A Method for the Analysis of the Controlling Step During Vacuum Decarburization of Steel

Most of the previous work on calculating fluxes of carbon during ultra-low carbon steel production under vacuum have relied on simplifications. Using Mathematica, the authors can generate better solutions with excellent visualizations.

by Renato Necco Castro and Andre Costa e Silva

During vacuum treatment of steel the reaction C+O=CO can be used for carbon deoxidation in high and medium carbon steels or to reduce the carbon content in low carbon steels such as stainless, electrical and Interstitial Free (IF) steels.  To predict the progress of this reaction and to define optimum conditions for its occurrence, mathematical models of different levels of complexity have been proposed.

One of the fundamental questions when modeling the carbon oxidation reaction is defining the rate-controlling step in the process kinetics.  When this is well established, model formulation can be significantly simplified without compromising the quality of the results.  Otherwise, more complex mixed-control models must be used, which increases the computational requirements, in particular when dynamic process control applications are envisaged.

In the present work a Mathematica notebook is used to define the conditions under which carbon transport will control the overall process kinetics. Using this notebook, limits for the validity of the application of simplified models can be set.  Furthermore, a better understanding of the conditions under which each step might play a significant role in the overall kinetics of the process can be achieved, without the use of simplifying assumptions

The reaction C + O = CO

The reaction C + O = CO, between carbon and oxygen dissolved in steel is one of the most important processes in steelmaking.  In particular, when performed under vacuum, this reaction can be used either to accomplish carbon deoxidation in medium and high carbon steels or to reduce carbon content in the case of stainless, electrical and Interstitial Free (IF) steels.  In the first case, improvements in steel cleanliness are achieved, as the amount of deoxidants that form liquid or solid products is significantly reduced [1].  In the production of IF steel, the reduction of CO pressure reduces the level of steel oxidation and makes possible the achievement of carbon contents lower than 20 ppm (weight).  Furthermore, the  oxidation of chromium in stainless steel and silicon in high silicon electrical sheet can be significantly reduced, when low carbon contents are aimed for [1].  The knowledge of the ideal conditions for the performance of these processes is essential for high productivity and yield.

Decarburization reaction kinetics

Heterogeneous reactions

Steel decarburization is an heterogeneous process.  As such, besides the chemical reaction proper, transport of the reactants (carbon and oxygen dissolved in steel) to the reaction site as well as transport of the product gas (CO) can interfere with the overall rate with which the process proceeds. The decarburization reaction can be described as:

C + O = CO 	(1)

ΔG = Δ(G^0) + R T Log(((Subscript[P, CO])/((Subscript[a, C]) (Subscript[a, O]))))        (2)

The driving force for the reaction can be quantified by the departure from equilibrium between steel and gas.  The simplest way to express this driving force is through the free energy change associated with the process, given by equation 2.  As the carbon and oxygen contents of the steel are in equilibrium with a given CO pressure (or chemical potential) different from that inside the vacuum degasser, the driving force can also be expressed by this potential difference.
When various steps take part in the overall decarburization process, two basic limitations will define the progress of the process: a) as there is no consumption or accumulation of mass at the reaction interface, the molar fluxes of C,O and CO are related and b) the total driving force available for the process will be distributed among the steps in inverse ratio to the resistance to the progress of each step.

In general, if the chemical reactions are taken to be sufficiently fast not to interfere with the overall kinetics at steelmaking temperatures [2], the mass transfer steps will use all the driving force available. The mass transfer fluxes can be expressed as:

(Subscript[(j ), O ])= (Subscript[k, O]) ((Subscript[C, O]^b) - (Subscript[C, O]^i))	(3)

(Subscript[(j ), C ])= (Subscript[k, C]) ((Subscript[C, O]^b) - (Subscript[C, O]^i))	(4)

(Subscript[j, CO])=-(Subscript[k, CO])((Subscript[C, CO]^b)-(Subscript[C, CO]^i)) = -(Subscript[k, CO]) (Subscript[C, T])((Subscript[X, CO]^b) - (Subscript[X, CO]^i))

(5)

Subject to the condition:

(Subscript[j, O]) = (Subscript[j, C]) = (Subscript[j, C O])		(6)

The use of molar concentrations in equations 3 to 5 is practical for flux calculation but is not a good way to visualize the division of the driving force. 
When the concentration difference in any one of the equations is very small, this indicates that this step is not the rate limiting one.  Conversely, when almost all the driving force is used in one step, this is said to be the rate controlling step [2].  Some simple methods of visualization of the division of the driving force among the different steps have been proposed (e.g. [3])  but their application is severely limited when two reactants are transported in the same phase, as in the present case.  These problems are discussed in detail elsewhere [4].

