Kinetic Analysis Considering Particle Size Distribution on Ca Elution from Slags in CaO–SiO₂–MgO–Al₂O₃–Fe₂O₃ System

Abstract

The understanding of the behavior of alkali elution from slags is important for their effective recycling and utilization. In a previous study, it was reported that the addition of iron oxide to steel slags significantly inhibited alkali elution. A lower modified basicity, i.e., CaO/(SiO₂ + Fe₂O₃) ratio, indicates a lower alkali elution from the slags. In addition to the effect of Fe₂O₃ content, particle size distribution is an important parameter to determine the elution of Ca quantitatively.

In this study, a kinetic model considering particle size distribution is developed and applied to the results of a dissolution experiment using samples of a slag, which is designated as SlagF4, with different particle size distributions. In the new kinetic model, an effective surface area and the effectiveness of total surface area α are introduced, and a kinetic analysis is performed.

The rate constant k obtained is a fixed value for one sample type; k decreased with the increase in Fe₂O₃ content.

The values of α increased significantly with the increase in the total surface area S_T (i.e., a decrease in particle diameter) in different particle size distributions. It is discovered that α typically represents the Ca elution tendency. Additionally, the change in α is small with the change in the Fe₂O₃ content. It is demonstrated that the developed kinetic model is valid for the analysis of Ca elution for samples with different particle size distributions.

1. Introduction

Many types of slags are produced in iron and steelmaking processes,¹⁾ and approximately 38 Mton/year (2017) are produced in Japan, in which steel slags constitute approximately 14 Mton/year, or 37% of the total slag. Although almost all blast furnace (BF) slags are utilized, steel slags might still be further utilized.

A better understanding of the characteristics of slags would contribute toward the reuse and recycling of slags significantly, in which the concept of molten slag crystallization is important. Time temperature transformation (TTT) and continuous cooling transformation (CCT) diagrams of BF slag have been measured by Kashiwaya et al.;²⁾ these diagrams are used for actual processes and research.³⁾ Meanwhile, TTT/CCT diagrams for steel slags are relatively difficult to measure because they have a high melting point and exhibit rapid crystallization; therefore, the nose point of the TTT diagram is typically less than 1 s. This will be an important issue for future research regarding steel slags.

Currently, steel slags are primarily used as roadbed and civil engineering materials; additionally, they are reused in steel works.^4,5,6,7,8) However, steel slags can be unstable, which hinders their reuse. Hence, many researchers are investigating and developing processes for the effective utilization of steel slags.⁷⁾

From 2016 to 2018, the authors conducted a study on the control of solidification structure of steelmaking slag for suppressing alkaline elution at the Iron and Steel Institute of Japan (ISIJ).

The behaviors of alkali (Ca) elution from dephosphorization slag and modified slags have been reported.⁹⁾ It was discovered that the relationship between Ca elution and pH can be expressed by Eq. (1). Furthermore, the addition of iron oxide to the steelmaking of slag significantly inhibited alkali elution. A lower modified basicity, C/(S + F) = CaO/(SiO₂ + Fe₂O₃), as shown in Eq. (2), indicates a lower alkali elution from the slags; therefore, it is important to increase the content of iron oxide in the slag to decrease the alkali elution.   

$$\left[{\text{Ca}}^{2+}\right]\left(\text{ppm}\right)=3.0\times {10}^{\left(\text{pH}-10\right)}$$(1)

  

$$\text{C}/\left(\text{S}+\text{F}\right)=\frac{{\text{X}}_{\text{CaO}}}{{\text{X}}_{{\text{SiO}}_2}+{\text{X}}_{{\text{FeO}}_{\text{x}}}}$$(2)

In the first part of this study, to elucidate the effects of Fe₂O₃ addition, Fe₂O₃ was added to an artificial slag modified from steelmaking slag. Dissolution experiments using the modified slags with different Fe₂O₃ contents were performed.

Many studies^10,11,12) reported that the diffusion through an intermediate product layer at the surface of particle controls the dissolution rate. It could be summarized that the reaction rate obeyed a first order rate equation at the initial stage of reaction and a Jander(three dimentional diffusion) type rate equation.¹³⁾ Lekakh, et al.¹⁴⁾ studied a leaching of steelmaking slag and showed a difference of particle size. However, they did not considere the effect of particle size distribution in the rate equation. Zhu, et al.¹⁵⁾ reported effects of mineralogical phases on alkaline dissolution from steelmaking slag. They assumed the formation of a surface layer on the particle of slag, and adopted the rate equation of diffusion control through this layer.

In the second part of this study, although it was confirmed that a dissolution rate was controlled by diffusion in the case of high dissolution rate for smaller diameter of particle, it was showed that a slower dissolution rate did not obey the diffusion control and new kinetic model was presented considering the particle size distribution.

2. Experimental

2.1. Procedure

2.1.1. Preparation and Composition of Samples

In this experiment, the slag composition designated as SlagF0 was used as the starting composition, and four levels (designated as SlagF1–SlagF4) of iron oxide (Fe₂O₃) were added to SlagF0, as shown in Table 1. The basicity (C/S=CaO/SiO₂) of the slags (SlagF0–SlagF4) was 1.29, and the modified basicity (C/(S + F)) decreased with the addition of Fe₂O₃ from 1.29 (SlagF0) to 0.85 (SlagF4).

Table 1. Chemical composition of SlagF0~SlagF4.

