Effects of Temperature and Carbon Content in Brick on MgO–C Reaction Behaviour

Abstract

The effects of the ambient temperature, gas flow rate and carbon content in the brick on the behaviour of the MgO–C reaction, which is an inherent phenomenon of MgO–C bricks, were investigated. As a result, the amount of the MgO–C reaction increased as the carbon content in the brick and the temperature increased, but was not significantly changed by increasing the gas flow rate. The apparent activation energies for the following reactions were determined from the results of this study and the results of previous reports.

CO(g) + MgO(s) = Mg(g) + CO₂(g) E = 348 kJ / mol

CO₂(g) + C(s) = 2CO(g) E = 296 kJ / mol

A new MgO–C reaction model was developed based on the shrinkage core model in order to discuss the effects of the temperature and carbon content in the brick quantitatively. The reaction model in this study could explain not only the results of the present study but also the results reported previously by other researchers. In addition, the effect of the particle diameters of MgO and carbon on theMgO–C reaction is also discussed using the reaction model proposed in this study.

1. Introduction

MgO–C brick is one type of unburned brick which is made from magnesia clinker, fine particle graphite and flake graphite as the main raw materials and phenol resin as a binder. The raw materials and binder are mixed together, pressed to the specified shape and then cured for an extended time. MgO–C bricks manufactured by this process are used widely in ironmaking and steelmaking facilities. For example, MgO–C bricks are mainly applied to the wear lining in the converter, which is the main facility in the steelmaking process.^1,2,3,4)

Various types of damage occur in wear lining bricks as the number of charges increases because these bricks are exposed to high temperature molten metal and a high temperature atmosphere. The life of the wear refractories is determined by the degree of damage, and refractory consumption increases as refractory life decreases, resulting in higher production costs. On the other hand, since raw material resources are limited and are not inexhaustible,^5,6) improvement of refractory life is also a very important issue from the viewpoint of saving resources as well as the economics of the steel production process.

Generally, the modes of damage that affect refractory life are divided into mechanical factors such as thermal spalling and mechanical spalling, and chemical factors such as the reaction between the refractories and the slag and furnace atmosphere (gas). In particular, it has been found that the MgO–C reaction, which is the reaction between the MgO and carbon in the brick, occurs in a reducing or inert gas atmosphere.^7,8) If this reaction occurs, the microstructure of the brick will deteriorate due to the increased number of pores in the brick. Because this can shorten the life of the furnace, it is important to control the MgO–C reaction so as to improve the durability of the refractory.

Many studies on the MgO–C reaction have been reported.^{9,10,11,12,13,14)} Ishibashi et al.¹¹⁾ investigated the effect of the raw material grade, such as the purity of MgO and graphite, on the behaviour of the MgO–C reaction. Nameishi et al.¹²⁾ investigated the behaviour of the MgO–C reaction by using the thermal balance under the high vacuum atmosphere (0.1 to 0.4 torr) assumed in VOD and RH refining. They reported that the MgO–C reaction became remarkable at temperatures over 1573 K, and the MgO–C reaction when using sintered MgO was accelerated in comparision with the reaction when using fused MgO. Tabata et al. also evaluated the behaviour of the MgO–C reaction under vacuum conditions by changing the carbon content and MgO particle size.¹³⁾ S. Jansson et al. studied the reaction behaviour of a MgO-5.5%C brick under various atmospheres, including Ar and CO.¹⁴⁾

As these examples suggest, the purity of the MgO particles (aggregates and fine particles), additives, carbon content, carbon purity, atmosphere, temperature, atmosphere pressure and gas flow rate are considered to be factors that affect the MgO–C reaction. However, because the effects of all these factors on the MgO–C reaction have not yet been explained, the construction of a comprehensive reaction model to explain their effects on the MgO–C reaction has been desired. Therefore, in this study, the effects of the temperature and the carbon content in the brick on the MgO–C reaction were investigated, and the reaction mechanism was discussed by suggesting a new reaction model in which the effects of these factors on the reaction can be explained quantitatively.

2. Experimental

A schematic illustration of the experimental apparatus is shown in Fig. 1. A horizontal tube furnace (electric resistance heating system) was used in the experiment. The centre of the furnace is an isothermal zone (defined as an experimental temperature with a range of ±5°C or less) with a length of about 60 mm. Both ends of the furnace are sealed with water-cooled chambers with a rod for moving the sample and have a gas inlet and gas outlet.

Fig. 1.

Schematic illustration of experimental apparatus. (Online version in color.)

