oxygen oxidizes the chromium, silicon, and manganese dis- Gibbs free energy for the reaction of 1/3^Cr2O3& 1 [Fe] 5
making process have been reviewed. The actual situations of the AOD process, including the competi- solved in the liquid steel, and the Cr2O3, SiO2, and MnO (FeO) 1 2/3[Cr] indicated that there is no possibility of the
tive oxidation of the elements dissolved in the molten steel and the changes in the bath composition, formed are reduced by carbon as they rise in the bath with oxidation of iron in terms of the reaction, even under their
as well as the nonisothermal nature of the process, have been analyzed. A new mathematical model the bubbles. Moreover, they noted the change in the partial experimental conditions (the corresponding equilibrium con-
for the AOD refining process of stainless steel has been proposed and developed. The model is based pressure of carbon monoxide with the bath height and intro- stant at 1700 8C is about 0.0245). So, this assumption regard-
on the assumption that the blown oxygen oxidizes C, Cr, Si, and Mn in the steel and Fe as a matrix, duced the heat balance to take account into the nonisothermal ing the oxidation of iron cannot be held.
but the FeO formed is also an oxidant of C, Cr, Si, and Mn in the steel. All the possible oxidation- nature of the bath, thus, proposing a mathematical model Using similar assumptions to those of Fruehan’s model,
reduction reactions take place simultaneously and reach a combined equilibrium in competition at for stainless steel making. The model was tested on plant a mathematical model for the AOD refining process of
the liquid/bubble interfaces. It is also assumed that at high carbon levels, the oxidation rates of data obtained in producing Type 304 stainless steel in a 45-t stainless steel was developed by Reichel and Szekely.[9]
elements are primarily related to the supplied oxygen rate, and at low carbon levels, the rate of AOD converter, and better results were obtained. However, According to this model, the decarburization rate is propor-
decarburization is mainly determined by the mass transfer of carbon from the molten steel bulk to tional to the supplied oxygen rate at high carbon levels and
the fact that there are different decarburization patterns at
the reaction interfaces. It is further assumed that the nonreacting oxygen blown into the bath does is related to the carbon content and the supplied oxygen rate
high and low carbon contents was also not reflected by the
not accumulate in the liquid steel and will escape from the bath into the exhaust gas. The model at low carbon levels, but the influence of the latter weakens.
model. Furthermore, they assumed that the oxygen distribu-
performs the rate calculations of the refining process and the mass and heat balances of the system. A critical state or a critical point exists in the refining process,
Also, the effects of the operating factors, including adding the slag materials, crop ends, and scrap, tion ratios near the jet entrance among the dissolved elements
were proportional to the molar concentrations of the ele- when the constant rate of decarburization transitions to a
and alloy agents; the nonisothermal conditions; the changes in the amounts of metal and slag during falling rate. The corresponding carbon concentration is the
the refining; and other factors have all been taken into account. ments. This assumption and the assumption of complete
oxygen utilization are potential sources of error in this model. critical carbon content. Essentially, this model has the same
Ohno and Nishida[7] proposed a bubble decarburization weaknesses as those of Fruehan’s model.
model for the AOD process. They also assumed that the In terms of the thermodynamics with the mass balance
of the system, Görnerup and Sjöberg[10] mathematically
injected oxygen primarily oxidizes chromium in the tuyere
I. INTRODUCTION the oxygen transfer from the FeO part of the slag to the modeled the AOD/Creusot–Loire Uddeholm process. The
zone and that the chromium oxide formed subsequently
THE keys for the refining of stainless steel are how to metal is quite rapid. Applying the model to the refining principal question to their model is that the state of thermody-
oxidizes the carbon as it rises in the bath with the argon
remove effectively the carbon in the steel and to raise the process in an oxygen-blown, electric furnace of 40-t capacity, namic equilibrium, in fact, cannot be reached and established
bubbles. At the same time, they assumed that the reductive
chromium recovery. In the various refining technologies the predictions and measurements showed quite good agree- in the whole bath during the refining process. So, it is hard
reaction of Cr2O3 by carbon is, all along, limited by the
of stainless steel, the argon-oxygen decarburization (AOD) ment. The first deficiency of this model is that the discrep- to say that their model is reliable and believable. Recently,
ancy of decarburization patterns at high and low carbon liquid-phase mass transfer of carbon to the bubble surface
process has a number of obvious advantages; thus, it has taking the rate equation of Fruehan’s model at the low-
levels was not noted. Second, according to this model, the in the whole refining process. Considering the influence of
been applied extensively and developed rapidly throughout carbon condition and the heat balance of the system as a
rate parameters used were the conductances determined by the bath height on the total pressure and partial pressure of
the world. At present, over 75 pct of the world’s stainless basis, a real-time online control model for the AOD process
consider all the resistances of the system to transfer, i.e., CO in the bubble, the rate equation of the decarburization was developed.[11] The results, applied to the refining process
steel output is produced using the process. process, characterized by the partial pressure of CO, was
Simultaneously, many studies on mathematical modeling mass transfer, chemical kinetics, mixing, etc. So, the values in a 68-t AOD vessel, indicated that the calculation and
of these conductance parameters for different components derived and obtained. As mentioned previously, it is not control precision of this model are not high.
for the AOD refining of stainless steel have been carried out, likely that the mass transfer of carbon becomes a limiting
and numerous models have been proposed and developed to in the bath must be different under different operating condi- To sum up, these studies, to different extents, offered
tions. However, it was assumed that the values of these link of the rate of the decarburization process at high carbon some useful information for understanding and improving
attempt to accomplish optimization and computer control of levels. In addition, the effects of the operating factors and
the process.[1–11] Taking the AOD refining of low-carbon parameters for the overall transfer of C, O, Cr, and Si under the process practice. However, all these available models
different operating modes were equal and constant. These changes in the bath composition and temperature in the have not reflected and described fully the real situations of
and high-chromium stainless steel as an example, Ray and
assumptions, combined with the assumption that all nonre- refining process were not considered. The activities of the the refining process of stainless steel and, to a certain degree,
Szekely[1] examined and analyzed the mathematical model-
ing of this process in the discussion of process optimization acting oxygen is accumulated in the molten stainless steel, elements in the steel were taken to be all 1, which must also have all those shortcomings. Using these models, in fact, it
methods. Following this work, based on the mass and heat are in clear contradiction to known phenomena and likely to bring about a considerable deviation from reality. is difficult to predict quantitatively and accurately the
balances of the system and the film theory of diffusion, lead to errors in predicting the kinetics of the AOD processes. Tohge et al.[8] investigated experimentally the refining changes in the chemical composition and temperature of the
Asai and Szekely[2,3] proposed a mathematical model for the Fruehan[4] assumed that most of the oxygen blown is process of austenitic-grade stainless steel with very a high bath during the practical process and the influence of the
decarburization of stainless steel. The essential assumption consumed in the oxidation of chromium in the tuyere zone, initial content of carbon (.2.5 mass pct) in a top-and-bot- relevant factors, as well as their interactions. It is still needed,
was that the oxygen supplied to the metal is either used to and the Cr2O3 formed oxidizes the carbon as it rises in the tom-combined blowing AOD vessel of 70-t capacity using and is of important theoretical and practical meaning, to
participate selectively in decarburization and the oxidative bath with the argon bubbles (FeO may also form, but it is the operating practice of a high gas bottom blowing rate study further and more deeply this process. Considering
reactions of Cr and Si, thus forming CO, Cr2O3, and SiO2, quickly reduced by Cr). It was further assumed that the with oxygen top blowing. On the basis of the mass and these conditions, the AOD refining of stainless steel has been
or it accumulates in the metal phase. The model did not allow oxidation of carbon by Cr2O3 is controlled by the liquid- heat balances, they developed a theoretical model which investigated. A new mathematical model for this process has
for the formation of FeO explicitly; namely, it postulated that phase mass transfer of carbon to the bubble surface at low considered the formation of some amount of FeO, noniso- been proposed and developed, which is expect to provide
carbon levels and is determined primarily by the rate of thermal conditions, and rate-controlling steps, as well as the more believable and useful information and a more reliable
oxygen blown at high carbon levels. Also, it was assumed addition of slag, scrap, and alloy agents. According to their basis for the optimization and computer control of the proc-
JI-HE WEI, Professor, is with the Department of Metallic Materials, that most of the silicon is preferentially oxidized to the model, the carbon, silicon, chromium, manganese, and iron ess. The model performs the rate calculations of the refining
Shanghai University, Shanghai, 200072, People’s Republic of China. chromium in the early stage of the blow. On the basis of in the steel are oxidized during the oxidation in the AOD process and the mass and heat balances of the system. Simul-
DE-PING ZHU, formerly Graduate Student, Department of Metallic Materi-
als, Shanghai University, is Engineer, Shanghai Wensi Sorftware Limited
these assumptions, a reaction model was developed to predict process, thus forming CO, SiO2, Cr2O3, MnO, and FeO. It taneously, the effects of the operating factors, including add-
Company. the rates of carbon and chromium oxidation in the AOD should be said that the conditions of the AOD refining were, ing the slag materials, crop ends, and scrap, and alloy agents;
Manuscript submitted January 9, 2001. process. It has been confirmed by numerous observations throughout, of quite concern in this model. However, it the nonisothermal conditions; the changes in the amounts

