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An improved mathematical model to describe the decarburization process in basic oxygen furnaces for steelmaking is presented in this work. This model takes into account those factors or parameters that determine the bath-oxygen impact area, such as the cavity depth, the lance height, the number of nozzles and the nozzles diameter. In the thermal issue, the model includes the targeted carbon content and temperature. The model is numerically solved, and is validated using reported data plant. The oxygen flow rate and the lance height are varied in the numerical simulations to study their effect on the carbon content and decarburization rate.

The basic oxygen furnace (BOF) is the world most important technology for producing raw steel from molten pig iron [

In many shops, a static charge model is used to calculate the amount of charge and the amount of oxygen re- quired. Some plants have static models that depend on the type of operation and product mix. The static charge model uses initial and final information about the heat, e.g. the amount of hot metal and scrap, the end-point carbon and temperature. The main information is sent to the process computer, and the static model performs calculations at the beginning of the heat. The static model determines the output of the amount of oxygen to be blown and the amount of fluxes to be added to get the targeted carbon and temperature [

In [_{c} is defined as the melt carbon content where the decarburization reaction rate during the main blow is equal to the decarburization rate at the end of the blow. In [_{c}. Some mathematical models have been recently reported to study oxygen steelmaking decarburization [

A less complex mathematical model previously reported in the literature [_{c} = 0.3 wt%, and two decarburization mechanisms prevail: in the first one for

two significant drawbacks: the bath-oxygen interface area is assumed constant and independent of the blowing conditions, and the employed thermal model, based on the thermodynamic carbon content-temperature equilibrium, is rather simple. In this work, the above model is improved in two ways: 1) taking into account those factors or parameters that determine the bath-oxygen interface area, such as the lance height, the cavity depth, the number of nozzles of the lance and the diameter of the nozzles, and 2) modification of the thermal model to include the targeted carbon content and temperature. The improved decarburization model is validated using da- ta plant reported in the literature. Besides, numerical simulations are carried out to analyze the influence of the oxygen flow rate and the lance height on the carbon content and the decarburization rate.

The fast decarburization rates in the BOF are due to the large surface area available for the chemical reactions. When the oxygen jet hits the metal bath, a great amount of gas composed by CO + CO_{2} + inert evolves and the formed gas-metal-slag emulsion increases the rate of the refining reactions [_{Si}. The total time during which oxygen is blown is named here as blowing time t_{b}. These are concepts that allow to tackle in an easy way the decarburization and the thermal modeling issues. On the other hand, the carbon content dynamics is different for low and high carbon contents given that the control- ling mechanism is different for each case.

For

where t is the time (min), k_{d} is the mass transfer coefficient (m/min), S is the area (m^{2}) of the oxygen-melt im- pact zone, and V_{m} is the melt volume (m^{3}).

For

where Q_{O} is the oxygen flow rate (Nm^{3}/min), x_{in} is the mol fraction (dimensionless) of inert gas in the decarbu- rization zone, e.g. nitrogen, x_{CO} is the relative content (dimensionless) of CO in the CO + CO_{2} mixture, and W_{m} is the melt weight (g). In Equation (2) x_{CO} is determined from the equilibrium constant of the chemical reaction _{CO} is derived:

where T is the melt temperature (K) and P is the pressure (atm). Besides, f_{c} is the carbon activity coefficient (dimensionless), which depends on the carbon content and the melt temperature. A least square fitting yields an expression to determine f_{c} [

where A_{1} = 0.1666, A_{2} = –0.01585, A_{3} = 9.9613 × 10^{–7}, A_{4} = 3.0246 × 10^{–5}.

The decarburization reactions take place both in the oxygen-melt impact zone and in the gas-metal-slag emul- sion. However, is in the impact zone where a significant proportion of the carbon removal in the BOF occurs [

where a is the cavity shape factor (dimensionless) and L is the cavity depth (mm). The cavity depth depends on the lance height, i.e. the distance from the lance tip to the initial melt level. In this work, L is estimated from the expression [

where h is the lance height (mm) and L_{0} is a parameter which depends on the oxygen flow rate and on the geo- metrical characteristics of the particular lance being used [

where k_{α} is a coefficient (dimensionless) according to the nozzle angle, n is the number of nozzles in the lance, and d is the nozzles diameter (mm).

The pig iron, scrap, iron ore and fluxes are charged into the BOF. The supersonic oxygen jet is injected at high flow rates through a water cooled lance. The generated gases and the iron oxide fumes exit from the mouth of the furnace. At tapping, the liquid steel and the molten slag are the remaining products. The oxidation reactions of the melt dissolved elements such as silicon, carbon, manganese, phosphorus and iron, occurring during the oxygen blow produce more thermal energy than that required to raise the temperature of the melt to the target temperature, and to melt the fluxes. The excess heat is used to melt the cold scrap and reduce the iron ore to metal; besides, some heat is lost to the surroundings [

In [

where A and B are constants whose values are 1873 and 50, respectively. The above assumption rests on the fact that when the scrap is added to the pig iron, the melt temperature follows a trajectory close and parallel to the liquidus line of the Fe-C equilibrium diagram. In this work a thermal model is proposed in which A and B de- pend on the initial and the end-point carbon content and temperature:

where T_{0} is the initial melt temperature (K), [C]_{0} is the initial melt carbon content, T_{ep} is the end-point melt tem- perature (K), and [C]_{ep} is the end-point melt carbon content. Solving the above pair of simultaneous equations yield

Substitution of the values of A and B obtained through Equations (11) and (12) in Equation (8) yields an im- proved value of the melt temperature.

