Introduction

The deoxidation equilibrium curve can be obtained depending on the calculation of the deoxidation equilibrium in liquid steel by using thermodynamic data for different elements, as shown in Figure 1 [1, 2]. According to Figure 1, there are minima for the Fe-O-Al, Fe-O-C and Fe-O-Cr systems, but no minimum exists in the deoxidation equilibrium curve of Fe-O-Si system.

Figure 1:

Deoxidation equilibrium curve of different elements in iron melt at 1,873 K.

Silicon–oxygen equilibrium in liquid iron was experimented by Gokcen and Chipman in the early 1950s [3]. Their results showed that there was no minimum for the Fe-O-Si system, even the silicon content up to 15 mass%. However, it was reported recently by Shibaev et al. [4] that the minimum occurred at 20 mass% Si, who measured the solubility of oxygen in iron–silicon melts in equilibrium with silica at 1,873 K, as shown in Figure 2. The criteria for a minimum developed by Hone et al. [1] and Wei [2] are not fit for the condition that the deoxidizer content is fairly high. The aim of the present study is to correct the criteria for the existence of the minimum of oxygen content in the deoxidation equilibrium curve.

Figure 2:

Oxygen solubility in Fe–Si melts in equilibrium with silica at 1,873 K.

Thermodynamic analysis

The reaction between deoxidizer and oxygen can be represented as follows:

where x and y are stoichiometric coefficients, [M] and [O] are metallic deoxidizer and oxygen in liquid iron, respectively, m is the reciprocal of equilibrium constant, aM and aO are the activities of [M] and [O]. By replacing aM and aO with fM[%M] and fO[%O], respectively, eq. (3) is obtained.

The dependence of f_M and f_O on composition can be expressed as eqs (4) and (5) [5].

Neglecting the higher order terms of [%O] and the second-order terms γMM,O[%M][%O] and γOM,O[%M][%O], eqs (4) and (5) reduce to eqs (6) and (7), respectively.

Substitution of eqs (6) and (7) into eq. (3) yields

If there is a minimum, the [%O] versus [%M] curve should meet the following two conditions:

Equation (11) is obtained by differentiating eq. (8) and further rearranging:

With the first derivative zero, the second derivative of eq. (8) is

According to eq. (11), eq. (13) is obtained.

If xγMM+yγOM=0, eq. (13) is a linear equation, to ensure the existence of a positive real solution for it, the following condition should be satisfied:

However, eq. (13) is a quadratic equation if xγMM+yγOM≠0. Equation (15) meets the condition that the quadratic equation has real solutions. And the corresponding real solutions are given in eq. (16).

For the case of xγMM+yγOM<0, there is always a positive real solution for eq. (13) no matter whether xeMM+yeOM is positive or negative. And eq. (15) is always satisfied if xγMM+yγOM<0.

But when it comes to the case of xγMM+yγOM>0, eq. (14) should be satisfied to ensure the existence of positive real solutions for eq. (13). In this case, there are two positive real solutions. This is similar to the work of St. Pierre [6], but in which the constraints were not discussed such as xγMM+yγOM>0 and the following eqs (18) and (19).

Equation (12) is needed to determine whether the positive real solution of eq. (13) is the minimum in the deoxidation equilibrium curve. Let [%M]_ex be the positive real solution of eq. (13) and [%O]_ex be the corresponding oxygen content obtained by substitution of [%M]_ex into eq. (8). Hence, eq. (12) can be written as

For the case of xγMM+yγOM≤0, since the denominator term of eq. (11) is positive, the denominator term of eq. (17) follows that

For the case of xγMM+yγOM≤0, it is reasonable that the smaller solution −(xeMM+yeOM)−(xeMM+yeOM)2−3.474x(xγMM+yγOM)4(xγMM+yγOM) of eq. (13) should be the minimum, since the oxygen in liquid iron decreases initially as the content of deoxidizer increases. As a consequence, the denominator term of eq. (11) should be positive. It means that it is also positive for the denominator term of eq. (17). Therefore the denominator term of eq. (11) should be greater than zero and satisfy the following equation:

From the thermodynamic analysis above, two criteria are developed for determining the existence of the minimum in the deoxidation equilibrium curve.

The first criterion is 0≤xγMM+yγOM≤min(x/4.606[%M]ex2,(xeMM+yeOM)2/3.474x) with xeMM+yeOM<0, or xγMM+yγOM<0.