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 33B, FEBRUARY 2002—111 112—VOLUME 33B, FEBRUARY 2002 METALLURGICAL AND MATERIALS TRANSACTIONS B
of slag and metal during the refining; and other factors were Another feature of the AOD process is that the bath is The reaction system is just one composed of a liquid alloy Correspondingly, the equilibrium concentration of carbon at
all considered. The details of the model and the determina- strongly agitated by the gas streams. The fluids in the bath (stainless steel) and a liquid slag phase with an atmosphere the reaction interface is
tion of its parameters with the computing procedure are undergo very vigorous stirring and circulatory motion during containing oxygen. The following independent reaction
presented in Part I of the present work. gas blowing, and there is no obvious dead zone in the bath.[13] equilibria in this system can be produced from combinations
!a
PCO 3 a 2[Cr]
This can very effectively promote and intensify the heat of reactions [1] through [4], respectively, with reaction [5]. [pct C]e 5 [16]
and mass transfer and is undoubtedly very advantageous in fC Cr2O3 KCr2C
II. ANALYSIS OF THE AOD PROCESS [C] 1 (FeO) 5 {CO} 1 [Fe];
accelerating the refining reactions and improving the homo-
It is well known that in AOD stainless steel making, the [6] where a[Cr] 5 fCr[pct Cr] and a(Cr2O3) 5 gCr2O3 NCr2O3. Clearly,
geneity of the bath composition and temperature. PCO
supplied oxygen is utilized to remove the carbon in the The conditions and characteristics mentioned previously DGC 5 DG8C 1 RT ln reaction [15] can be reached by a combination of reactions
molten steel. The argon (or nitrogen) blown simultaneously must all be considered and noted in mathematical modeling a[C]a(FeO) [1] and [2] or reactions [6] and [7]. Paying attention to
can decrease the partial pressure of the carbon monoxide of the AOD refining process of stainless steel. the diluting role of the inert gas (argon or nitrogen) and
2[Cr] 1 3(FeO) 5 (Cr2O3) 1 3[Fe]; nonreacting oxygen, the partial pressure of carbon monoxide
and promote decarborization, thus achieving the effective- [7]
ness and objective of removing carbon and reducing the loss III. MATHEMATICAL MODEL OF THE aCr2O3 (PCO) should be
of chromium. However, the silicon and manganese dissolved PROCESS DGCr 5 DG8Cr 1 RT ln
in the molten steel can also absorb the blown oxygen and a 2[Cr]a 3(FeO) nCO
A. Basic Assumptions of the Model PCO 5 Pt
restrict the oxidation reactions of carbon and chromium. nCO 1 n8O 1 nsub
[Si] 1 2(FeO) 5 (SiO2) 1 2[Fe];
Due to their low initial contents in the steel, their oxidative In order to propose and develop a new mathematical [8]
reactions during the refining process will rapidly reach the model to deal with the refining process, the following initial a(SiO2) Relevantly,
relevant dynamic equilibrium. After that, the supplied oxy- assumptions were made for the process, based on the previ- DGSi 5 DG8Si 1 RT ln
a[Si]a 2(FeO)
1 2
gen will, apparently, all be consumed by the oxidation of ous analysis. Wm d[pct C]
nCO 5 2 dt
carbon and chromium, except the part escaped from the (1) The oxygen blown into the molten steel simultaneously 100 MC dt
[Mn] 1 (FeO) 5 (MnO) 1 [Fe];
bath. That is to say, there exists throughout the competitive oxidizes the carbon, chromium, silicon, and manganese [9]
oxidation of the carbon, chromium, silicon, manganese, and dissolved in the steel and the iron as a matrix; the iron a(MnO) and n8O 5 QO(1 2 h )dt/22,400 and nsub 5 Qsubdt/22,400.
DGMn 5 DG8Mn 1 RT ln Thus, the following expression can be reached:
other elements dissolved in the steel during the whole refin- oxide formed is also an oxidant for the other elements a[Mn]a(FeO)
ing process. and is, essentially, an intermediate product of the refin-
Moreover, at high carbon concentrations, the driving force These all belong among the possible reactions which occur d[pct C]
ing process. 2
for the mass transfer of carbon in the liquid steel to the (2) All the possible oxidation-reduction reactions take place in the system. Thermodynamically, the reaction schemes dt
PCO 5 Pt [17]
reaction interface would be high enough at the oxygen blow simultaneously and reach and establish a combined equi- presented by reactions [1] through [5] and reactions [6] 100 MC QO(1 2 h ) 1 Qsub d[pct C]
rates usually used in the AOD practice. In this case, there through [9] can all characterize the chemical-equilibrium 2
librium in competition at the liquid/bubble inter- Wm 22,400 dt
would be insufficient oxygen to oxidize the carbon trans- faces.[14–18] feature of the refining system but, kinetically, they are differ-
ferred to the reaction interface from the bulk of the molten (3) At high carbon contents, the oxidation rates of elements ent, the former being direct, and the latter being indirect. Substituting Eqs. [16] and [17] into Eq. [14], the following
steel. This means that at high carbon concentrations, the rate are primarily related to the supplied oxygen rate; at low can be obtained:
of decarburization would primarily be related to the rate of carbon concentration levels, the rate of decarburization
oxygen blow. When the carbon content in the steel is is mainly determined by the mass transfer of carbon in d[pct C] 1
decreased to a certain low level, the rate of decarburization
C. Rate Equations of the Process 5 (2s1 2 !s21 1 s2) [18]
molten steel. dt 2
may change to being controlled by the mass transfer of (4) The unabsorbed oxygen blown into the liquid steel will At high carbon contents, the average loss rates of the where
carbon to the reaction interface from the liquid bulk. Numer- escape from the bath and form CO2 with CO in the carbon, chromium, silicon, and manganese dissolved in the
ous observations of the AOD refining process suggest that exhaust gas, rather than dissolving and accumulating in steel in the competitive oxidation are, separately, 100 MC QO(1 2 h ) 1 Qsub
there is a critical point or a critical state in the process like s1 5 2
the steel. Wm 22,400
that in oxygen-converter steelmaking. This point corres- (5) The bath composition and temperature are continually Wm d[pct C] 2hQO [19]
2 5 x [10]
ponds to the critical carbon concentration and to the transi- changing and are uniformly distributed at any moment 100 MC dt 22,400 C
1 !a 2
ArearmkC Pt a2[Cr]
tion from the decarburization rate being related to the during the whole refining process. 1 2 3 1 [pct C]
supplied oxygen amount to being the rate controlled by the (6) The oxidation of elements in the steel other than C, Cr, Wm d[pct Cr] 2hQO Wm fC Cr2O3KCr2C
mass transfer of carbon in the liquid phase. 21.5 5 x [11]
Si, and Mn is temporarily not taken into account; i.e., 100 MCr dt 22,400 Cr
1 2
The oxygen molecules entering the bath would also con- the oxygen consumed by the other elements is ignored ArearmkC 100 MC QO(1 2 h ) 1 Qsub
s2 5 4 [pct C]
tact the iron atoms as a matrix of stainless steel and form in the present work. Wm d[pct Si] 2hQO Wm Wm 22,400
iron oxide, but most of the iron oxide formed would, subse- 22 5 xSi [12]
100 MSi dt 22,400
quently, quickly be reduced by the carbon, chromium, sili- [20]
B. Refining Reaction Schemes
con, manganese and other elements in the molten steel. This Wm d[pct Mn] 2hQO
means that the iron oxide formed also would be an oxidant The oxidative reactions of the carbon, chromium, silicon, 2 5 xMn [13]
and manganese dissolved in the molten steel and the iron 100 MMn dt 22,400
for them and would be mainly an intermediate product of D. Heat Balance of the System
the gas-blowing refining. In addition, their oxidation, to a as a matrix of the steel by the blown oxygen can be written as
At low carbon concentration levels, the average rate of
certain extent, would be related to the supplied oxygen rate 1 decarburization can be expressed as The molten steel, slag melt, and gases (including the
even at low carbon concentration levels. [C] 1 O2 5 {CO} [1] blown oxygen and argon (or nitrogen) and the exhaust gas)
2
Furthermore, the bath always demonstrates an obvious d[pct C] all carry heat. The oxidation reactions of elements can release
3 2Wm 5 ArearmkC ([pct C] 2 [pct C]e) [14]
nonisothermal characteristic during the refining process. The 2[Cr] 1 O2 5 (Cr2O3) [2] dt heat. Also, the heat of the system can be lost by conduction
oxidation reactions of the elements dissolved in the steel 2 and adsorption of the refractory lining and shell during the
make the bath temperature continuously increase; the addi- [Si] 1 O2 5 (SiO2) [3] At this time, the refining reactions in the bath mainly are rising-temperature process of the bath; by radiation; by the
tion of slag materials, crop ends, scrap, and alloy agents, as 1 the oxidation of carbon and chromium, and the bath tempera- operations of taking the samples and measuring the tempera-
well as the heat loss of the system, cools the bath. The [Mn] 1 O2 5 (MnO) [4] ture is also raised to a higher level. Thus, the following ture; by adding the slag materials, crop ends, scrap, and
2
nonisothermal nature of the bath can directly and strongly reaction can appropriately be considered: alloy agents; and by other factors. Moreover, the heat can
influence the equilibrium and rates of the various refining 1 be obtained or lost due to some uncertain reasons. According
[Fe] 1 O2 5 (FeO) [5]
reactions. 2 (Cr2O3) 1 3[C] 5 2[Cr] 1 3{CO} [15] to these, the heat-balance equation is