The experimental method was as follows: First, a refractory sample is set at one end of the furnace while the temperature is low. Argon gas is introduced into the furnace and the atmosphere in the furnace is replaced with argon gas, and the furnace is then heated at a rate of 0.083 K/s (0.03 K/s at temperatures of 1473 K and higher). Second, after heating to the experimental target temperature, the sample is quickly (nearly instantaneously) moved from the end to the centre of the furnace by using the rod, and is held at that temperature for the specified time. Third, after the specified time has passed, the sample is moved quickly to the other end of the furnace using the rod. Finally, the sample is quenched in an inert gas atmosphere. The weight of the sample is measured before and after heating, and the weight loss ratio is evaluated and compared under the respective test conditions. The weight loss ratio is defined as shown in Eq. (1).   

$$\begin{array}{l}(Weight\mathrm{\hspace{0.17em} } loss\mathrm{\hspace{0.17em} } ratio\mathrm{\hspace{0.17em} } (\%))=\\ \frac{(The\mathrm{\hspace{0.17em} } weight\mathrm{\hspace{0.17em} } of\mathrm{\hspace{0.17em} } sample\mathrm{\hspace{0.17em} } before\mathrm{\hspace{0.17em} } heating\mathrm{\hspace{0.17em} } (g))-(The\mathrm{\hspace{0.17em} } weight\mathrm{\hspace{0.17em} } of\mathrm{\hspace{0.17em} } sample\mathrm{\hspace{0.17em} } after\mathrm{\hspace{0.17em} } heating\mathrm{\hspace{0.17em} } (g))}{(The\mathrm{\hspace{0.17em} } weight\mathrm{\hspace{0.17em} } of\mathrm{\hspace{0.17em} } sample\mathrm{\hspace{0.17em} } before\mathrm{\hspace{0.17em} } heating\mathrm{\hspace{0.17em} } (g))}\\ \times 100\end{array}$$(1)

The experimental conditions are shown in Table 1. Sintered MgO aggregates (purity: 98.5%), MgO fine graphite (purity: 98.5%) and flake graphite (purity: 99%) as a carbon source were used as the main raw materials. In general, this type of refractory is constituted by MgO aggregates (particle diameter: ≧1 mm) and MgO fine particles (particle diameter: <1 mm), flake graphite and a binder. As the ratio of the MgO aggregates in this study was 40 mass% in all cases, the average particle diameter was calculated as 0.28 mm. The average particle diameter was calculated considering the interfacial area of all particles on the assumption that all particles are involved in the MgO–C reaction. The average particle diameter of the carbon, which was calculated in the same way, was 0.20 mm.

Table 1. Experimental conditions.

Sample conditions	Bricks	MgO-10%C	MgO-15%C	MgO-20%C
	Ratio of MgO aggregates(*1) (%)	40%
	Average particle diameter of MgO(*2) (φ mm)	0.28
	Average particle diameter of carbon(*2) (φ mm)	0.20
	Carbon content in brick	10 mass%	15 mass%	20 mass%
	Apparent porosity	8.07%	8.60%	9.69%
	Shape (mm)	20 × 20 × 20
	Pretreatment	1623 K with a holding time of 3 hours in a coke-filled atmosphere
Experimental conditions	Temperature	1673 K–1823 K
	Heating rate (K/s)	0.083 (−1673 K) 0.03 (1673 K−)
	Atmosphere	Ar
	Gas flow rate (105 Nm3/s)	0.30–5.83
	Holding time (s)	0–10800

*1  Aggregates of MgO whose diameter is over 1mm

*2  Derivation by assuming similarity of interfacial area

In general, phenolic resin is used as a binder in carbon-containing refractories such as MgO–C brick because this type of binder offers a combination of desirable properties, namely, a good mixing property, good dispersibility, good adhesiveness, higher base strength after pressing, no significant expansion of the dried compact and formation of a rigid carbon bond.¹⁵⁾ Phenolic resin was also used as the binder in this study for this reason. The sample bricks were prepared by mixing the materials at ratios of MgO: C = (100 − x): x (where x = 10, 15 or 20 mass%). The mixed batch powders were fed into a steel die and pressed using a friction press to form brick samples with standard sizes, i.e., 114 mm (width) × 65 mm (height) × 230 mm (length). The bricks were further cut into smaller cubic samples with dimensions of 20 mm × 20 mm × 20 mm, and then heat-treated at 1623 K for 3 h in a carbon monoxide atmosphere prior to testing in order to remove the influence of the volatiles contained in the phenol resin binder and prevent oxidation of the carbon in the brick during heating, as well as to simulate the conditions inside an actual steelmaking furnace. For reference, the distribution of pores in each sample was in the range between 0.01 μm and 500 μm, and the average pore diameter was about 44 μm.