(mass%)
	CaO	SiO2	Al2O3	MgO	Fe2O3	C/S	C/(S+F)
SlagF0	40.9	31.8	15.0	12.3	–	1.29	1.29
SlagF1	39.4	30.6	14.5	11.9	3.6	1.29	1.15
SlagF2	38.6	30.0	14.2	11.6	5.6	1.29	1.08
SlagF3	37.1	28.8	13.6	11.1	9.4	1.29	0.97
SlagF4	35.2	27.3	12.9	10.6	14.0	1.29	0.85

The composition of SlagF0 is plotted in the CaO–SiO₂–MgO–15%Al₂O₃ phase diagram (Fig. 1).¹⁶⁾ In a previous study,⁹⁾ dephosphorization slag was modified to low melting point slag for a complete melting at approximately 1300°C. In this process, fly ash was added to dephosphorization slag and the final composition was controlled to a region near 1300°C, as shown in Fig. 1. As it is difficult to obtain a complete melting of slag near tricalcium silicate, the starting composition used in this experiment was SlagF0, which had approximately the same basicity (C/S) as that of BF slag. The purpose of this experiment was to clarify the effect of iron oxide addition on alkali elution behavior. The compositions of SlagF1 to SlagF4 are plotted in Fig. 2, which is the CaO–SiO₂–FeOx phase diagram reported by Kimura, et al.,¹⁷⁾ although the contents of Al₂O₃ and MgO were 5%.

Fig. 1.

Phase diagram of CaO–SiO₂–MgO–15 mass%Al₂O₃ and position of slag F0 composition. (Online version in color.)

Fig. 2.

Effect of 5 mass%Al₂O₃ and 5 mass% MgO on liquidus lines in the CaO–SiO₂–Fe₂O₃–FeO system¹⁷⁾ at 1573 K and points of slag composition used. (Online version in color.)

As shown in Fig. 2, only SlagF3 and SlagF4 might be located in the liquid region. Because the contents of Al₂O₃ and MgO were more than 5%, all samples were melted completely at 1673 K. Whereas precipitation occurred during cooling to the ambient temperature for SlagF0 and SlagF1, the other three types of slags (F2, F3, and F4) were in the amorphous state. From X-ray diffraction (XRD) analyses, it was confirmed that the crystals that precipitated in SlagF0 were merwinite (Ca₃Mg(SiO₄)₂, 00-035-0591), and the crystals in SlagF1 were merwinite (Ca₃Mg(SiO₄)₂, 00-035-0591) and gehlenite (Ca₂Al(AlSi)O₇, 01-089-5917); the XRD results are omitted herein because of space limitations.

The preparation method of the slag samples and the details of the dissolution experiment were the same as those of a previous study.⁹⁾ In this study, the effect of dissolved CO₂ on alkali elution was examined through Ar gas bubbling during the dissolution experiment, in addition to the effect of temperature from 296 K (23°C) to 301 K (28°C). Although the amount of dissolved CO₂ imposed a relatively large effect in a relatively low alkali elution of less than 10 ppm, the effect of CO₂ in this experiment was negligible. In addition, the effect of temperature was negligible when it was less than 300 K (27°C); however, it must be accounted for at 301 K and beyond (28°C). It was discovered that the most prominent effect was the distribution of particle size; therefore, small particles that appeared irregularly affected the results over the range of experimental error, unlike the dissolved CO₂ content and temperature.

2.1.2. Measurement of Particle Size Distribution and Calculation of Total Surface Area

Particle size distributions were investigated because they affected the results. The particle size distributions were measured using a laser diffraction particle size analyzer (HORIBA LA-950). The sample, SlagF4, was selected and sieved into five distributions, based on particle sizes, designated as (1) −25 μm, (2) −53 μm, (3) 25–53 μm, (4) 53–75 μm, and (5) 75–100 μm. The results are compared and shown in Fig. 3. Generally, the distribution depends on the crushing method, which is affected by the characteristics the crusher. For example, when 1 g sample is to be sieved to less than 25 μm (hereinafter denoted by −25 μm), one person may crush continuously until the entire sample became −25 μm, while another person may crush the same sample by sieving several times and separating the sample to −25 μm. In the former method, the proportion of −25 μm increases, whereas in the latter method, particles larger than those obtained using the former method increase in proportion; therefore, even for same particles of −25 μm, the actual distribution of particle size changes and hence the elution behavior changes.

Fig. 3.

Comparison of particle distributions among different sieving processes. (Online version in color.)

In this experiment, the crushing and sieving of the sample was performed as similarly as possible.

As shown in Fig. 3, as a general tendency, the proportion of particles larger than the size of the sieve was higher because the peak diameter at the distribution was larger than the sieve size. In addition, the frequency (vol%) in the lower particle size distribution was the largest in the −25 μm sample, whereas the frequency in the larger size distribution was the largest in the sample from 75 to 100 μm. The maximum size of the particle was almost approximately three times the sieve size, implying that the shape of larger particles was long and narrow, as confirmed under a microscope (the photograph is omitted herein).

The frequency (f_d, vol%) is expressed by Eq. (3), where n_d is the number of particles; d (μm) is the particle diameter; V_d (μm³) is the particle volume (Eq. (4)); V_T (μm³) is the total volume of the sample (1 g), which can be calculated using Eq. (5); d (μm) is assumed to be spherical; m (g) is the mass of the sample; and ρ (g/cm³) is the density.   

$${\text{f}}_{\text{d}}\left(\text{vol\%}\right)=\frac{{\text{n}}_{\text{d}}{\text{V}}_{\text{d}}}{{\text{V}}_{\text{T}}}\times 100$$(3)

  

$${\text{V}}_{\text{d}}\left(\mu {\text{m}}^3\right)=\frac{4}{3}\pi {\left(\frac{\text{d}}{2}\right)}^3=\frac{1}{6}\pi {\text{d}}^3$$(4)

  

$${\text{V}}_{\text{T}}\left(\mu {\text{m}}^3\right)=\frac{\text{m}\left(\text{g}\right)}{\rho \left(\text{g}/{\text{cm}}^3\right)\times {10}^{-12}}$$(5)

Based on Eqs. (3), (4), and (5), the number of particles, n_d can be obtained using Eq. (6).   

$${\text{n}}_{\text{d}}=\frac{6\text{m}}{\rho \pi }\times \frac{{\text{f}}_{\text{d}}}{{\text{d}}^3}\times {10}^{10}$$(6)

Therefore, the total surface area S_T (m²) can be obtained, which is expressed in Eq. (7).   

$$\begin{array}{l}{S}_T\left({\text{m}}^2\right)=\sum {\text{n}}_{\text{d}}{\text{S}}_{\text{d}}\\ =\sum \left\{{\displaystyle \frac{6\text{m}}{\rho \pi }}\times {\displaystyle \frac{{\text{f}}_{\text{d}}}{{\text{d}}^3}}\times {10}^{10}\times 4\pi {\left({\displaystyle \frac{\text{d}}{2}}\right)}^2\times {10}^{-12}\right\}\\ ={\displaystyle \frac{6\text{m}}{\rho }}\times {10}^{-2}\times \sum {\displaystyle \frac{{\text{f}}_{\text{d}}}{\text{d}}}\end{array}$$(7)

In this experiment, m = 1.0 g and ρ = 3.0 g/cm³ (generally, the density of a slag containing an iron oxide is within 2.8–3.1 g/cm³).