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 33B, FEBRUARY 2002—113 114—VOLUME 33B, FEBRUARY 2002 METALLURGICAL AND MATERIALS TRANSACTIONS B
Wmcp,mT 1 QOdtrOcp,OTg,0 1 Qsub dtrsubcp,subTg,0 1 Wscp,sT oxygen and inert gas (argon or nitrogen); the temperatures Table I. Interaction Coefficients ei(j) used in the present work[19]
of the shell and exhaust gas; the amounts of various addi-
1
Wm d[pct C] d[pct Cr] tion agents. The third element j
1 2 DHC 2 DHCr
100 dt dt Element i C Cr Mn Si O Ni Mo
C 0.14 20.024 20.012 0.08 20.34 0.012 20.0083
2
d[pct Mn] d[pct Si] Cr 20.12 20.0003 — 20.0043 20.14 0.0002 0.0018
2 DHMn 2 DHSi dt
dt dt IV. DETERMINATION AND ESTIMATION OF Mn 20.0538 0.0039 — — 20.083 — —
PARAMETERS FOR THE MODEL Si 0.24 0.015 0.002 0.37 — 0.005 —

1 1
d[pct C] d[pct Cr] d[pct Mn]
5 Wm 1 1 1 1 A. Distribution Ratios of Blown Oxygen among Elements
dt dt dt
This is a very important parameter for the model. It has
2 2
d[pct Si] dt 4130 C. Equilibrium Constants of Reactions
1 cp,m (T 1 dT ) [21] been either treated using different ways or simply evaded lggFeO 5 (NCaO 1 NMgO)(NSiO2 1 0.25 NAlO1.5)
dt 100 in the literature, and it has not been reasonably determined T For reaction [6], from
1 QO (1 2 h )dtrOcp,OTg 1 Qsubdtrsubcp,subTg until now. Tohge et al.[8] have used the free energy of each 1720
oxide and the concentration difference of each element to 1 NMnO(NSiO2 1 0.45 NCrO1.5) [27b] [C] 1 [O] 5 {CO}, lg KC-O 5 2525/T 1 1.43[21]
T
1 2
Wm d[pct C] MCO determine the parameter. Relatively, that is more reasonable, [Fe] 1 [O] 5 (FeO), lg KFe-O 5 6320/T 2 2.734[22]
1 2 dt c T but the concentration difference has to be determined. It
100 dt MC p,CO g 1
1246
NAlO1.5 NSiO2 1
42
N N
may be believed that the distribution ratios of the blown T T MnO AlO1.5 it can be obtained that
Wmdt d[pct Cr] MCr2O3 oxygen among the elements dissolved in the steel would be
1
1 Ws 2
100 dt 1 2MCr proportional to the Gibbs free energies of their oxidation
reactions at the interface. From this consideration, the fol-
1
692
N N
T CrO1.5 SiO2
lg Kc 5 23795/T 1 4.164 [31]
For reactions [7] through [9] and [15], the equilibrium con-
d[pct Mn] MMnO d[pct Si] MSiO2 lowing relationships should be held: stants were taken, respectively, as
1
dt MMn
1
dt MSi 22 c p,s (T 1 dT )
DGC
lggCr2O3 5 lggFeO 2
1594
T
(NCaO 1 NMgO) lg KCr 5 24,025/T 2 10.566[18] [32]
xC 5 [23] [28a]
1 (qloss 1 q5)dt DGC 1 DGCr /3 1 DGMn 1 DGSi /2 lg KSi 5 17,770/T 2 6.122[16] [33]
664 593
2 N 2 N
The appropriate rising rate of the bath temperature is T MnO T SiO2 lg KMn 5 8695/T 2 3.93 [16]
[34]
DGCr /3
MCr2O3 d[pctCr] xCr 5 [24] lg KCr-C 5 35,410/T 2 23.058 [35]
1 1
dT MMnO d[pctMn] DGC 1 DGCr /3 1 DGMn 1 DGSi /2 1859
5 cp,s T 1 lggCr2O3 5 lggFeO 2 (NCaO 1 NMgO)
dt 2MCr dt MMn dt T
DGSi /2 [28b]
D. Mass-Transfer Coefficient of Carbon in Liquid Steel
MSiO2 d[pctSi] xSi 5 [25] 774 692
2 1
d[pctC] d[pctCr] DGC 1 DGCr /3 1 DGMn 1 DGSi /2 2 NMnO 2 NSiO2
1 2cp,mT 1 T T The following expression was used to calculate the mass-
MSi dt dt dt
DGMn transfer coefficient of carbon in the liquid steel (kC):[23]
xMn 5 [26] 3540
2
d[pctMn] d[pctSi] DGC 1 DGCr /3 1 DGMn 1 DGSi /2 lggSiO2 5 lggFeO 2 (NCaO 1 NMgO) kC 5 0.8 r eq21/4 DC1/2g1/4 [36]
1 1 T
dt dt [29a] At a sufficiently high velocity of gas stream, each bubble
1475 1068 593
100 2 NMnO 2 NAlO1.5 2 N in the bubble group roughly has a uniform size.[24] On the
2 (Q r c ((12 h )Tg 2Tg,0) [22] T T T CrO1.5 basis of the results of water modeling in a prior work,[13]
Wm O O p,O B. Activity Coefficients the average diameter of a bubble (db) was taken to be 2.5
MCO d[pctC] 4130 1720 cm, with DC 5 7.46 cm2/s.[5]
The activity coefficients of the components in the molten lggSiO2 5 lggFeO 2 (NCaO 1 NMgO) 2 NMnO
1qloss 1q5)1cp,COTg T T
MC dt steel can be estimated using the interaction parameters e(j)i . [29b]
The values of e(j) i were all taken from Reference 19 in the
1246 692 E. Estimation of the Area of Reaction Interface
1
d[pctC] d[pctCr] d[pctMn] 2 NAlO1.5 2 NCrO1.5
2 DHC 1DHCr 1DHMn present work (Table I). T T
dt dt dt There is no CaF2 in the slag, and the temperature was During the AOD refining process, the oxidative or reduc-
lower at the early stage of the refining. Based on the expres- 1475 tive reactions of elements will occur at the bubble surface.