Thus, the most common alternative for the evaluation of the decarburization process [5] [6] [7] is the solution of equations 3, 4 and 6, through the elimination of the unknown interfacial concentrations.  In the present case, transport in the gas phase is assumed not to play an important role in the process.  The resulting flux solution is obtained through Mathematica, defining auxiliary functions to perform the conversion of concentration units and equating the unknown interfacial concentrations:

(Subscript[j, c]^global) = (((Subscript[C, c]^b)(Subscript[k, c]) + (Subscript[C, c]^b)(Subscript[k, c]) - (Sqrt[((((Subscript[C, c]^b)(Subscript[k, c])))^2) - 2  (Subscript[C, c]^b)(Subscript[k, c]) (Subscript[C, c]^b)(Subscript[k, c]) + ((((Subscript[C, c]^b)(Subscript[k, c]) ))^2)+ 4 (Subscript[k, c])(Subscript[k, o])(Subscript[K, E  Q])(Subscript[p, c  o])]))/2)

Flux Calculation

The following Mathematica functions are defined as auxiliary functions.

The function cb is defined to convert  concentration in steel from weight percent (#1) in mole fraction (cb) given the atomic mass of the element (#2):

cb=((7*#1)/(#2*100))&;

The interface concentration ci can be calculated if the bulk concentration (#1), the molar flux (#2) and the mass transfer coefficent (#3) are known :

ci=((#1-(#2/#3)))&;

The flux is calculated taking in consideration the carbon oxygen equilibrium at the interface and selecting the valid solution of the quadratic equation, part [1,1,2]:

Clear[pctob,pctcb,fluxomolar,kc,ko,KEQ,pco]

fluxo=Simplify[Solve[ci[cb[pctcb,12],fluxomolar,kc]*
               ci[cb[pctob,16],fluxomolar,ko]==
                  KEQ*pco,
                  fluxomolar][[1,1,2]]];

The analysis of the effect of different parameters on the resulting global flux is somewhat complex, so it is usually restricted to limiting condition cases [6].  The application of Mathematicab makes possible the visualization of the effect of all parameters, without simplifying assumptions.  In particular, the effect of chemical composition and CO pressure can be evaluated.  To perform this analysis, some experimental data is required, however.

Experimental Data

The processes that involve carbon and oxygen absorption and removal from liquid steel are among the most investigated in steelmaking [8] [5] [6] [7] [9].  Several experimental techniques have been used to evaluate the kinetics of these processes and to determine the rate controlling steps.  A detailed overview of these experiments can be found elsewhere [4].  There is agreement that in the usual conditions that prevail in the refining of medium and low carbon steels the chemical reaction is sufficiently fast so that transport phenomena control the process kinetics.  Suzuki and co-workers [6] have performed experiments with sufficient gas flow to guarantee that transport in the metal was the controlling step.  Under these conditions they have noticed that above  a   ratio of 0.6910.11, oxygen transport controls the overall process.  Below this ratio, carbon transport controls.  For the very low CO pressures used, Suzuki and co-workers did not observe the occurrence of mixed c!
ontrol.  They reported values of

kc=0.051;
ko=0.035;
KEQ=2.43*10^-3*49/(12*16*10^4);

Decarburization modeling

Three strategies have been used for the modeling of steel decarburization, in special for the RH (Rhurstahl-Heraus) process [1].  One strategy consists in assuming a priori that under the conditions prevailing in the process, the overall decarburization kinetics is controlled by carbon transport in the liquid steel [10, 11].  An alternative is to assume that either carbon or oxygen transport in the liquid steel control the process and there exists a pre-established criterion for control switch [12-14].  Finally, one can assume a mixed control model with the inherent difficulties [15].
If the mass transport coefficients are known, the third option leads to the best results but requires more computational resources.  To use the second strategy, a criterion for the transition from oxygen to carbon transport control must be defined.  Yamaguchi and co-workers [13], have started with the 0.69 ratio proposed by Suzuki and co-workers [6], and later adjusted a ratio of 0.66 to the experimental data;  Kouroki [14] proposed the use of a  ratio of 1.  This strategy will only be acceptable if the assumption of the absence of a region of mixed control is valid.  Evidently all strategies assume that the transport of CO in the gas is sufficiently fast so that this stage does not interfere with the overall kinetics.