The calculated total surface, S_T for five different sieving are shown in Table 2 and Fig. 4. S_T decreased continuously with the increase in the sieving size. The maximum S_T was approximately 0.096 m² for the −25 μm sample, which was approximately five times that of the 75–100 μm sample (0.02 m²).

Table 2. Total surface area and fitted dissolution parameters for SlagF4.

Sample	XCa	ST (m2)	α (m−2)	[Ca]0	δ (μm)	βT	k (s−1)
SlagF4 −25 μm	0.2514	0.0956	1.800	0.1721	0.6000	23.4	1.13E-05
SlagF4 −53 μm	0.2514	0.0657	0.400	0.0263	0.1333	23.4	1.13E-05
SlagF4 25–53 μm	0.2514	0.0434	0.170	0.0074	0.0567	23.4	1.13E-05
SlagF4 53–75 μm	0.2514	0.0280	0.130	0.0036	0.0433	23.4	1.13E-05
SlagF4 75–100 μm	0.2514	0.0228	0.115	0.0026	0.0383	23.4	1.13E-05

Fig. 4.

Comparison of total surface area S_T among different sieving samples.

2.1.3. Dissolution Experiment

The experimental apparatus and conditions are the same as those in a previous study.⁹⁾ Based on the JIS method (JIS A 5015-2013),¹⁸⁾ Kitamura et al.¹⁹⁾ adopted a modified condition for a university laboratory, which was 1 g of −53 μm slag sample. The solvent was 400 cc. of distilled water. The temperature from 297 K (23.8°C) to 299 K (26°C) was controlled to ±0.5 K. The dissolution experiment in this study was performed for 3 h (10800 s).

After the experiment was completed, an aqueous solution with slag powder was filtered through a paper filter (ADVANTEC No. 1 (filtered over 6 μm), JIS3801). Ca, Si, Al, Mg, and Fe in the filtered aqueous solution were analyzed quantitatively by inductively coupled plasma (ICP) optical emission spectrometry.

3. Results and Discussion

3.1. Effect of Particle Size Distribution on pH Variation of SlagF4

The pH variation during the experiment was monitored and recorded every 2 s using a pH meter. The results obtained for different particle size distributions are shown in Fig. 5.

Fig. 5.

pH variation in dissolution experiments with SlagF4 sieved from −25 to 75–100 μm in 180 min. (Online version in color.)

The pH variations in the initial stage (less than 5 min on the left-hand of Fig. 5) did not show consistency with the particle size. For example, the −53 μm sample showed the highest pH rate increase at the beginning of dissolution; however, the pH rate increase of the −25 μm sample soon overtook that of the −53 μm sample and showed the highest value until the end of the experiment (at 180 min on the right-hand size of Fig. 5). The final values showed the order of the particle size. The scattered results at the initial stage may be affected by the difference in the particle size distribution, especially in small size particle. The small particle less than 40 μm will be largely affected by electrostatic force and will stick to the surface of larger particle. Consequently, the smaller particles remained and yielded a high elution rate, which contributed to an irregular value at the initial elution stage.

3.2. Results of ICP Analysis of Aqueous Solution

The aqueous solution after the dissolution experiment was analyzed, and Ca, Si, Al, Mg, and Fe were quantified. The elemental contents are plotted in Fig. 6, and the values are listed in Table 3. The Ca contents were the largest compared with the other elements and decreased with increasing particle size. It is clear that S_T is important for understanding the elution of alkali from the particle surface. The relationship between the element content in the aqueous solution and S_T is shown in Fig. 7.

Fig. 6.

Comparison of Ca, Si, Al, and Mg contents eluted in aqueous solutions from SlagF4 with different particle size distributions. (Online version in color.)

Table 3. The concentration in aqueous solutions of SlagF4 −25 μm ~ 75–100 μm.

(ppm)
Sample	Ca	Si	Al	Mg	Fe
SlagF4 −25 μm	6.72	0.34	0.98	0.97	0.04
SlagF4 −53 μm	5.85	0.22	1.44	1.59	0.05
SlagF4 25–53 μm	3.02	0.07	0.58	0.82	0.03
SlagF4 53–75 μm	2.54	0.04	0.09	0.60	0.01
SlagF4 75–100 μm	2.11	0.02	0.10	0.48	–

Fig. 7.

Relationship between contents of elements (Ca, Si, Al, and Mg) eluted and total surface area of SlagF4 with different particle size distributions. (Online version in color.)

A correlation existed between Ca content and S_T. The contents of Mg and Al show a similar tendency, and those of samples with larger particle sizes (> −53 μm) increased with the total surface area. However, at the greatest S_T, which corresponded to the smallest particle size of −25 μm, the Mg and Al contents were low, i.e., approximately 1.0 ppm. Meanwhile, the Si content was the lowest for all S_T. The lines in Fig. 7 were drawn according to the results of the kinetic analysis, which are presented below. It was assumed that the points distant from the lines were related to the mechanism of the element elution. The details will be discussed in a later section. The Fe content was low, i.e., less than 0.05 ppm, and Fe will hardly dissolve into the water solution under this condition.

3.3. Effect of Iron Oxide Content on pH Variation for SlagF0 and SlagF1–SlagF4

The dissolution experiments using SlagF0 and SlagF1–SlagF4 were performed based on the procedure mentioned above, and the results are shown in Fig. 8.