22 /(100c
d[pctSi] lggMnO 5 lggFeO 2 (NSiO2 1 0.45 NCrO1.5) Therefore, the area of the reaction interface (Area) will be
1DHSi p,m 1100cp,sWs /Wm) sions of the activity coefficients of the components in the T
dt molten slag used for electroslag remelting of stainless [30a] approximately the total surface area of the bubbles. Using
steel,[18] Eqs. [27a] through [30a] were employed to calculate 36 the expression for estimating the total number of bubbles,
where qloss 5 q1 1 q2 1 q3 1 q4 1 qu . In the present 2 N given by Diaz et al.[25] as nb 5 6 QHb /(pd b3 ub), the follow-
the activity coefficients of the oxide components in the slag T AlO1.5
work, the refractory lining with the shell was referred to ing can be obtained:
approximately as a multilayer plate; q1, q2, q3, and q4 were, for the early period of the refining.
1720
respectively, determined in terms of the one-dimensional lggMnO 5 lggFeO 2 (NSiO2 1 0.45 NCrO1.5) Area 5 6 QHb /(dbub) [37]
3540 T
transient heat-conduction problems; q5 was taken to be lggFeO 5 (NCaO 1 NMgO)(NSiO2 1 0.25 NAlO1.5) [30b] where Hb 5 95 cm for an 18-t AOD vessel and ub can be
W1Cp,1 DT and T 42
2 NAlO1.5 found from Eq. [38]:[26]
qu 5 (q1 1 q2 1 q3 1 q4) 3 15 pct. 1475 T
It should be pointed out that Eq. [22] included not only 1 NMnO(NSiO2 1 0.45 NCrO1.5) [27a] ub 5 1.02 (gdb /2)1/2 [38]
the influence of the changes in the amounts of molten steel T
For the latter period of the refining, at which the bath
and slag, but also the heat needed for the rise of the refrac- 1068 36 temperature has been evidently heightened, Eqs. [27b]
tory-lining temperature during the refining process. The ini- 1 NAlO1.5NSiO2 1 NMnONAlO1.5 F. Oxidation Enthalpies of Elements
T T through [30b] were taken in this work. Also, the solubility
tial and boundary conditions of the model involve the initial of Cr2O3 in the slag is about 5 mass pct.[20] Consequently, The oxidation enthalpies of the elements in the steel were
amounts, chemical composition, and temperature of the mol- 593 aCr2O3 was taken to be 1 if (Cr2O3) $ 5 mass pct in the estimated in terms of the direct oxidation reactions by the
1 NCrO1.5 NSiO2
ten steel and slag; the blowing rates and temperatures of the T present work. blown oxygen. The standard enthalpies at (298 K) of the