The atmosphere temperature during holding in the experiment was 1673 K, 1773 K or 1823 K. The holding time was varied in the range from 0 s to 10800 s, and the Ar gas (purity: 99.99%) flow rate was varied in the range from 3.0×10⁻⁶ Nm³/s to 5.83×10⁻⁵ Nm³/s.

3. Experimental Results

3.1. Basic Reaction Behaviour and Effect of Carbon Content in Brick on MgO–C Reaction Rate

As a typical example, SEM photographs of the surface of the MgO-20%C brick before and after the experiment are shown in Fig. 2. Although some graphite particles were seen among the MgO particles before experiment, they were not observed after experiment, and numerous pores were observed on the surface of the refractory. Figure 3 shows a X-ray CT scan image of the MgO-20%C brick after the experiment. The presence of porosities inside the refractory (black points and area) could be observed.

Fig. 2.

SEM images of sample before and after experiment. (Online version in color.)

Fig. 3.

X-ray CT scan image of MgO-20%C brick after experiment. (Online version in color.)

Figure 4 shows the change of the weight loss ratio of the MgO–C bricks with time as a representative result of the experiment in this study. Because of the experimental procedure used here, the MgO–C reaction occurs slightly during the heating and cooling periods, even if the holding time is 0 s. Therefore, the net value of the weight loss ratio was expressed by deducting the weight loss ratio for the holding time of 0 s from the experimental results. From Fig. 4, the weight loss ratio increased greatly during the initial stage of the experiment, but the rate of increase gradually became stagnant with the passage of time. Figure 5 shows the relationship between the weight loss ratio and the carbon content of the bricks at a constant test time of 3 h. From Fig. 5, a tendency in which the weight loss ratio increased with increasing carbon content was observed. However, the difference of the weight loss ratios of the MgO-15%C and MgO-10%C bricks was not particularly large.

Fig. 4.

Change of weight loss ratio of MgO–C bricks with time.

Fig. 5.

Relationship between weight loss ratio and carbon content in brick.

3.2. Effect of Temperature and Gas Flow Rate on MgO–C Reaction Rate

Figure 6 shows the relationship between the weight loss ratio and temperature for the MgO-20%C brick. The holding time was 3600 s, and the Ar gas flow rate was 8.3×10⁻⁶ Nm³/s.. Although the weight loss ratio at 1673 K was less than 1%, the ratio increased at temperatures over 1773 K. Figure 7 shows the relationship between the weight loss ratio and the Ar gas flow rate for the MgO-15%C brick at 1773 K. The weight loss ratio did not change and was roughly constant when the gas flow rate increased. In this figure, the weight loss ratio seemed to increase slightly at the higher gas flow rate. It was conjectured that a direct oxdation reaction of graphite by impurities such as oxygen or H₂O contained in the Ar gas might occur under the condition of the higher gas flow rate.

Fig. 6.

Relationship between weight loss ratio and temperature.

Fig. 7.

Relationship between weight loss ratio and gas flow rate.

Based on these results, the reaction mechanism of the MgO–C reaction was discussed quantitatively.

4. Discussion by Kinetic Reaction Model

4.1. Outline, Concept and Assumptions of Reaction Model

The formula of the MgO–C reaction is shown as Eq. (2).   

$$MgO(s)+C(s)=CO(g)+Mg(g)$$(2)

Some reports concerning the mechanism of the MgO–C reaction have been published, although the number is relatively small compared to research on the direct oxidation reaction of carbon in bricks.^16,17,18) R. J. Leonards et al.¹⁰⁾ suggested that the mechanism of the MgO–C reaction comprises the following series of reactions: 1) Chemical decomposition of MgO, 2) Absorption of oxygen on the surface of the carbon, 3) CO gas generation on the surface of the carbon and 4) reduction of MgO by CO gas. They reported that the rate-determining step was the CO gas generation reaction. Tabata et al.¹³⁾ investigated the behaviour of the MgO–C reaction under vacuum conditions for different carbon contents and MgO particle distributions, and discussed the reaction mechanism by suggesting a reaction model in which the chemical reaction at the interface and the mass transfer of the gas phase were assumed to be the rate-determining steps. According to their discussion, the chemical reaction at the interface is the rate-determining step in the initial stage of the reaction (within the first 1200 s), and is then replaced by mass transfer of gas as the rate-determining step. Because it is difficult to explain the effects of various factors (i.e., temperature, carbon content in the brick, average particle diameter and so on) on the MgO–C reaction mathematically and quantitatively by using their model, it is necessary to construct a new model in which the effects of several factors such as the carbon content and temperature on the MgO–C reaction are considered comprehensively. Here, according to the results shown in Fig. 3 and previous reports on the MgO–C reaction, many voids were observed between the MgO aggregate and the carbon particle matrix after the MgO–C reaction.⁸⁾ This phenomenon was presumed to be the result of reduction of the particle diameters of the MgO and carbon particles by the reaction.