Equations (1) and (2) are ordinary differential equations that are numerically integrated using the fourth order Runge-Kutta method [^{–4} min. The computer simulations for verification purposes showed that this time step is small enough to assure the stability and the convergence of the numerical solution. On the other hand, Equation (3) is a nonlinear algebraic equation that is numerically solved for x_{CO} by means of the first order Newton-Raphson procedure [^{–6} which satisfies the re- quired accuracy. FORTRAN programming language was employed for the elaboration of the computer program, and the executable file was obtained using an Absoft compiler. Calculation flow chart is shown in

Numerical simulations are carried out to: 1) validate a particular numerical solution of the mathematical model with data plant taken from the literature; 2) analyze the influence of oxygen gas flow rate and lance height on the carbon content and the decarburization rate. Numerical results are discussed and compared, as much as possible, with results previously reported by other authors.

The mathematical model was validated using the plant data reported in [_{m} = 200 × 10^{6} g, Q_{O} = 620 Nm^{3}/min, T_{0} = 1623 K, T_{ep} = 1923 K, [C]_{0} = 4 wt%, [C]_{ep} = 0.1 wt%, h = 2500 mm, n = 6, d = 45 mm, P = 1 atm, t_{Si} = 2 min, t_{b} = 18 min. Calculated values of A and B were 1930.7 and 76.9, respectively. Besides, the following parameter values were employed in the computer simulations [_{c} = 0.3 wt%, k_{d} = 3.98 m/min, a = 4.1632, x_{in} = 0, k_{α} = 1.2. Melt volume is calculated from V_{m} = W_{m}/ρ_{m}, where ρ_{m} = 7.1 g·cm^{–3}. The results of validation are shown in

Calculation flow chart

Model validation using the plant data reported in [1] and [13] . Plant data (filled circles), model results (solid line)

The oxygen flow rate controls the decarburization rate if

Considering the plant data of [^{3}/min. The results are depicted in

_{c} is exhibited by the three flow rates considered, and this inflexion point corresponds to a transition in the decarburization mechanism: the carbon oxidation process passes from being controlled by the oxygen flow rate to a process controlled by the mass transfer, the area of the impact zone and the lance height. The time at which this transition of decarburization mechanism occurs is named here as transition time t_{t}, and is depicted in _{t} decreases from 15.85 to 11.89

Melt carbon content as a function of the blowing time and the oxy- gen flow rate. 500 (solid), 620 (dashed), 700 Nm^{3}/min (dotted)

Decarburization rate as a function of the blowing time and the oxygen flow rate. 500 (solid), 620 (dashed), 700 Nm^{3}/min (dotted)

min as the oxygen flow rate increases from 500 to 700 Nm^{3}/min. In accordance to _{t} has a slightly nonlinear dependence on the oxygen flow rate.

Given that t_{Si} = 2 min, during this time the decarburization rate is null given that the silicon has a preferential oxidation than the carbon, as is seen in

The lance height is a trade-off between a faster carbon removal rates and a right slag forming. When the lance is too high, the rate of carbon removal is reduced and erratic; besides, the slag will be over-oxidized and higher iron losses occur by oxidation. When the lance is too low, the carbon removal is increased [

Time for the decarburization mechanism shift as a function the of oxygen flow rate

Decarburization rate as a function of the oxygen flow rate

oxygen blow. For

In spite of the exponential time decay of the carbon content, at the end of the blow this variable has a linear dependence on the lance height, as is seen in

Finally, as reported in [

An improved mathematical model of decarburization in basic oxygen furnaces is presented in this work. Im- provements consist in taking into account those factors or parameters that determine the bath-oxygen impact area, such as the depth of cavity, the lance height, the nozzles number and the nozzles diameter, and by modifi- cation of the thermal model to include the carbon content and temperature end-points specifications. The im- proved model is validated using data plant reported in the literature. Conclusions are as follows:

1) The oxygen flow rate controls the decarburization rate for carbon contents above a critical value. As the oxygen flow rate is increased, the decarburization rate is increased too. The decarburization rate has a linear de- pendence on the oxygen flow rate.

2) The lance height controls the decarburization rate for carbon contents below a critical value. As the lance height is decreased, the decarburization rate is increased. The carbon content at the end of the blow has a linear dependence on the lance height.

Melt carbon content as a function of the blowing time and the lance height. 1800 (solid), 2200 (dashed), 2500 mm (dotted)

Carbon content at the end of the blow as a function of the lance height

Decarburization rate as a function of the blowing time and the lance height. 1800 (solid), 2200 (dashed), 2500 mm (dotted)

3) The present model has a reasonable reliability given that it is able to reproduce in a proper manner the data plant considered. Then, in spite of its simplifications, this improved model can be used as a static or dynamic model to train plant technicians and operators.