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 33B, FEBRUARY 2002—115 116—VOLUME 33B, FEBRUARY 2002 METALLURGICAL AND MATERIALS TRANSACTIONS B
 carbon in the molten steel (dC2/dt) is also calculated. Com- pt average absolute pressure in AOD vessel, atm
Table II. Change in Temperature Caused by Added 1 kg Alloy for 1 t Molten Steel, K[29] paring these two rates, if .dC1/dt. , .dC2/dt., the rising Q total gas flow rate, cm3?s21
High carbon Middle carbon Low carbon Middle carbon High carbon Electrolitic Metallic rate of the bath temperature is computed with .dC1/dt., the Qph physical heat of lime melting and dissolving in
Alloy Fe-Cr Fe-Cr Fe-Cr Fe-Mn Fe-Mn Mn Ni Mn-Si 75 Si-Fe oxidation rates of the other elements, and the corresponding molten slag, J?g21
DT, K 22.3 22.0 21.8 22.01 22.26 21.98 21.39 21.58 10.57
oxidative enthalpies. Otherwise, the relevant amount of oxy- Qch chemical heat of lime dissolving in slag melt,
gen consumption to carbon is achieved using .dC2/dt., and J?g21
the oxidation rates of chromium, silicon, and manganese are Qi flow rate of i gas, cm3?s21
found using the surplus oxygen amount and the oxygen Qsub total flow rate of inert gas, cm3?s21
respective oxide formation involved in the following equa- distribution ratios obtained from the Gibbs free energies of q1 heat loss by conduction from bottom of the ves-
tions were all taken from Reference 27, and the relevant their oxidation reactions. Then, the appropriate rising rate sel, J?s21
heat capacities at constant pressure with the enthalpies of of the bath temperature is also calculated. Finally, the con- q2 heat loss by conduction from the lower of the
solution formation were taken from Reference 28. centrations of the components in the steel and the bath tem- vessel, J?s21
perature can be obtained by numerical integrating, and the q3 heat loss by conduction from the upper of the
DHC 5 DHCO 2 DH[C] 2 1/2 DHO concentrations of the components in the slag melt can be vessel, J?s21
5 11,852 2 (2.367 Tg 1 1.708 3 1024T g2 [39] determined from the mass balance of the system. The effect q4 heat loss by conduction from top of the ves-
of FeO on the mass and heat balances can be neglected, sel, J?s21
1 3.835 3 103 /Tg) because it is referred to as an intermediate product of the q5 heat loss absorbed by refractory lining of the
gas-blowing refining. The ultimate output results are the vessel during bath rising temperature, J?s21
DHC 5 DHCr2O3 2 2DH[Cr] 2 3/2 DHO changes in the chemical composition, temperature and qu uncertain heat loss of the system, J?s21
5 11,519 2 (1.148T 1 4.4 3 1025 T 2 [40] amount of the liquid steel and slag melt, the distribution R gas constant (58.314), J?mol21?K21
ratios of oxygen among the elements, the decarburization rb mean radius of bubble, cm
1 1.5 3 104 /T ) rate with the blowing refining time. req mean equivalent radius of bubble, cm
DHSi 5 DHSiO2 2 DH[Si] 2 DHO [41] The model has been used to deal with and analyze the T bath temperature, K
austenitic stainless steel making (including ultralow carbon Tg , Tg0 temperature of gas and its initial value, K
5 30,658 2 (2.15 T 1 1.45 3 1024T 2) steel) in an 18-t AOD vessel and tested on data of 32 heats ub velocity of bubble, cm?s21
obtained in producing 18Cr9Ni grade steel. The application Walloy mass of alloy agents added, g
DHMn 5 DHMnO 2 DH[Mn] 2 1/2 DHO of the model to the AOD industrial practice and the results WCaO mass of lime added, g
5 7581 2 (0.845 T 1 7.38 [42] will be reported in Part 2 of the present work. Wm mass of liquid steel, g
Ws mass of slag, g
3 1025 T 2 1 6.69 3 104 /T ) x distribution ratio of oxygen for i component in
LIST OF SYMBOLES
The other physical constants were, respectively, taken to be liquid steel
rm 5 7.37 g/cm3,[18] rs 5 3.1, rO2 5 1.4277 3 1023, rAr 5 Area total reaction interface, cm2 g Raoultian activity coefficient of j component in
1.7821 3 1023, rN2 5 1.2499 3 1023 g/cm3, cp,m 5 0.8159, ai activity of i component slag melt
cp,s 5 1.1966, cp,scrap 5 0.7113, cp,O2 5 0.9184, cp,Ar 5 cp,i specific heat of i substance at constant pres- h utilization ratio of oxygen
0.5238, cp,N2 5 1.0376, cp,CO 5 1.0447, and cp,CaO 5 0.9205 sure, J?g21?K21 l heat conductivity of i material, W?cm21?K21
J?g-1?K21, from Reference 28. The thermal conductivities DC diffusion coefficient of carbon in molten steel, r density of i material, g?cm23
of the various refractory materials and shell (steel plate) cm2?s21