From the above discussion, it is believed that the reaction occurs on the surface of MgO and graphite particles, and the reaction proceeds with a decrease in the unreacted layer, i.e., a decrease in the diameters of the residual MgO or graphite particles (when assuming the particles are spherical). In order to comprehensively investigate the behaviour of the MgO–C reaction, in this study, we focused on the shrinkage core model,^{19,20,21,22,23)} because the reaction between particles such as the solid MgO and graphite that comprise MgO–C bricks and the gas phase can be treated and the decrease in particle size as the reaction proceeds can be considered. Shrinkage core models include a model in which the constituent particles of the bulk reaction are regarded as unreacted cores.¹⁹⁾ In this study, the reaction rate was analysed by applying the shrinkage core model to the MgO and graphite constituting the refractory, and a MgO–C reaction model which can calculate the total reaction from the reduced mass of each particle was constructed based on this analysis.

The concept of the shrinkage core model in this study is shown schematically in Fig. 8. In the conventional model, a product layer is assumed to exist outside the unreacted core. However, in the present study, it was assumed that the diameter of the unreacted core diameter is identical with the particle diameter because all the reaction products of the MgO–C reaction are gaseous. Furthermore, both the MgO and carbon particles were assumed to be spherical.

Fig. 8.

Schematic diagram of shrinkage core model.

In general, although the MgO–C reaction is thought be a direct reduction reaction of MgO by carbon, this reaction can be divided into the following two reactions, namely, the reduction of MgO by CO gas and the oxidation of carbon by CO₂ gas, as shown in Eqs. (3) and (4).   

$$CO(g)+MgO(s)=Mg(g)+C{O}_2(g)$$(3)

  

$$C{O}_2(g)+C(s)=2CO(g)$$(4)

L. Rongti et al.^24,25) investigated the reduction behaviour of MgO particles by carbon and reported that the reactions shown in Eqs. (3) and (4) mainly occur. Ishibashi et al.¹¹⁾ reported that the MgO–C reaction is promoted when a CO atmosphere exists. Here, supposing the main reaction is the direct reduction of MgO by solid carbon shown in Eq. (2), the reaction would be suppressed by a CO atmosphere according to Le Chatelier’s law. Therefore, the results reported by Ishibashi et al. suggest that reduction by CO gas is very likely to occur. Considering these previous results, the reactions shown in Eqs. (3) and (4) are also considered to occur in this model, as in the discussion by L. Rongti et al.

Next, the rate-determining steps of the reaction were examined. The sequence of processes of the reaction in Eqs. (3) and (4) is shown below. Here, “bulk” was defined as pores in the refractory in this study.

1) Mass transfer of CO gas from the bulk to the boundary layer

2) Chemical reaction of Eq. (3) at the interface

3) Mass transfer of Mg gas from the boundary layer to the bulk

4) Mass transfer of CO₂ gas from the boundary to the bulk

5) Mass transfer of CO₂ gas from the bulk to the boundary layer

6) Chemical reaction of Eq. (4) at the interface

7) Mass transfer of CO gas from the boundary layer to the bulk

In the results of this study, according to Fig. 6, the effect of temperature on reaction behaviour was comparatively large. From this result, it was thought that either the diffusion of gas phases or the chemical reactions described in the above 2) and 6) could be regarded as a rate-determining step. Here, assuming that the temperature dependence of the weight loss ratio was attributable to the diffusion of gas in the pores in the MgO–C brick, the difference of the weight loss ratio with the change of temperature would be regarded as the difference of the temperature dependence of the diffusion coefficient of the gas. According to the study by S. Ueda,²⁶⁾ the change of the diffusion coefficient with temperature change can be expressed as the following equation.   

$$D=D_0{\left(\frac{T}{T_0}\right)}^{1.75}\frac{P_0}{P}$$(5)

Where, D, T, P stand for the diffusion coefficient, temperature and pressure, respectively, and the subscript (₀) means the standard state (273 K, 1 atm).

On the assumtion that the change of atmospheric pressure is negligible, the degree of the increase of the diffusion coefficient with the change of temperature from 1673 K to 1823 K was estimated to be 1.16 times from Eq. (5). On the other hand, the degree of the increase of the weight loss ratio with the change of temperature from 1673 K to 1823 K was calculated to be 22.9 times from the experimental results in Fig. 6, which is a much larger value. Therefore, from the above, it is considered that the effect of the diffusion coefficient on the temperature dependence of the weight loss ratio was no longer large.