Controlling Steps in the decarburization process

Plotting the global flux as a function of the carbon and oxygen content of steel for a given CO pressure, shows that there are three regions concerning flux behavior: the first one, where the flux is not a function of carbon content, indicating that oxygen transport controls the overall kinetics of decarburization; the second region is equivalent to the first, but the flux is only a function of carbon content, indicating that its transport controls the process.  Each one of these regions is a plane in the graph.  The third region is the transition between the planes and corresponds to the range of compositions where mixed control exists.  The extent of this region defines the extent of the mixed control region of compositions.
A series of plots with varying CO pressures indicates that increasing this pressure results in smaller flux values and wider mixed control regions, as suggested by Suzuki and co-workers [6].  Animating this sequence of graphs shows the effect of pressure on the process.

Effect of CO Pressure,%C and %O on flux

To evaluate the effect of the composition variables, an animation is prepared with CO pressure varying in the usual vacuum range.  For low pressures (0.001) one sees the two planes that characterize control by C or O transport. The mixed transport region increases for high pressures.  Negative fluxes (i.e. %C and %O in steel that are below equilibrium with pco) are forced to zero.  Observe that the intercept of the flux surface with the zero plane is the typical equilibrium hyperbole %C %O=K pco.

Table[
      Plot3D[fluxo,
            {pctcb,0.001,0.07},{pctob,0.001,0.1},
    ViewPoint->{-0.795,-3.030,1.280},
         BoxRatios ->{1,1,1},
    AxesLabel-> {"Cb%", "Ob%","Fluxo"},
    PlotRange-> {{0,0.07},{0,0.1},{0,0.000015}},
    PlotLabel-> "pco=" pco],
     {pco,0.001,0.30,0.04}]

The Pi Function

The transition between planes is better characterized if one observes that the planes in the previous graphs satisfy the conditions:

((∂ (Subscript[j, c]^global))/(∂ (Subscript[C, c]^b))) = 0  and ((∂ (Subscript[j, c]^global))/(∂ (Subscript[C, c]^b)))  = 0

respectively.
Thus, the mixed control region can be characterized as the region where both partial derivatives are non-zero.  This is easily seen using an auxiliary function:

∏ =((∂ (Subscript[j, c]^global))/(∂ (Subscript[C, c]^b))) .((∂ (Subscript[j, c]^global))/(∂ (Subscript[C, c]^b)))

Regions where Pi is non-zero are regions where mixed control prevails.  The effect of CO pressure on the extent of the mixed control region is clearly seen through the use of the Pi function.

Visualization using the Pi Function

The regions of composition where mixed controls exist are the regions where the flux is a function both of %C and of %O.  In these regions the product of the partial derivatives of the flux with respect to to %C and %O is non-zero.  In regions where there is no mixed control one of the partial derivatives vanishes.  Thus Pi can be defined to help the visualization.

We need the following function:

derivs=D[fluxo,pctob]*D[fluxo,pctcb];

We can then plot the Pi function:

Table[
      Plot3D[Evaluate[derivs], {pctob,0.001,0.1},{pctcb,0.001,0.07},
       BoxRatios->{1,1,0.5},ViewPoint->{-0.911,-2.686,1.845},
       PlotPoints->40,
       PlotRange->{{0.001,0.1},{0.001,0.07},{0,1*10^-8}},
       PlotLabel-> "pco" pco ],
       {pco,0.001,0.01,0.009}]

Application to Process Conditions

Superimposing actual steel initial compositions and decarburization paths on contour plots of the Pi function shows to what extent mixed control will affect the decarburization process.  This way, the applicability range of simplified models such as the one proposed by Cordeiro and co-workers [10] or Yamaguchi and collaborators [13] can be verified.  For each pressure, two paths are plotted: a) a constant oxygen path, simulating a decarburization process with continuous oxygen addition (RH-KTB [16]) and b) a path in which carbon and oxygen decrease according to the stoichiometry of the decarburization reaction.  The sooner the paths reach the flat regions of the plots, the more accurate the prediction of the carbon transport control models.  It is evident that conditions prevailing in RH-KTB will more adequately meet this objective.  Furthermore, it is clearly seen that mixed control can occur in the vacuum decarburization of steel depending on the CO pressure and the initial composition of the steel.

Although only one starting point for the degassing process has been plotted, it is clear that if the starting composition lies in the region above the crest of the P graph, either mixed control or oxygen transport control will be experienced, both of which are less favorable for fast decarburization.