Fig. 8.

pH variation in dissolution experiments of SlagF0 and SlagF1–SlagF4. (Online version in color.)

The pH of SlagF0 increased rapidly from the beginning of experiment and was approximately 10.0 and 10.5 at 0.5 and 3.5 min, respectively. Subsequently, the pH increased gradually to 11.08 at 180 min.

Meanwhile, for samples SlagF1–SlagF4, in which Fe₂O₃ was added to SlagF0, the increase in pH was slower and the final values were lower than those of SlagF0. Within 5 min, the increase rates of SlagF1–SlagF4 were not in the order of Fe₂O₃ content owing to the particle size distribution, as mentioned above.

At 180 min, the pH values of SlagF1–SlagF4 decreased with increasing Fe₂O₃ addition. The order was SlagF1 (3.6 mass% Fe₂O₃), SlagF2 (5.6 mass% Fe₂O₃), SlagF3 (9.4 mass% Fe₂O₃), and SlagF4 (14.0 mass% Fe₂O₃), whereas the pH values were 10.63, 10.59, 10.53, and 10.27, respectively.

Figure 9 shows the results of ICP analysis of the aqueous solution after the dissolution experiments using SlagF0 and SlagF1–SlagF4. The values are summarized in Table 4. The elution of Ca in all slags was the highest, followed by that of Si for SlagF0 and then Mg for SlagF1–SlagF4, in which Fe₂O₃ was added. The elution of Si from SlagF0 was 17.5% that of Ca, whereas the elutions of Al and Mg were 10% and 1.5%, respectively. The elutions of Mg from SlagF1–SlagF4 were 22% to 27%, whereas those of Si to Ca were 0.5% to 3.8%. The elutions of Al were scattered from 1.7% to 25%. Hence, it can be concluded that Ca affects the pH value significantly.

Fig. 9.

Comparison of Ca, Si, Al, and Mg contents eluted in aqueous solutions of SlagF0 and SlagF1–SlagF4. (Online version in color.)

Table 4. The content of element in aqueous solutions for Slag F0 and SlagF1 to SlagF4.

(ppm)
Sample	Ca	Si	Al	Mg	Fe
SlagF0	28.10	4.92	2.70	0.52	–
SlagF1	8.39	0.31	0.14	2.15	–
SlagF2	9.09	0.80	1.20	2.06	0.08
SlagF3	7.44	0.04	0.16	1.60	–
SlagF4	5.85	0.22	1.44	1.59	0.05

As reported in a previous study,⁹⁾ Fe₂O₃ addition decreased the elution of Ca significantly. In this study, the inhibition tendency of Ca elution was confirmed, and the modified basicity C/(S + F) was valid for understanding the elution of Ca from slags. Although the data of SlagF1 and F2 have relatively large errors from Eq. (8), which were obtained in a previous study,⁹⁾ they remained within the error of the difference in particle size distributions.   

$${\left[Ca\right]}_{elute}(\%)=2.154\times {10}^{-2}{e}^{4.04x}$$(8)

where x = C/(S + F), and [Ca]_elute(%) = [Ca²⁺]/X_Ca × 100

3.4. Kinetic Analysis of Calcium Elution from Slags

As described above, Ca elution was significantly affected by the particle size distribution. Furthermore, the crushing procedure is important for obtaining consistent data. A kinetic model considering particle size distribution was developed and applied to the results of the dissolution experiment of SlagF4 with different particle size distributions.

3.4.1. First-order Reaction of Ca Elution Considering Particle Size Distribution

In a previous study,⁹⁾ it was reported that the relationship between [Ca²⁺] (ppm) and pH can be expressed by Eq. (9). Using Eq. (9), the Ca concentration, [Ca²⁺] (mol/L) can be expressed by Eq. (10).   

$$\left[{\text{Ca}}^{2+}\right]\left(\text{ppm}\right)=3.0\times {10}^{\left(\text{pH}-10\right)}$$(9)

  

$$\begin{array}{l}\left[{\text{Ca}}^{2+}\right]\left(\text{mol}/\text{L}\right)={\displaystyle \frac{\left[{\text{Ca}}^{2+}\right]\left(\text{ppm}\right)\times {10}^{-3}}{{\text{M}}_{\text{Ca}}(\text{g}/\text{mol})}}\\ \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}}={\displaystyle \frac{3.0\times {10}^{\left(\text{pH}-10\right)}\times {10}^{-3}}{40.08}}\\ \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}}=7.5\times {10}^{\left(\text{pH}-15\right)}\end{array}$$(10)

As mentioned above, the crushed sample of SlagF4 was sieved into five gradations (−25, −53, 25–53, 53–75, and 75–100 μm). Dissolution experiments using the respective samples were performed, and the variations in [Ca²⁺] (mol/L) were obtained based on Eq. (10) with the experimental time t. The results are shown in a later section.

For the rate equation, it is assumed that the first-order reaction can be written as shown in Eq. (11).²⁰⁾ The region related to the elution of Ca in a particle is extremely small in this study. As illustrated in Fig. 10, the thickness was assumed to be δ, which differed from the particle size because the curvature increased with smaller particle radii.

Fig. 10.

Schematic illustration of dissolution interface of slag to aqueous solution. (Online version in color.)

The rate of dissolution reaction is almost proportional to the surface area of the particle; however, the percentage of volume related to Ca elution V_δ ($=\frac{4}{3}\pi \left[{\left(\frac{d}{2}\right)}^3-{\left(\frac{d}{2}-\delta \right)}^3\right]$) to the original particle volume V₀ ($=\frac{4}{3}\pi {\left(\frac{d}{2}\right)}^3$) increased with the decrease in particle size, as shown in Fig. 11. When the curvature increased, the stability of the crystal structure decreased. Therefore, one may infer that the effective surface area S_eff is not constant with the particle radius.

Fig. 11.

Variation of percentage of volume related to Ca elution in a constant thickness of d(mm). (Online version in color.)