were lMg-Cr 5 0.0198, lMg-Al 5 0.0233, lcl 5 0.0177, las 5 db average diameter of bubble, cm Subscripts
0.0016, lmag 5 0.0243, and lsh 5 0.50 W?cm21?K21, also fi Henrian activity coefficient of component i in as asbestos board
from Reference 28. molten steel cl clay brick
Fig. 1—Flow chart for computer program. Mg-Al alumina-magnesite brick
DGi Gibbs free energy for oxidation reaction of i
element, J?g21 Mg-Cr chrome-magnesite brick
G. Estimation of the Cooling Effects of Addition Agents g acceleration due to gravity, cm?s22 Mag magnesite
Tmelt 5 1538 2 65[pct C] 2 30[pct P] 225[pct S] Hb rising height of bubble, cm m, s, l metal, slag phase and lining, respectively
The temperature drop caused by added alloy agents is
DHi oxidation enthalpy of i element, J?g21 sh shell
DT 5 zalloyWalloy 3 1023 /(Wm 3 1026) [43] 2 20[pct Ti] 2 8[pct Si] 2 7[pct Cu] [ ]—metal phase; ( )—slag phase; { }—gaseous phase; and
[46] [pct i] mass percent concentration of i solute in molten
2 5[pct Mn] 2 4[pct Ni] 2 3[pct Al] 2 2[pct V] steel, mass pct ^ &—solid phase
where zalloy is the chill factor, in K?kg21. The values of the
factor for some alloys are shown in Table II.[29] [pct i]e equilibrium concentration of i solute in molten
2 2[pct Mo] 2 1.5[pct Cr] 2 1.5[pct Co] 2 [pct W] steel at reaction interface, mass pct
The temperature drop caused by added lime is
(pct j) mass percent concentration of j component in ACKNOWLEDGMENTS
DT 5 WCaO(Qph 2 Qch)/(Wmcm 1 Wscs) [44] V. NUMERICAL SOLUTION FOR THE MODEL slag melt, mass pct
Ki equilibrium constant for indirect oxidation reac- The authors gratefully acknowledge the support of the
Here, Qph 5 cp,CaO(T 2 TCaO) 1 Qmelt,CaO, Qmelt,CaO ' 1419.46 The model can be used to deal with the AOD refining National Natural Science Foundation of China (Grant No.
tion of i solute in molten steel
J?g21,[30] and Qch ' 1280.23 J?g21.[31] process of stainless steel, including the first and second 5947016). The Shanghai No. 5 Iron and Steel (Group) Cor-
KCr-C equilibrium constant of [C](Cr2O3) reaction
The following equation was used to estimate the tempera- blowing periods (also the third period, for the ultralow- poration kindly offered the data on AOD heats.
kC mass transfer of carbon in molten steel, cm?s21
ture drop caused by added crop ends and scrap: carbon steel). The flow chart for the computer program is
Mi mole mass of i substance, g?mole21
presented in Figure 1.
(Tmelt 2 Tscrap)cp,scrap 1 Qmelt,m 1 (T 2 Tmelt)cp,m Nj mole fraction concentration of j component in
DT 5 It can be seen from Figure 1 that, first, the appropriate
Wmcp,m 1 Wscp,s slag REFERENCES
distribution ratios of blown oxygen are determined from the
obtained Gibbs free energies of the oxidative reactions of nb total number of bubble
[45] ni mole flow rate of i component, mole?s21 1. W.H. Ray and J. Szekely: Process Optimization with Applications in
elements. Further, the decarburization rate related to the Metallurgy and Chemical Engineering, John Wiley & Sons, Intersci-
21 [28]
where Qmelt,m ' 251 J?g ; the melting point of the steel oxygen flow rate (dC1/dt) is calculated; on the other hand, PCO partial dimensionless pressure of carbon ence, New York, NY, 1971, pp. 310-19.
(Tmelt) can be determined by the following expression:[32] the decarburization rate controlled by the mass transfer of monoxide 2. S. Asai and J. Szekely: Metall. Trans, 1974, vol. 5, pp. 651-57.
Pt total dimensionless pressure in AOD vessel 3. J. Szekely and S. Asai: Metall. Trans, 1974, vol. 5, pp. 1573-80.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 33B, FEBRUARY 2002—117

118—VOLUME 33B, FEBRUARY 2002 METALLURGICAL AND MATERIALS TRANSACTIONS B
 used to predict phosphorus content of liquid steel, but each model giving a different result. The
FUNDAMENTALS OF STEELMAKING METALLURGY CHAPTER 9 strength of thermodynamics in choosing a desulphurizer is also demonstrated through worked
THERMODYNAMICS AS TOOL FOR PROCESS ANALYSIS out example. An example of deep thermodynamic analysis of deoxidation of steel with
aluminum is also given.
Brahma Deo
Visiting Professor A detailed background reading material on thermodynamic fundamentals related to dilute
School of Minerals, Metallurgical and Materials Engineering solutions in steel, optical basicity, ionic theory of slag, cell model of slag, and molecular theory
of slag can be found in Chapter 2 of [1]. Details of quasi regular solution and it application to
IIT Bhubaneswar, India calculation of activity of iron oxide in slag are given in [2,3].