Moreover, assuming that the mass transfer of the reactant gas or the generated gas was a rate-determining step (as in Tabata et al.), a boundary layer would be present around the surface of the MgO and graphite particles, and the gradient of the gas concentration between the bulk and interface would become the driving force of the reaction. At this time, it is thought that the CO and CO₂ gas concentrations in the bulk would be lower than those in the interface. Therefore, it is presumed that the diffusion of CO and CO₂ gas shown as the processes 1), 4) and 5) would be impossible. It is also predicted that, if the the reaction expressed as Eq. (4) occurred, the occurrence of processes 1) and 5) would be difficult because the gas pressure at the interface would be higher than that in the bulk. Furthermore, the gas flow rate dependence of the reaction behaviour could not be confirmed clearly from the results in Fig. 7. From the above discussion, it is difficult to regard the mass transfer of gas expressed as processes 1), 3), 4), 5) and 7) as the rate-determining-step. Hence, it was thought that the rate-determing step of Eqs. (3) and (4) should be regarded as the chemical reaction at the interface, and analyses of those reactions were carried out.

4.2. Reaction Rate

4.2.1. Reduction of MgO

The rate of decrease of the MgO particle diameter by the reaction in Eq. (3) was considered. Equation (6) is materialized by considering the mass balance, in which the change of the molar amount accompanying the reaction is equal to the amount of the product.   

$$\mathrm{\Delta }{n}_{MgO}={\frac{4}{3}\pi r_{Mg}^3{C}_{MgO}|}_{t+\mathrm{\Delta }t}-{\frac{4}{3}\pi r_{Mg}^3{C}_{MgO}|}_t={\gamma }_{MgO}{V}_{Mg}\mathrm{\Delta }t$$(6)

Where, γ_MgO (mol/m³/s) means the reaction rate of a MgO particle, V_Mg (m³) means the volume of the unreacted core, C_MgO (Mol/m³) means the molar amount of MgO and r_Mg (m) means the radius of the MgO particle. Equation (7) can be obtained by expansion of Eq. (6), and Eq. (8) can be obtained by a further expansion of Eq. (7).   

$$\frac{d}{dt}\left(\frac{4}{3}\pi r_{Mg}^3{C}_{MgO}\right)={\gamma }_{MgO}{V}_{Mg}$$(7)

  

$$4\pi r_{Mg}^2{C}_{MgO}\frac{d{r}_{Mg}}{dt}={\gamma }_{MgO}{V}_{Mg}$$(8)

The reaction at the surface of the unreacted core was formulated for each process according to the above-mentioned assumptions concerning the rate-determining steps. When the chemical reaction at the interface is the rate-determining step, the reaction rate can be expressed as Eq. (9).   

$$-{\gamma }_{MgO}=\frac{4\pi r_{Mg}^2}{V_{Mg}}\cdot \left(k_{Mg}{a}_{MgO}{P}_{CO}-k_{Mg}^{\prime }{P}_{Mg}{P}_{C{O}_2}\right)$$(9)

Where, P_CO2,P_Mg (atm) mean the partial pressures of CO₂ and Mg gas, respectively, and a_MgO means the activity of MgO, which is regarded as 1 in this study. Due to the assumption of a small value of the product of solubility between Mg and CO, the terms P_MgP_CO2 can be considered negligible, and Eq. (10) can be substituted for Eq. (9).   

$$-{\gamma }_{MgO}=\frac{4\pi r_{Mg}^2}{V_{Mg}}\cdot k_{Mg}{a}_{MgO}{P}_{CO}^{*}\cong \frac{4\pi r_{Mg}^2}{V_{Mg}}\cdot k_{Mg}{P}_{CO}$$(10)

Where, k_Mg (m/s) means the rate constant for reaction (3). Substituting Eq. (10) into Eq. (8) and integrating this result, Eq. (11) can be obtained. Where, R_Mg (m) means the initial radius of a MgO particle.   

$$r_{MgO}=R_{MgO}-\frac{P_{CO}}{C_{MgO}}{k}_{Mg}\cdot t$$(11)

In this connection, the number of MgO particles in a refractory sample can be expressed by using R_Mg and the density ρ (kg/m³) and weight of the initial sample W_MgO(initial) (kg).   

$$W_{MgO(initial)}=N\cdot {\rho }_{MgO}\cdot \frac{4}{3}\pi R_{MgO}^3$$(12)

Hence, the weight loss change after a lapse of time Δt (s) can be calculated as shown in Eq. (13).   

$$\begin{array}{l}\mathrm{\Delta }{W}_{MgO}=N\cdot {\rho }_{MgO}\cdot C_{MgO}\cdot \frac{4}{3}\pi \left({r_{MgO}^3|}_{t+\mathrm{\Delta }t}-{r_{MgO}^3|}_t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}=\frac{W_{MgO(initial)}}{R_{MgO}^3}\cdot C_{MgO}\cdot \left({r_{MgO}^3|}_{t+\mathrm{\Delta }t}-{r_{MgO}^3|}_t\right)\end{array}$$(13)

By solving Eqs. (11) and (13) simultaneously, the cumulative weight loss change of MgO can be calculated.