Example: Decarburizing a steel

Given the initial composition of the steel, what would be the decarburization path if oxygen and carbon reacted according to stoichiometry, or if oxygen was kept constant (oxygen blow during vacuum treatment)?

We use the following Mathematica code to obtain the visualization.

cinit=0.04;
oinit=0.08;
ofinal=oinit-((cinit-0.001)/12)*16;
stoich=Graphics[Line[{{oinit,cinit},{ofinal,0.001}}]];
ob=Graphics[Line[{{oinit,cinit},{oinit,0.001}}]];

Table[
      p1=ContourPlot[derivs, {pctob,0.001,0.1},
      {pctcb,0.001,0.07},
      ContourShading->False,
      Contours->15,
      PlotPoints->40,
      PlotLabel-> pco "pco",
      DisplayFunction->Identity];
      Show[p1,ob,stoich, DisplayFunction-> $DisplayFunction,
      PlotRange->All], {pco,0.001,0.01,0.009}]

Comparision with flux calculated using carbon control

Finally, the magnitude of the error incurred in calculating the flux, assuming carbon transport controls the process, can be evaluated if the relative difference between this and the mixed control model flux is plotted as a function of composition.

Definition of flux controlled by Carbon

fluxcarbon=kc*(cb[pctcb,12]-KEQ*pco/cb[pctob,16]);

pco=0.001
graf=Plot3D[(fluxcarbon-fluxo)/fluxcarbon,{pctcb,0.001,0.07},
          {pctob,0.001,0.1},
     ViewPoint->{1.108,3.026,1.033},
         BoxRatios ->{1,1,1}, PlotPoints ->20,
          AxesLabel-> {"Cb%", "Ob%","DF"},PlotLabel->pco,
         ColorOutput->Automatic,DefaultColor->Automatic]

Using these visualization methods it is possible not only to employ a simple and efficient model to describe decarburization in steel refining under vacuum, but to avoid the use of the model under conditions where its application would not be correct. Finally, it is possible to recommend measures to improve decarburization kinetics.

Conclusions

A simple Mathematica notebook can be used to visualize the conditions under which carbon transport control, oxygen transport control and mixed transport control prevail in the vacuum decarburization of steel.  The analysis performed using this notebook leads to the following observations related to the decarburization process:
a) The best conditions for fast carbon removal in vacuum decarburization are achieved when carbon transport controls the overall process kinetics.  Using this notebook, the composition/ pressure combinations where this occurs can be determined without simplifying  assumptions.
b) The concept that a sharp transition between carbon transport control and oxygen transport control exists, used in some models, does not seem to apply, in particular for the higher pressures that would prevail during the initial stages of vacuum treatment.
c) The advantages of using techniques such as RH-KTB were that oxygen content is kept at a high value so as to avoid its influence in the decarburization kinetics. This process can be easily demonstrated with the present notebook.
One of the limitations of the present technique is that the actual concentration- pressure- time path followed by the steel during the treatment cannot be easily visualized.  As the actual concentrations and pressures vary in different points of the steel being treated, this limitation might be very difficult to eliminate.

Acknowledgments

The support of CAPES-Brazil, FUGEMSS and Companhia Siderurgica Nacional-CSN are gratefully acknowledged.  The authors thank Prof. N. Tokumitsu and Mr. N. Cordeiro for helpful discussions.

Symbols

DG	Gibbs free energy change (KJ/mol)
R	Gas constant (J/mol K)
T	Temperature (K)
p	Pressure (atm)
a	Chemical activity
j	Molar flux (mol/m2 s)
k	Mass transfer coefficient (m/s)
C	Molar concentration (mol/ m3)
K	Equilibrium constant
M	Molar mass (kg/mol)
P	Mixed control factor
X	Molar fraction