bdeo@iitk.ac.in; bdeo@iitbbs.ac.in The set of worked out examples have been chosen to cover a wide spectrum and are essentially
of educative value in process analysis of steelmaking. Otherwise, to a first time student learner,
ABSTRACT researcher, teacher, and steelmaker alike, wanting to learn and apply thermodynamics, it
Thermodynamics is a powerful tool to analyze the conditions required for a reaction to go in a appears like a maize and one does not where to start from and focus upon, which model to
forward or backward direction. But when the chemical reactions are written, the students and chose or rely upon. For actual application on shop floor, only after learning the basics one can
researchers often make errors in choice of standard state of different reacting components and proceed to tune the chosen model to process data, because model by itself is inadequate and
reaction products. The steps involved in conversion of one standard to another are explained different models may predict widely different results.
through worked out examples. In steelmaking, the activity of iron oxide in slag plays a key role
in oxidation of elements across slag metal interface. Over past five decades, a large number
of approaches have been published on calculation of iron oxide activity in slag through several 2.0 Selection and change of standard state in chemical reactions
models, like ionic theory of slag, optical basicity, regular solution model, cell model of slag,
and a number of empirical relations developed thorough actual experiments. However, each of The first step in equilibrium calculations is to choose standard state or reference state of
these methods result in different activity of iron oxide in slag, sometimes differing by a factor reactants and products. The reference states are: are pure solid (<>, with crystal structure
mentioned ), pure liquid ({ }), gaseous ( ( )g), dissolved state of 1 Mass% and ( [ ]1 mass %), and
of two or more, and the researcher is at a loss to choose which one. A worked out example is
dissolved state of infinite dilution( [ ] ID ]. For example, in the reaction given below, the
given to bring out this aspect. Similarly, several approaches can be used to predict phosphorus
reference state of Al, by sign convention, is pure Al liquid (using { } brackets), of oxygen is
in steel, like ionic theory of slag, Healy’s model, molecular theory of slag, optical basicity, and
pure gas (using ( )g), and of Al2O3 is pure solid (using <>).
quasi-regular solution model. Again, each of these approaches result in different phosphorus
content of liquid steel for the same slag composition. A worked out example is given to bring
2 {Al} + 3/2 (O2)g = <Al2O3> (9.1)
out these differences to show that in all cases, with one model or another, the calculated results
indicate only a possible trend. What is required in an actual application to steelmaking process It is to be noted that Al2O3can be present in different crystalline states and that has also to be
is to tune the model to a practical situation. The usefulness of simple thermodynamic mentioned, depending upon temperature, say α Al2O3, or β Al2O3, etc.
calculations in selecting a proper desulphurization agent, like CaO, CaO+ Al, Cao+ C+ Al, is
shown through a worked out examples. A worked out example is also given on deoxidation of For concentrated solutions the Raoults is stated as ai= γi* Xi where Xi is mole fraction, ai is
steel with Al in order to show the power of thermodynamic analysis through simple Raoultian activity and γi is Raoultian activity coefficient. For ideal solutions γ i = 1.
thermodynamic calculations. The strength of present work is essentially of its educative nature
in analyzing and applying thermodynamic fundamentals to steelmaking. The students, For dilute solutions in steel 𝑎𝑖= γoi ∗ Xi . Henery’s law has two forms:hi= fi*mass% (i) or ℎ𝑖′ =fi*
researchers and teachers can use it to understand or to explain the concepts involved. Xi.Here,both hi and ℎ𝑖′ are Henerian activity, but with different reference states. For solutions
obeying Henery’s law, fi=1. And hence hi=1 for 1 mass% in dissolved state. ℎ𝑖′ is Henerian
activity defined at infinite dilution when concentration is expressed in mole fraction and when
1.0 Introduction fi=1 at infinite dilution, and ℎ𝑖′ is equal to Xi and thus ℎ𝑖′ is never unity whereas hi=1 iffi=1
Thermodynamics is fundamental tool for process analysis of steelmaking. The students and For dilute solutions in steel [1], the general relationship between Raoult’s and Henrey’s law is
researchers often make error in calculation of appropriate equilibrium constant because the in terms of equality
standard states are not properly chosen for the process involved. Further, iron oxide activity
plays a crucial role in steel refining. But several approaches to calculate iron oxide activity are 𝑎𝑖 ℎ𝑖 ℎ𝑖′
= = = 𝑓𝑖 (9.2)
available. When we use these approaches different values are obtained, sometimes differing by γ𝑜
𝑖 𝑋𝑖 𝑚𝑎𝑠𝑠%𝑖 𝑋𝑖
a factor of two or more. This is demonstrated through a worked out example by using ionic
theory of slag, regular solution model, optical basicity model, and empirical relation based on The Henrian activity coefficient fi can be more than one (positive deviation), equal to 1 (ideal
experiments. Similarly, worked out example is given to show how different models can be condition) and less than 1 (negative deviation). Since, mathematically, a i= γi* Xi= fi∗ γoi ∗ Xi
the product 𝛾𝑖𝑜 ∗ 𝑓𝑖 is equal to γi which is Raoultian activity coefficient when reference state 2 [Al]1 mass% + 3 [O]1 mass% = <Al2O3>
is pure substance, and 𝛾𝑖𝑜 is defined for infinite dilution. Obviously, γi = 𝛾𝑖𝑜 when fiis unity. γiis
always positive and can be equal to 1 (ideal solution) more than 1 (positive deviation) and less is as follows.
than 1 for negative deviation. So, activity and activity coefficients will always be
positive.Activity itself cannot be more than one, except for the case of Henrian activity hi, Soultion
(reference state 1 mass%); for example if Henery’s law is obeyed up to 2 mass % then hi=2.
In the reaction
By convention, the dissolved state is represented by square brackets. In case of 1 mass % as
reference state, 2 {Al} + 3/2 (O2)g = <Al2O3> G1o ==-1685627 + T*11.5368
2 [Al]1mass% + 3 [O]1mass% = <Al2O3> (9.3)
As indicated by appropriate symbols, the chosen standard state of Al is pure liquid Al, standard
If the square brackets do not have a subscript then it is assumed by default that the standard state of oxygen is pure oxygen gas and standard state of alumina is pure solid alumina; one can
state of the dissolved element is 1 mass%. In the case of infinite dilution as reference state, Eqn define in addition the crystallographic structure of alumina, α, β, etc. We cannot straight away
(9.1) is written as use the above equation to calculate the dissolved oxygen content of liquid steel containing, for
example, 0.02% Al in equilibrium with solid alumina. It is necessary to go through a series of
2 [Al]ID + 3 [O]ID = <Al2O3> (9.3) steps to finally get the equation