It is assumed that the increase rate of Ca²⁺ in an aqueous solution ($\partial \left[{\text{Ca}}^{2+}\right]/\partial \text{t}$) is equal to the decrease rate of Ca in the mass fraction (–)(= g/g) at the surface of the slag particle ($-\partial {\left[\text{Ca}\right]}_{\text{in}\mathrm{\hspace{0.17em} }\text{slag}}/\partial \text{t}$), as expressed in Eq. (11).   

$$\frac{\partial \left[{\text{Ca}}^{2+}\right]}{\partial \text{t}}=-\frac{\partial {\left[\text{Ca}\right]}_{\text{in}\mathrm{\hspace{0.17em} }\text{slag}}}{\partial \text{t}}=S_{eff}k{\left[\text{Ca}\right]}_{\text{in}\mathrm{\hspace{0.17em} }\text{slag}}=k^{\prime }{\left[\text{Ca}\right]}_{\text{in}\mathrm{\hspace{0.17em} }\text{slag}}$$(11)

where [Ca]_{in slag} means the Ca related to a dissolution reaction. k is the rate constant (s⁻¹); k’ (= S_eff k) is the apparent rate constant (s⁻¹); S_eff (–) is the effective surface area, which will be discussed in detail later. Eq. (11) can be integrated between t = 0 and t = t, and Eq. (12) is obtained when t = 0 and [Ca]_{in slag} = [Ca]₀.   

$${\left[\text{Ca}\right]}_{\text{in}\mathrm{\hspace{0.17em} }\text{slag}}={\left[\text{Ca}\right]}_0\text{exp}\left(-{\mathrm{k}}^{\prime }\mathrm{t}\right)$$(12)

Because the total calcium in the slag particle and solution is a constant, Eq. (13) can be obtained as follows:   

$$\begin{array}{l}{\left[Ca\right]}_0\times 1.00\left(\text{g}\right)={\left[Ca\right]}_{in\mathrm{\hspace{0.17em} }slag}\times 1.00\left(\text{g}\right)+\left[C{a}^{2+}\right]\left(\text{mol}/\text{L}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times 40.08(\text{g}/\text{mol})\times 0.4\left(\text{L}\right)\\ \hspace{1em}\hspace{1em}\hspace{0.5em}{\left[Ca\right]}_{in\mathrm{\hspace{0.17em} }slag}={\left[Ca\right]}_0-16.0\times \left[C{a}^{2+}\right]\end{array}$$(13)

Substituting Eq. (13) into Eq. (12), Eq. (14) can be obtained,where [Ca]₀ means Ca in the volume concerning to the elution reaction.   

$$\left[C{a}^{2+}\right]\left(\text{mol}/\text{L}\right)=\frac{{\left[Ca\right]}_0}{16.0}\left(1-exp\left(-k^{\prime }t\right)\right)$$(14)

  

$${\left[Ca\right]}_0=\frac{\delta S_T}{V}=\frac{\rho \delta S_T}{m} =3\times {10}^{-6}\delta S_T=\alpha S_T$$(15)

  

$${\left[Ca\right]}_0^d=\alpha n_d{S}_d,\hspace{1em}\alpha =3\times {10}^{-6}\delta$$(16)

  

$${\left[Ca\right]}_0=\sum {\left[Ca\right]}_0^d=\sum \alpha n_d{S}_d=\alpha S_T$$(15)’

In Eq. (15), [Ca]₀ is a function of S_T, and α (m⁻²) means a parameter containing a thickness from surface, i.e., α is the effectiveness of S_T. Hereinafter, α is referred to as the “effectiveness of the total surface.” As shown in Eq. (7), S_T is equal to Σn_dS_d, where n_d and S_d are the number of particles and the surface area of particle having diameter d (m), respectively. Then, at the each size of diameter, d, ${\left[Ca\right]}_0^d$ can be written as Eq. (16) on the diameter d considering the particle size distribution. Finally, ${\left[Ca\right]}_0^d$ can be related to [Ca]₀ as shown by Eq. (15)’.

Since n_d means the number of particles of diameter d, total Ca elution on the particle of diameter d can be expressed by Eq. (17). The overall rate of elution can be expressed as the summation of elution of the respective particles as shown by Eq. (18). At first, calculation using Eq. (18) was performed for fitting to experimental data, however, it was impossible. The modification of Eq. (18) was done for fitting to experimental data, the variable in the exponential term, $S_{eff}^d$ was introduced instead of n_dS_d as shown in Eq. (19). The values of $S_{eff}^d$ were obtained for respective experiments using different particle size distribution. It was found that the $S_{eff}^d$ can be expressed as exp(β_Tn_dS_d) as shown by Eq. (20), when β_T equals to 23.4.

Particle d μm & number of n_d   

$$n_d{\left[C{a}^{2+}\right]}^d=\frac{\alpha n_d{S}_d}{16.0}\left(1-exp\left(-S_{d}kt\right)\right)$$(17)

Summation for all size of particles   

$$\sum \left(n_d{\left[C{a}^{2+}\right]}^d\right)=\sum \frac{\alpha n_d{S}_d}{16.0}\left(1-exp\left(-S_{d}kt\right)\right)$$(18)

Modification of Eq. (17)   

$$\left[C{a}^{2+}\right]\left(\frac{mol}{L}\right)=\sum \left(n_d{\left[C{a}^{2+}\right]}^d\right)=\sum \frac{{\left[Ca\right]}_0^d}{16.0}\left(1-exp\left(-S_{eff}^{d}kt\right)\right)$$(19)

  

$$S_{eff}^d=exp\left({\beta }_T{n}_d{S}_d\right),\mathrm{\hspace{0.17em} }{\beta }_T=23.4$$(20)

Although Eqs. (19) and (20) might be necessary to verify theoretically, herein it was adopt as an empirical equation. $S_{eff}^n$ were calculated using particle size distributions and are shown in Fig. 12. The range of value was from 1 to 1.18. It is important to note that the value of k was constant for a sample with same chemical composition as shown Table 2. According to this type of equation, it could be possible to separate between intrinsic rate constant and effect of surface condition.