4.2.2. Oxidation Reaction of Carbon

The rate of decrease of the carbon particle diameter can also be considered in the same manner as that of the MgO particle diameter. Equation (14) can be obtained by considering the change of the mass balance accompanying the reaction (change of molar number accompanying the reaction = amount of formation of reaction product by the reaction).   

$$4\pi r_C^2{C}_C\frac{d{r}_C}{dt}={\gamma }_C{V}_C$$(14)

Where, γ_C (mol/m³/s) means the reaction rate of a carbon particle, V_C (m³) means the volume of the unreacted core of carbon, C_C (Mol/m³) means the molar number of carbon and r_C (m) means the radius of carbon. The reaction rate of the carbon particles was formulated as similar to the reaction in Eq. (3). When the chemical reaction is the rate-determining step, the reaction rate is expressed as Eq. (15).   

$$-{\gamma }_C=\frac{4\pi r_C^2}{V_C}\cdot \left(k_C{a}_C{P}_{C{O}_2}-k_C^{\prime }{P}_{CO}{}^2\right)$$(15)

Where, k_C (m/s) means the rate constant for the reaction in Eq. (4) and a_C means the activity of carbon, which is regarded as 1 in this study. Transforming this equation in the same manner as in the case of MgO, Eq. (16) can be obtained.   

$$-{\gamma }_C=\frac{4\pi r_C^2}{V_C}\cdot k_C\left(P_{CO2}-\frac{P_{CO}{}^2}{K_C}\right)$$(16)

Where, K_C represents the equilibrium constant for reaction in Eq. (4) and is expressed as Eq. (17).²⁷⁾   

$$log{K}_C=\frac{20\mathrm{\hspace{0.17em} }557}{T}+0.113$$(17)

Substituting Eq. (16) into Eq. (14) and integrating the result of the substituted equation, Eq. (18) can be obtained. Where, R_C (m) is the initial radius of a carbon particle.   

$$r_C=R_C-\frac{1}{C_C}{k}_C\left(P_{CO2}-\frac{P_{CO}{}^2}{K_C}\right)\cdot t$$(18)

Therefore, the weight loss change is calculated as shown in Eq. (19).   

$$\mathrm{\Delta }{W}_C=\frac{W_{C(initial)}}{R_C^3}\cdot C_C\cdot \left({r_C^3|}_{t+\mathrm{\Delta }t}-{r_C^3|}_t\right)$$(19)

By solving Eqs. (18) and (19) simultaneously, the cumulative weight loss change of carbon can be calculated. Where,   

$$W_{C(initial)}=N\cdot {\rho }_C\cdot \frac{4}{3}\pi R_C^3$$(20)

Finally, the total weight loss change of the MgO–C brick can be calculated as shown in Eq. (21)   

$$\mathrm{\Delta }{W}_{brick}=\mathrm{\Delta }{W}_{MgO}+\mathrm{\Delta }{W}_C$$(21)

4.3. Determination of Reaction Rate Constants

In order to calculate the weight loss ratio by using the reaction model, the reaction rate constants expressed by Eqs. (3) and (4), which were unknown parameters, were determined. These constants were formulated by fitting by using the results in this study and quoting previously reported results.

Generally, the temperature dependence of a rate constant can be expressed by the Arrhenius equation, as shown in Eq. (22). Where, E (J/mol) means apparent activation energy.²⁸⁾   

$$ln{k}_{Mg,C}=ln{k}_0-\frac{E}{RT}$$(22)