And the second criterion is (xeMO+yeOO)+y/2.303[%O]ex>0 where min(A,B) is the minimum between A and B. It should be noted that there is always a minimum no matter whether xeMM+yeOM is positive or negative, in the case of xγMM+yγOM<0 and (xeMO+yeOO)+y/2.303[%O]ex>0.

Discussion

The limitation of the criteria for the minimum in terms of first-order activity interaction parameters

Since the amount of deoxidizer addition is not large in steelmaking practice in general, the first-order activity interaction parameters are adopted to determine the existence of the minimum in the deoxidation equilibrium curve [1, 2, 7, 8]. In other words, γMM and γOM are both regarded as zero. The first criterion for the minimum is xeMM+yeOM<0 in this condition, which is consistent with eq. (14). It can be noted that this criterion is the special case of xγMM+yγOM=0 in Section 2.

In the work of Wei [2], the second criterion is given as

(20)xeMO+yeOO<0

It is considered that eq. (20) is questionable. Of course, xeMO+yeOO>0 satisfies eq. (18). On the other hand, it is not guaranteed that the value of −(xeMO+yeOO) is smaller than that of y/2.303[%O]ex in the case of xeMO+yeOO<0. Thus, the second criterion is eq. (18) rather than eq. (20). [%O]_ex is obtained by eqs (21) and (22) in terms of first-order criterion parameters. Then, the existence of the minimum is confirmed by substitution of [%O]_ex into eq. (18).

(21)[%M]ex=−x2.303(xeMM+yeOM)

(22)lgm=[%M]ex(xeMM+yeOM)+[%O]ex(xeMO+yeOO)+xlog[%M]ex+ylog[%O]ex

It is worth noting that only the first criterion is not sufficient for the existence of the minimum. It is not persuasive enough to confirm the existence of a minimum in the work of St. Pierre and Blackburn [7] and the work of Feldman and Kirkaldy [8].

The value of activity interaction parameter can be judged by the existence of the minimum in the deoxidation equilibrium curve. It can be concluded that the value −0.13 of eOC is incorrect [2], since there is no minimum for the Fe-O-C system with eOC=−0.13. This is the contribution of the criteria in terms of the first-order activity interaction parameters.

However, there are limitations of criteria for the minimum in terms of first-order activity interaction parameters.

First, it is only suitable to the range that lgfMM and lgfOM have a linear relationship with [%M] due to the neglect of second-order interaction parameters. As [%M] increases, it is also lager for the degree of lgfMM and lgfOM deviating from the linear relationship with [%M]. The influence of Ti elements on the oxygen solubility in liquid iron at 1,873 K is shown in Figure 3 [8], in which the full line is the theoretical solubility curves calculated in terms of first-order activity interaction parameters, and the dots are experimental data from the work of Fruehan [9]. It can be seen from Figure 3 that the position of the minimum is accurately predicted by the criteria in terms of the first-order activity interaction parameters for some elements. But the deviation between the calculated value and the experimental value is larger as [%M] increases.

Figure 3:

Influence of Ti element on the oxygen solubility in liquid iron at 1,873 K (reproduced from Ref. [8]).

Second, it is concluded that there is no minimum in the case of xeMM+yeOM>0 in Refs. [1] and [2]. Thus the minimum is considered to be inexistence for the Fe-O-Si system. According to Figure 2, there is a minimum in the deoxidation equilibrium curve at high silicon content. Thus, even if xeMM+yeOM>0, there is still a minimum when it satisfies the condition of xγMM+yγOM<0 and (xeMO+yeOO)+y/2.303[%O]ex>0.

Conclusions

Two criteria are developed for determining the existence of the minimum in the deoxidation equilibrium curve in liquid iron in the present paper. The first criterion is 0≤xγMM+yγOM≤min(x/4.606[%M]ex2,(xeMM+yeOM)2/3.474x) with xeMM+yeOM<0 or xγMM+yγOM<0. And the second criterion is (xeMO+yeOO)+y/2.303[%O]ex>0.

The criteria in terms of first-order activity interaction parameters are the special case of present thermodynamic analysis with neglecting the second-order activity interaction parameters. They are not fit for the case of xeMM+yeOM>0. In this case, the criteria in terms of second-order activity interaction parameters are taken into account to determine the existence of the minimum.

The value 0.11 of eSiSi is smaller based on the existence of the minimum in the deoxidation equilibrium curve for the Fe-O-Si system. It is guaranteed that the minimum value of oxygen content on the deoxidation equilibrium curve existed at silicon content 20 mass%, when the value 0.32 of eSiSi is chosen, and the second-order activity interaction coefficients γSiSi and γOSi satisfy the condition γSiSi+2γOSi=−1.54×10−3.