Fig. 12.

Comparison of effective surface area, S_eff (−) for different surface distributions. (Online version in color.)

The variations in $S_{eff}^n$ were similar to the distribution of particle size as shown in Fig. 3; however, they indicated an opposite tendency in magnitude to frequency f_d. The highest values of $S_{eff}^n$ was for the −25 μm sample and the lowest one was for the 75–100 μm sample.

The rate constant k and the factor β_T should be constant when the same sample is used as mentioned above. Meanwhile, one of the pre-exponential terms of α should differ from the particle size. The k, β_T, and α were determined by trial and error and are listed in Table 2. These parameters were determined so that the correlation factor between observation and calculation was close to one. For example, the correlation factor for the result of −53 μm sample was about 0.997. Calculations using Eq. (19) for the respective particle size distributions were performed and compared with experimental results, which are shown in Figs. 13, 14, 15, 16, 17.

Fig. 13.

Variation of [Ca²⁺] in dissolution experiment using SlagF4 of −25 μm. (Online version in color.)

Fig. 14.

Variation of [Ca²⁺] in dissolution experiment using SlagF4 of −53 μm. (Online version in color.)

Fig. 15.

Variation of [Ca²⁺] in dissolution experiment using SlagF4 of 25–53 μm. (Online version in color.)

Fig. 16.

Variation of [Ca²⁺] in dissolution experiment using SlagF4 of 53–75 μm. (Online version in color.)

Fig. 17.

Variation of [Ca²⁺] in dissolution experiment using SlagF4 of 75–100 μm. (Online version in color.)

In Fig. 13, the calculation result for the −25 μm sample could not fit the observed result over the entire experimental period of 10800 s (3 h). Only until 800 s did the calculation result agree with the observation result. This was understood from the results of other samples (−53, 25–53, 53–75, and 75–100 μm); therefore, the other samples were able to fit in the entire experimental period. When the rate of reaction increased, the rate-controlling step changed from reaction control to diffusion control in the solid phase.

The diffusion equation, Eq. (21) can be derived by assuming a one-dimensional infinite plate.¹²⁾   

$$\left[{\text{Ca}}^{2+}\right]=2\frac{{\text{X}}_{\text{Ca}}}{16.0}\left(\frac{\text{S}}{\text{V}}\right)\sqrt{\frac{\text{Dt}}{\pi }}=\text{U}\sqrt{\text{t}}$$(21)

where X_Ca/16.0 means the total elution, D is the diffusion coefficient of Ca in the solid phase, and U is a constant. The equation of Ca elution under diffusion control in the solid phase can be simplified to Eq. (22).   

$$\left[{\text{Ca}}^{2+}\right]=\text{U}\sqrt{\text{t}}+\text{V}$$(22)

where U and V are constants. In Fig. 13, these constants are U = 2.8 × 10⁻⁶ and V = 1.01 × 10⁻⁴. The starting time was assumed to be 800 s when the calculated result was apart from the observed one. The calculation result using Eq. (21) agreed well with the observed results after 800 s. When the particle diameter decreased, the rate of elution increased. This is generally caused by an increase in S_T; however, the kinetic analysis already considers the effect of S_T. Even if S_T was considered, the calculated result could not express the observed one. Hence, the thickness of δ (Table 2), which is related to Ca elution, will increase with the decrease in diameter. Consequently, the reaction control will become diffusion control in the solid phase.

As shown in Fig. 14, the calculation results for the −53 μm sample are compared with the observed ones. For the −53 μm sample, the calculation result agreed well with the observed results over the entire experimental period (10800 s). The −25 μm sample could not fit in the entire experimental period because of diffusion control in the solid phase, while the −53 μm sample can. An overlap region of less than −40 μm between the −53 and −25 μm samples might be the boundary region between reaction control and diffusion control.

Figure 15 shows the comparison between the calculated and observed values of a 25–53 μm sample. The 25–53 μm had little region of −40 μm, from which the behavior of Ca elution differed from those of the two samples mentioned above. The variation in Ca elution increased linearly with time. However, in the initial period of less than 800 s, the observed rate was higher than that in the later region, and α was 0.25. This result might be due to an irregularly remaining smaller particle that was adhering to the larger sample by electrostatic force.

Figures 16 and 17 show the results of 53–75 and 75–100 μm samples. The elution rate was almost constant, and α was 0.115 and 0.13 for 53–75 and 75–100 μm in most of the experimental period, respectively. However, the initial rate of Ca elution at less than 1000 s was slow, and α was 0.03 and 0.01 for 53–75 and 75–100 μm, respectively. When the particle size increased to beyond 53 μm, the proportion of small particles less than 40 μm was small. The elution of the larger particle required a longer time to reach a steady state compared with the smaller particle.

The values of α are summarized in Table 2, and a graph of α vs. S_T is plotted in Fig. 18. α becomes significantly large with the increase in S_T (a decrease in particle diameter). It was discovered that α typically represented the Ca elution tendency.

Fig. 18.

Relationship between effectiveness of total surface area a and S_T. (Online version in color.)

The obtained values for parameters, k, α and β_T could be applicable to the same kind of sample under the range of experimental conditions such as temperature, agitation intensity and liquid/solid ratio. As mentioned above, effect of temperature was small in the range from 20°C to 30°C. The liquid/solid ratio will be applicable to higher value, however, when the Ca concentration in the solution is close to a saturation, the situation will change. In addition, the agitation intensity will also change the situation. In the case of lower agitation intensity, it will cause to a diffusion control in aqueous solution near particle surface. While in the case of higher agitation intensity, it will cause the increase in collision between particles.

As describe in next section, the k and α must be change for a different kind of sample with different chemical nature, but the same value of β_T can be used. The applicable range of β_T is unknown in this stage, it would be necessary to perform further experimental and theoretical works for understanding of meaning of β_T.