The chemical reaction rate constants k_Mg and k_c were determined by using the experimental results in this study shown in Fig. 4 and changing those values as fitting parameters. The effect of temperature on MgO–C reaction behaviour was discussed based on the chemical reaction rate constants obtained in this manner. Figures 9 and 10 show the Arrhenius plots of k_Mg and k_c, respectively. The figures also show the chemical reaction rate constants k_Mg and k_c derived from the results reported by Tabata et al. and Ishibashi et al. From Figs. 9 and 10, temperature dependence, which corresponded to the slope of each plot, namely, the apparent activation energy in this study, was similar to that in other reported results. From these results, the apparent activation energies for reactions (3) and (4) were determined from the regression lines. The values of the apparent activation energies for reactions (3) and (4) were 348 kJ/mol and 296 kJ/mol, respectively. For comparison, the same values according to Tabata et al. were 369 kJ/mol and 307 kJ/mol, and the values reported by Ishibashi et al. were 418 kJ/mol and 278 kJ/mol. Thus, the values in the present study were close to the previously reported values, and were also close to the theoretical value of the activation energy for the MgO–C reaction in an inert gas atmosphere (317 kJ/mol)²⁹⁾ calculated by S.C. Carniglia and the value reported by L. Rongti et al. (374 kJ/mol).²⁴⁾ However, the apparent activation energy for reaction (3) obtained in this study was somewhat lower than the reported value (470 kJ/mol), and as an absolute value, that for reaction (4) was higher than the reported value (180 kJ/mol).³⁰⁾

Fig. 9.

Arrhenius plots of k_Mg.

Fig. 10.

Arrhenius plots of k_c.

From above estimation, the chemical reaction rate constants can be expressed by the following equation. The parameters k₀ and E are shown in Table 2.   

$$k=k_0\cdot exp\left(-\frac{E}{RT}\right)$$(23)

Table 2. Values of parameters k_Mg, k_c and E.

		kMg	kc
Constant	k0 (mol/m2/s/atm)	483360	29502
Apparent activation energy	E (kJ/mol)	348	296

4.4. Calculation Results

Applying the experimental conditions in this study to the reaction model based on the results of the above discussion, a calculation was carried out, and the consistency between the experimental results and the calculation results, that is, the validity of the reaction mechanism, was discussed. From the previous chapter, the initial diameters of the MgO and carbon particles were given the average diameter taking into account the interfacial reaction area. Concerning the reaction at the time of 0 s (including just before the start of the experiment), the partial pressure of CO gas was calculated from the Gibb’s free energy change expressed by Eq. (24),²⁷⁾ assuming that the direct oxidation of carbon expressed as Eq. (2) occurs. The calculation procedure was as follows: The amounts of the weight loss changes of MgO and carbon were calculated by using Eqs. (13), (20) and (21), after which the volume of gas generation and the partial pressure of the gas phase were calculated by assuming that they would be stoichiometric. Using the obtained partial pressure, the amount of the reaction of MgO and carbon after a lapse of Δt (s) was calculated. These cycles were then repeated until the predetermined time.   

$$\mathrm{\Delta }{G}_{(2)}^{\circ }=818\mathrm{\hspace{0.17em} }808+30.84TlogT-578.84T\hspace{1em}(J)$$(24)

Figure 11 shows a comparison of the experimental and calculation results for the change of the weight loss ratio of the MgO–C bricks. Figure 12 shows the relationship between the weight loss ratio of a MgO–C sample and the carbon content in the brick at 10800 s, comparing the calculation results and experimental results. From these figures, the calculation results approximately corresponded to the experimental results, showing that the effect of the carbon content in the brick on MgO–C reaction behaviour can be explained by this model simulation. Furthermore, from the results of the model calculations, it was predicted that the difference in the reaction ratio (weight loss ratio) after several elapsed times would become smaller in the case of a MgO–C brick whose carbon content was less than 10 wt%. That is, the weight loss ratio would stagnate in MgO–C bricks with low carbon contents.

Fig. 11.

Comparison of experimental results and calculation results for change of weight loss ratio of MgO–C bricks.

Fig. 12.

Relationship between weight loss ratio of MgO–C sample and carbon content in brick.

Figure 13 shows a comparison of the experimental and calculation results for the effect of temperature on the behaviour of the MgO–C reaction. Although there was a slight difference between the calculation results and the experimental results in the higher temperature range, the effect of temperature on MgO–C reaction behaviour could be explained roughly as an overall tendency.

Fig. 13.

Comparison of experimental results and calculation results for effect of temperature on MgO–C reaction behaviour.

Next, this model was applied to other previously reported results. Figure 14 shows a comparison of the actual experimental results reported by Ishibashi et al.¹¹⁾ and the calculation results by this model for the temperature dependence of the weight loss ratio when the experimental conditions used by Ishibashi et al. were applied to the model. As mentioned previously, the model still does not consider the effect of temperature on the mass transfer coefficient, but in spite of this issue, the calculation results roughly corresponded to the actual results.

Fig. 14.

Comparison of actual experimental results and calculation results when experimental conditions of Ishibashi et al.¹²⁾ were applied to proposed model.