The selection of standard state (or reference state)of reactants and products should be done 2 [Al]1 mass% + 3 [O]1 mass% = <Al2O3>
depending upon the actual physical state of the reactants in the system. Otherwise the
equilibrium constant calculated will not correspond to the conditions which actually exist in a The steps to arrive at above equation are as follows. The corresponding free energy values are
system. For example, if a particular component in a system exists in liquid state then the taken from Table 1.
standard state for that component should be chosen as pure liquid, and if it is present is in a
dissolved state then reference state should be dissolved state [ ] 1 mass %,or [ ]ID. 2[Al]1 mass%= 2{Al} G o2 = 2* (63118+ T*27.8806)
This represents the free energy involved in change of standard state of Al from 1 mass% to
If the reference state is pure liquid then activity in pure liquid state would be unity, and similarly {Al}.
the activity in pure solid state will also be unity. But it does not mean that, all other things
remaining same, the equilibrium constant with reference state chosen as pure liquid will be Similarly for oxygen, from Table 1
same as the equilibrium constant with reference state chosen as pure solid. This is because a 3
free energy change is involved when solid state is changed to liquid state and this is reflected 3[O]1mass%  (O2 ) g G 3o  3 * (11740  T * 2.8842)
in the equilibrium constant value.In fact equilibrium constant is solely dependent on the
2
particular reference states chosen for the set of reactants and products in a particular reaction.
Each of the reactants can have a different reference state in a given reaction or same state if where G 3o is the overall equation representing free energy of dissolution and also the free
they are actually present in same state. It is clear that the standard state can be changed (i.e. energy involved in the change of standard state of pure gaseous oxygen to 1 mass%.
from one state to another, say from pure solid to pure liquid) only by including the free energy
involved, say from sold to liquid, or liquid to gas, or from [ ]1 mass % to [ ]ID etc. Now we can write all the steps together

It is important to note that for the components dissolved in slag only Raoult’s law is applicable. 3
Both Raoult’s and Henery’ law are applicable only for the elements dissolved in the metal and 2 {Al}pure liquid + (O2)g = <Al2O3> G1o ==-1685627 + T*11.5368
2
difference is the choice of standards state.
2 [Al]1 mass% = 2{Al}pure liquid Go2 = 2 * (63118  T * 27.8806)
The examples given below clarify how one can change the standard state of reactants and 3
3[O]1mass %  (O 2 ) g G 3o  3 * (11740  T * 2.8842)
products. 2
2 [Al]1 mass% + 3 [O] 1 mass% = <Al2O3> G o4 = G1o + G o2 + G 3o
2.1 Change of standard state for Al dissolved in metal
= -1524171 +75.9606 *T
If the free energy data for the reaction is (from Table 1)
2 {Al} + 3/2 (O2)g = <Al2O3> is G1o = =-1685627 + T*11.5368, j/mol 2.2 Change of standard state of manganese from pure liquid to Mn dissolved in metal
then the procedure for finding out the free energy for the reaction Suppose the reaction is
{Mn} + [O]1 mass% = (MnO).
It is required to change the standard state of Mn from {Mn}, ie. pure liquid, to [Mn] 1 mass% so
as to find out the free energy for the reaction
[Mn]1mass% + [O]1mass% = (MnO).

Assume that Mn follows ideal Raoult’s law when dissolved in liquid steel.

Solution
The reference state of dissolved Mn in the equation {Mn} + [O] 1 mass% = (MnO) is pure Mn The reference state of MnO is now liquid MnO and the free energy of solution of MnO in slag
liquid. Now, if we want to change the standard state from {Mn} to [Mn] 1 mass% then the steps is incorporated into G12
o
through G10
o

needed are Care must be taken in writing the equilibrium constants as well. The equilibrium constant for
the reaction
{Mn}pure liquid + [O]1 mass% = (MnO) G 5o
{Mn} + [O]1 mass% = <MnOα>, G o7
 

[Mn]1mass% {Mn}pure liquid  G i
Mn
  RT ln  io 0.5585 / M Mn  is to be written as
 a MnO
G 13
o
=-RTlnK where K 
[Mn]1mass% + [O]1 mass% = (MnO) G o7  G 5o   G i
Mn
a Mn * [h o ]

It is to be noted that if since MnO is present as pure solid aMnO = 1.If Raoult’s law is obeyed
In the equation [Mn]1mass% {Mn} the value for change of standard state is (see Chapter 2)is
by metal then aMn=XMn because the reference state of Mn is pure Mn liquid. if Mn did not
follow ideal behavior (Raoult’s law) in iron then a Mn=γMnXMn where γMn is the activity
 

partial molar free energy of solution of Mn in liquid iron =   GiM   RT ln  io 0.5585 / M Mn  coefficient of Mn. Similarly, in case of dilute solution, ho= mass%O if oxygen follows ideal
. MMn is atomic mass of Mn (= 55.5). Henery’s law, otherwise ho= fO*mass%O where fO is the Henrian activity coefficient. Now
consider the reaction
In principle, whenever a free energy equation is written it must be clearly mentioned as to what
the chosen reference states are. As per convention, the term (MnO) indicates that MnO is
[Mn]1 mass% + [O]1 mass% = (MnO), G14
o
present in dissolved from liquid slag and in writing G 5o the free energy of dissolution of MnO
The equilibrium constant is to be written as
in slag has been taken into account. The example given below illustrates the point.
a MnO
G14o
=-RTlnK where K 
[h Mn ] * [h o ]
2.3 Change of standard state for MnO in slag and Mn in metal for the given reaction Here aMnO cannot be assumed to be unity because MnO is present in the dissolved form, in fact
{[Mn} + [O]1 mass% = <MnOα> aMnO= XMnO if it follows Raoult’s law otherwise aMnO=γMnO* XMnO where γMnO is the Raoultian
activity coefficient of MnO in slag. Similarly, h Mn=mass% Mn and hO=mass %O if both Mn
Solution and O fobey Henery’s law, otherwise hMn=fMn*mass%Mn and hO=fO*mass%O, where fMn and
fO are the Hnerian activity coefficients for Mn and O in metal, respectively.
The steps involved are as follows

{Mn} + [O]1 mass% = <MnOα> G 8o

<MnOα>= {MnO} G 9o
{MnO}= (MnO) G10
o

[Mn]1mass%={Mn}pure liquid G11

o

[Mn]1 mass% + [O]1 mass% = (MnO) G12

o
= G 8o + G8o  G 9o  G10
o
 G11
o