3.4.2. Effect of Fe₂O₃ Content on Rate Constant

In the section above, the effect of particle size distribution on Ca elution was investigated using the same sample (SlagF4). In this section, the effect of Fe₂O₃ content is analyzed using the same particle size distribution. For comparison, SlagF0, in which Fe₂O₃ was not added, was selected (Table 1). The content of Fe₂O₃ from SlagF1–SlagF4 increased stepwise from 3.6 mass% to 14.0 mass%.

SlagF0–SlagF4 samples were sieved through a −53 μm mesh; however, the particle size distribution was measured only for SlagF4 and then applied to SlagF0–SlagF3 (Table 5). In these samples, the rate constants k should be changed because the chemical nature changes with Fe₂O₃ content. Meanwhile, the pre-exponential term α should be constant if the particle size distributions are exactly the same. In fact, α was not constant in the experiment, the reason of which will be discussed later.

Table 5. The list of fitted dissolution parameters at SlagF0-SlagF4.

Sample	Particle size distribution (μm)	XCa	C/(S+F)	ST (m2)	α	k (/s)
SlagF0	−53*	0.2923	1.29	0.0657*	0.40	6.0×10−4
SlagF1	−53*	0.2817	1.15	0.0657*	0.08	2.8 ×10−4
SlagF2	−53*	0.2767	1.08	0.0657*	0.08	1.8 ×10−4
SlagF3	−53*	0.2649	0.97	0.0657*	0.10	9.0 ×10−5
SlagF4	−53	0.2514	0.85	0.0657	0.40	1.13×10−5

*  SlagF0 - SlagF4 were sieved by mesh of −53 μm, the particle size distribution was measured only for SlagF4.

A kinetic analysis of the dissolution experiment using SlagF0–SlagF3 was performed based on Eq. (19). The results of SlagF0 are shown in Fig. 19. The rate of Ca elution of SlagF0 was the highest in the samples used in the present study. As mentioned above (Fig. 13, SlagF4, −25 μm), the calculated value can fit in the initial period only (< 488 s). Beyond 488 s, rate control changed to diffusion control in the solid phase, which can be expressed by Eq. (22) (U = 7.0 × 10⁻⁶, V = 2.0 × 10⁻⁴). The calculated values using Eq. (22) agreed with the observed values.

Fig. 19.

Variation of [Ca²⁺] in dissolution experiments with SlagF0. (Online version in color.)

Figures 20, 21, 22 show the comparisons between the calculated and observed values for SlagF1 (3.6 mass%Fe₂O₃, X_Ca = 0.2817, C/(S + F) = 1.15), SlagF2 (5.6 mass%Fe₂O₃, X_Ca = 0.2767, C/(S + F) = 1.08), and SlagF3 (9.4 mass%Fe₂O₃, X_Ca = 0.2649, C/(S + F) = 0.97), respectively. The calculated values agreed well with the observed values, except for the initial stage of the reaction. The behavior of the initial stage was caused by the characteristics of the particle size distribution of the −53 μm sample, as mentioned above; therefore, the proportion of small particles less than 40 μm was higher.

Fig. 20.

Variation of [Ca²⁺] in dissolution experiments with SlagF1. (Online version in color.)

Fig. 21.

Variation of [Ca²⁺] in dissolution experiments with SlagF2. (Online version in color.)

Fig. 22.

Variation of [Ca²⁺] in dissolution experiments with SlagF3. (Online version in color.)

The rate constants k and pre-exponential term α were obtained by trial and error, and the values are listed in Table 5. The value of α was not constant, contrary to expectations. The obtained rate constants k were plotted against the modified basicity C/(S + F), as shown in Fig. 23.

Fig. 23.

Variation of rate constant for samples with different Fe₂O₃ contents. (Online version in color.)

The relationship between k and C/(S + F) is expressed by Eq. (23).   

$$log\left(k\right)=7.4(C/(S+F))-7.89\hspace{1em}{\mathrm{r}}^2=0.919$$(23)

The correlation coefficient was not high, i.e., approximately 0.919.

Using Eq. (23) and assuming α = 0.1 and 0.2 as well as a distribution of −53 μm particles, [Ca]elute (%) was calculated using Eq. (24) and compared with the results of a previous study (Eq. (8))⁹⁾ in Fig. 24.   

$${\left[Ca\right]}_{elute}(\%)=\left[C{a}^{2+}\right]\times 40\times 0.4\left(L\right)/X_{Ca}\times 100$$(24)

Fig. 24.

Calculated values of Ca elution for samples with different Fe₂O₃ contents. (Online version in color.)

The parameters used are summarized in Table 6. In the case of α = 0.1, the calculated values agreed well with the observed values until the modified basicity C/(S + F) was 1.15. When C/(S + F) exceeded 1.15, the calculated result saturated and [Ca]_elute = 2.25%. The k value is considered inadequate in this C/(S + F) range because Eq. (23) involves a relatively low correlation coefficient. Meanwhile, when α = 0.2, the calculated value is greater than the value obtained using Eq. (8) for C/(S + F) = 0.8. The range of α might be from 0.1 to 0.2, when k is expressed by Eq. (23).

Table 6. Calculation results of Ca elution from different content of Fe₂O₃.