Superscripts

0	Standard state
CO	CO gas
C	Carbon
O	Oxygen
b	Bulk
i	Interface
global	Mixed control
eq	Equilibrium

Subscripts

CO	CO gas
C	Carbon
O	Oxygen
b	Bulk
eq	Equilibrium
T,t	Total
L	Liquid

References

1.	Costa e Silva, A. and Mei, P.R., Steels and Specialty Alloys (in Portuguese). 2 ed. 1988, Sumare, Brasil: Eletrometal. 528pp.
2.	Richardson, F.D., Physical Chemistry of Melts. Metallurgy. 1974: Academic Press, London.
3.	Rist, A., Analogue diagrams for process metallurgists. ISIJ International, 32 (1992), 1034-1043.
4.	Castro, R.N., Cordeiro, N.B., Costa e Silva, A., and Tokumitsu, N. Metodologias para a analise da etapa controladora na cinetica de remocao de carbono do aco durante o tratamento sob vacuo. in XXVIII Seminario sobre Fusao, Refino e Solidificacao dos Acos. 1997. Campinas SP: ABM SP.
5.	Suzuki, K., Mori, K., Kitagawa, T., and Shibayama, T., Rate of Removal of Carbon and Oxygen from Liquid Iron. Testsu-to-Hagane, 62 (1976), 42-49.
6.	Suzuki, K. and Mori, K., Rate of desorption of CO from liquid iron. Transactions ISIJ, 17 (1977), 136-142.
7.	Ito, K., Amano, K., and Sakao, H., On the Kinectics of CO Degassing from Molten Iron by Argon Flow. Trasactions of the ISIJ, 17 (1977), 685-692.
8.	Oeters, F., Metallurgy of Steelmaking. 1989, Duesseldorf: Verlag Stahleisen. 512pp.
9.	Sano, M., Yetao, H., Sawada, T., and Kato, M., Decarburization and oxygen absorption of molten iron of low carbon concentration with blowing Ar-O2 mixture of low oxygen pressure. ISIJ International, 33 (1993), 855-861.
10.	Cordeiro, N.B., Tokumitsu, N., and Costa e Silva, A., Modelamento matematico da descarburagco no desgaseificador a vacuo com circulagco.  51 Congresso Anual da ABM. 1996. Porto Alegre: ABM, Sao Paulo, Brasil.
11.	Kuwabara, T., Umezawa, K., Mori, K., and Watanabe, H., Investigation of decarburization behavior in RH-reactor. Transactions ISIJ, 28 (1988), 305-314.
12.	Deo, B. and Gupta, S., Metal phase mass transfer and kinetic model for decarburization for RH process. Steel Research, 67 (1996), 7-11.
13.	Yamaguchi, K., Kishimoto, Y., Sakuraya, T., Fujii, T., Aratani, M., and Nishikawa, H., Effect of refining conditions for ultra low carbon steel in decarburization reaction in RH degasser. ISIJ International,32 (1992), 126-135.
14.	Kouroki, S., Obana, T., and Shirota, Y., Decarburization Behavior at the Low Carbon Region in the RH-Degassing Process. The Sumitomo Search, 44 (1990), 237-243.
15.	Takahashi, M., Matsumoto, H., and Saito, T., Mechanism of decarburization in RH degasser. ISIJ International, 35 (1995), 1452-1458.
16.	Kobayashi, H. and Donahue, F. Start up of KTB (Oxygen top blowing in RH) at National Steel Great Lakes division. 78th Steelmaking Conference Proceedings, 1995. Nashville: ISS-AIME, Warrendale PA.

ABOUT THE AUTHORS

Renato Necco Castro is a graduate student at the School of Metallurgical
Engineering (EEIMVR) of the UFF (Universidade Federal Fluminense).  His
research interests include the modeling of primary steelmaking processes.  He
plays chess and basketball.

Renato Necco Castro
Av. dos Trabalhadores 420
27260-740 Volta Redonda RJ
Brazil
Phone: +55 243 44 3014
FAX: +55 243 44 3029
email: necco@metal.eeimvr.uff.br

Andre Costa e Silva is a professor of metallurgy at the School of Metallurgical
Engineering (EEIMVR) of the UFF (University Federal Fluminense). He teaches
undergraduate numerical methods, fundamentals of steelmaking both at
undergraduate and graduate level and graduate thermodynamics.  His research
interest include computational thermodynamics, reaction and transformation
kinetics and the modeling of steelmaking processes.  Most recent publications
include, "Synthesis of MoSi2-Boride composites through in-situ displacement
reactions", Intermetallics, 5 (1997), 1-15 and "The application of thermodynamic
modeling to the prediction of slag-metal equilibrium", in Proc. of  XXVIII
Meeting on Steelmaking and Solidification of the Brazilian Society for
Metallurgy, ABM, Sao Paulo, SP, 1997, 439-455.  Andre is married, has one
teenage daughter, and his hobby is mountain biking.

Andre Costa e Silva
Av. dos Trabalhadores 420
27260-740 Volta Redonda RJ
Brazil
Phone: +55 243 44 3011
FAX: +55 243 44 3029
email: andrecs@omega.lncc.br

ELECTRONIC SUBSCRIPTIONS

Included in the distribution for each electronic subscription is the file VacuumDecarburization.nb containing Mathematica code for the material described in this article.