S. Jansson et al.¹⁴⁾ investigated the MgO–C reaction behaviour of a MgO-5.5%C brick in an inert gas (Ar) atmosphere. Although the model was also applied to the results reported by Jansson et al., it is possible that the heat treatement condition (773 K) which they used in sample preparation may have been insufficient to remove the volatile component of the binder.^31,32,33) Therefore, their data were revised by estimating the amount of the volatile component of the binder from their results, which were obtained at a condition under 1173 K, and deducting that amount from the original results. These revised data were then compared with the results calculated by the proposed model. The results of application of the model to their results is shown in Fig. 15. The calculation results were close to the revised results, showing that their results could also be explained by applying this model.

Fig. 15.

Comparison of actual experimental results and calculation results when experimental conditions of Jansson et al.¹⁰⁾ were applied to proposed model.

From the simulation results described above, the effects of the temperature, gas flow rate and carbon content in the brick on MgO–C reaction behaviour could be explained quantitatively, and the reaction behaviour could also be predicted by applying this reaction model, which was constructed based on the shrinkage core model. Furthermore, examples of the MgO–C reaction behaviour reported previously by other researchers could also be explained by applying this model.

4.5. Simulation Using This Reaction Model

The effect of the particle diameters of MgO and carbon on MgO–C reaction behaviour was discussed using the reaction model in this study.

Figure 16 shows the calculation result of the weight loss ratio, that is, the reaction ratio, when the average particle diameter of MgO is changed. As a precondition, it is assumed that the carbon content in the brick is 20%, the temperature is 1873 K, the holding time is 3600 s and the average particle diameter of carbon is 0.2 mm. From this simulation, it is anticipated that the reaction ratio is strongly dependent on the average particle diameter of MgO and the reaction ratio increases as the average particle diameter of MgO decreases.

Fig. 16.

Predicted reaction ratio as function of average particle diameter of MgO.

On the other hand, Fig. 17 shows the calculation result for the reaction ratio when the average particle diameter of carbon is changed. As a precondition, the carbon content in the brick is assumed to be 20% in order to clarify the difference, and the temperature is 1873 K and the holding time is 3600 s. The simulation is carried out with 0.28 mm as the average particle diameter of MgO. From this figure, the effect of the particle diameter of carbon is less remarkable than that of MgO. For example, the rise of the reaction ratio is at most a few percent (4%).

Fig. 17.

Predicted reaction ratio as function of average particle diameter of carbon.

This model makes it possible to explain the effect of the particle diameters of MgO and carbon on the MgO–C reaction quantitatively. From these figures, it is predicted that the effect of the particle diameter of MgO on the MgO–C reaction rate is comparatively larger than that of carbon. In particular, it is expected to be necessary to consider the particle size distribution of MgO, as the mean particle size of MgO should be increased in order to suppress the MgO–C reaction.

Furthermore, this reaction model cannot consider the effect of the particle size distribution on the reaction, and the calculations were carried out under the assumption that the entire refractory sample has a constant average particle diameter. However, actual bricks have a wide particle size distribution. The effect of this particle size distribution³⁴⁾ is considered to be one cause of error between the experimental and calculated results. Reconstruction and improvement of the reaction model considering the effect of this particle size distribution are issues for future study.

5. Summary

The effects of the ambient temperature, gas flow rate and carbon content in the brick on the behaviour of the MgO–C reaction, which is an inherent phenomenon of MgO–C bricks, were investigated by a method in which MgO–C bricks were heated in an inert gas. A reaction model based on a shrinkage core model was constructed, and the MgO–C reaction behaviour of the brick was discussed. The results obtained in this study may be summarized as follows:

(1) The amount of the MgO–C reaction, that is, the weight loss ratio, increased as the carbon content in the brick increased.

(2) The weight loss ratio was not changed significantly by increasing the gas flow rate.

(3) The weight loss ratio increased as the ambient temperature increased. The apparent activation energies for the following reactions were determined from the results of this study and the results published in previous reports.   

$$CO(g)+MgO(s)=Mg(g)+C{O}_2(g)\hspace{1em}\mathrm{E}=348\mathrm{\hspace{0.17em} } \mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$$

  

$$C{O}_2(g)+C(s)=2CO(g)\hspace{1em}\mathrm{E}=296\mathrm{\hspace{0.17em} } \mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$$

(4) A MgO–C reaction model was constructed, in which a shrinkage core model was applied and the mass transfer of the gas phase and the chemical reaction at the interface were considered as the rate-determining steps, and the effects of the ambient temperature and carbon content in the brick were discussed quantitatively. The reaction model proposed in this study could explain not only the results of the present study, but also the results reported previously by other researchers.

(5) According to the simulation using this model, it is expected to be necessary to consider the particle size distribution of MgO, as the mean particle size of MgO should be increased in order to suppress the MgO–C reaction.

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