C/(S+F)	Fe2O3 (%)	CaO (%)	XCa	k (1/s)	α=0.1		α=0.2
					[Ca2+]cal (mol/L)	[Ca2+]elute (%)	[Ca2+]cal (mol/L)	[Ca2+]elute (%)	[Ca2+]elute (%), Eq. (8)
1.29	0.0	40.90	0.292	8.60E-04	4.10E-04	2.25	8.19E-04	4.49	3.95
1.28	0.2	40.80	0.291	7.89E-04	4.10E-04	2.25	8.19E-04	4.50	3.79
1.2	2.3	39.97	0.286	3.96E-04	4.00E-04	2.24	8.04E-04	4.51	2.75
1.1	5.2	38.88	0.278	1.67E-04	3.40E-04	1.96	6.86E-04	3.95	1.83
1	8.5	37.70	0.269	7.08E-05	2.17E-04	1.29	4.38E-04	2.60	1.22
0.8	16.6	35.06	0.250	1.26E-05	5.17E-05	0.33	1.05E-04	0.67	0.55
0.6	28.4	31.85	0.228	2.26E-06	7.96E-06	0.06	1.98E-05	0.14	0.24
0.4	47.7	27.68	0.198	4.04E-07	1.76E-06	0.01	3.57E-06	0.03	0.11
0.2	90.6	21.45	0.153	7.21E-08	3.16E-07	0.00	6.39E-07	0.01	0.05

4. Conclusion

Fe₂O₃ was added to a slag sample in a CaO–SiO₂–MgO–Al₂O₃ system from 3.6 mass% to 14 mass%. Dissolution experiments were performed using (1) different particle size distributions and (2) different Fe₂O₃ contents. A new kinetic analysis considering particle size distribution was developed and performed. The results obtained were as follows:

(1) The elution of elements (Ca, Si, Al, and Mg) increased with S_T (i.e., a decrease in particle diameter). Among the elements, Ca elution was the highest among all particle size distributions.

(2) When Fe₂O₃ was added to the slag, the elution of elements decreased significantly. In addition, the elution of Fe was the lowest and under the limit of detection of ICP.

(3) A kinetic model considering particle size distribution (Eq. (18)), where an effective surface area $S_{eff}^n$ and the effectiveness of the total surface area α were introduced, was developed.   

$$\left[C{a}^{2+}\right]\left(\frac{mol}{L}\right)=\sum \left(n_d{\left[C{a}^{2+}\right]}^d\right)=\sum \frac{{\left[Ca\right]}_0^d}{16.0}\left(1-exp\left(-S_{eff}^{d}kt\right)\right)$$

  

$$S_{eff}^d=exp\left({\beta }_T{n}_d{S}_d\right),\mathrm{\hspace{0.17em} }{\beta }_T=23.4$$

  

$${\left[Ca\right]}_0^d=\alpha n_d{S}_d,\hspace{1em}\alpha =3\times {10}^{-6}\delta$$

(4) Kinetic analysis was performed, and the parameters for the rate equation were obtained.

(5) The obtained rate constant k was a fixed value for one type of sample (SlagF4), while k decreased with increasing Fe₂O₃ content.

(6) The values of α increased significantly with S_T (a decrease in particle diameter) in different particle size distributions. It was discovered that α typically represented the Ca elution tendency.

(7) The change in α was small with the change in Fe₂O₃ content.

(8) It was discovered that the developed kinetic model was valid for the analysis of Ca elution for samples with different particle size distributions.

Acknowledgments

The authors appreciate the financial support by the Iron & Steel Institute of Japan (ISIJ) for “The control of solidification structure of steelmaking slag for suppressing alkaline elution” study.

References

• 1)  Annual Statistics of Iron and Steel Slags in 2018, Nippon Slag Association, Tokyo, (2018), 2 (in Japanese).
• 2)   Y.  Kashiwaya,  T.  Nakauchi,  P.  Khanh Son,  S.  Akiyama and  K.  Ishii: ISIJ Int., 47 (2007), No. 1, 44.
• 3)   Y.  Ta,  T.  Hoshino,  H.  Tobo and  K.  Watanabe: Tetsu-to-Hagané, 104 (2018), 708 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2018-032
• 4)   T.  Teratoko,  N.  Maruoka,  H.  Shibata and  S.  Kitamura: High Temp. Mater. Process., 31 (2012), 329.
• 5)   C.  Du,  X.  Gao,  S.  Kim,  S.  Ueda and  S.  Kitamura: ISIJ Int., 56 (2016), 1436.
• 6)  Division of Basic Research on Slag Utilization: Tetsu-to-Hagané, 65 (1979), 1787 (in Japanese).
• 7)   F.  Tsukihasi and  H.  Matsuura: J. MMIJ, 129 (2013), 185. https://doi.org/10.2473/journalofmmij.129.185
• 8)   T.  Kato,  T.  Kusui,  C.  Kosugi and  T.  Fukushima: Tetsu-to-Hagané, 106 (2020), 50 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2019-026
• 9)   S.  Tauchi and  Y.  Kashiwaya: Tetsu-to-Hagané, 104 (2018), 107 (in Japanese).
• 10)   C. F.  Dickinson and  G. R.  Heal: Thermochim. Acta, 340–341 (1999), 89.
• 11)   S.  Benita and  M.  Donbrow: Int. J. Pharm., 12 (1982), 251.
• 12)   S. N.  Lekakh,  C. H.  Rawlins,  D. G. C.  Robertson,  V. L.  Richards and  K. D.  Peaslee: Metall. Mater. Trans. B, 39 (2008), 125.
• 13)   C. F.  Dickinson and  G. R.  Heal: Thermochim. Acta, 340–341 (1999), 89.
• 14)   S. N.  Lekakh,  C. H.  Rawlins,  D. G. C.  Robertson,  V. L.  Richards and  K. D.  Peaslee: Metall. Mater. Trans. B, 39 (2008), 125.
• 15)   Z.  Zhu,  X.  Gao,  S.  Ueda and  S.  Kitamura: ISIJ Int., 59 (2019), No. 10, 1908.
• 16)  Slag Atlas, 2nd ed., ed. by VDEh, Verlag Stahleisen, Düsseldorf, (1995), 157.
• 17)   H.  Kimura,  T.  Ogawa,  M.  Kakiki,  A.  Matsumoto and  F.  Tsukihashi: ISIJ Int., 45 (2005), 506. https://doi.org/10.2355/isijinternational.45.506
• 18)  JIS K 0058: 2005, Test methods for chemicals in slags--Part 1: Leaching test method.
• 19)   M.  Numata,  N.  Maruoka,  S. J.  Kim and  S.  Kitamura: ISIJ Int., 54 (2014), 1983.
• 20)   T.  Igarashi,  T.  Izutsu and  Y.  Oka: J. Jpn. Soc. Eng. Geol., 43 (2002), 208 (in Japanese).