Temperature Dependence and Prediction of Density of Alkali Silicate Slag Melts

Abstract

We measured the density of eight types of molten alkali silicate slag within the range of 1673–1823 K using the Archimedean double-bob method. The melt density of binary alkaline silicate glass linearly increased with a decrease in temperature. The temperature coefficient of the density ranged from −21.5 to −13.2 × 10⁻⁵ g·cm⁻³·K⁻¹, and its order corresponded to the order of the ionic radii of the alkali oxide components. We applied Doolittle’s free volume theory by combining the reported values of melt viscosity with the measured densities and volume expansion coefficients of melts and glass solids to propose an equation for predicting melt density from slag composition and temperature. Melt densities in the range of 1273–1823 K were predicted and compared with experimental values.

1. Introduction

In metal and steel production, molten slag plays a role in protecting metal melts from the atmosphere and extracting unwanted impurities.^1,2,3) To achieve carbon neutrality by 2050, the development and introduction of high-temperature melting technologies, fuel conversion, and the use of slag by-products⁴⁾ are being considered. Owing to the aforementioned advanced melting processes, it is important to understand the physical properties of oxide slag melts of various compositions.^5,6) This study focuses on the density of high-temperature melts.⁷⁾ This density is directly related to the weight and volume of molten slag. The coefficient of volume expansion can be calculated from the temperature dependence of the density. Highly reliable melt densities over a wide temperature range are required during the simulation of the melting process of multicomponent silicate glass because it is similar to slag.⁸⁾ The temperature dependence of melt density is also related to the homogenization of multi-component melts due to thermal convection current and to the phase separation process of slag/metal and slag/matte melts. Additionally, the density is closely concerned with the structure of the melt at the atomic level in microscopic terms. Thus, the density of high-temperature molten slag is an important physical property.

Methods for measuring the density of molten slag at high temperatures mainly include the Archimedean double-bob,^9,10) maximum bubble pressure,¹¹⁾ and static sessile drop methods.¹²⁾ The Archimedean double-bob method is a highly accurate measurement method.^9,10,13,14) However, the apparatus used is typically made by researchers. Therefore, the shapes and characteristics of the furnace, heating element, and temperature control device differ. Thus, the error and reproducibility in each measurement must be evaluated. The density of oxide slag melt at high temperatures is highly dependent on the chemical composition and temperature.^9,10,13,14) The composition of multicomponent silicate slag melts used in industrial applications varies, and a wide temperature range is typically used during the production process. Thus, a highly accurate prediction equation for the density of the melt from a chemical composition over a wide temperature range is desired. Conventional prediction formulas have been proposed based on reported values and partial molar fractions of slag components.^15,16) However, high-accuracy prediction equations have not necessarily been introduced. Highly accurate measured values are required in the development of density prediction equations. In glass solids formed by the rapid cooling of molten slag, lattice oscillations of ion pairs cause thermal expansion.¹⁷⁾ Furthermore, for molten slag melt, the contribution of free volume is considered.

This study attempted to create a prediction equation for the density of molten slag based on Doolittle’s free volume theory.^18,19) According to Doolittle, the temperature dependence of the viscosity of organomolecules and liquids is associated with free volume. The temperature dependence of density can be converted into the coefficient of volume expansion (temperature dependence of volume). Thus, it is assumed that the density of molten slag can be predicted from the temperature dependence of viscosity, according to Doolittle’s free volume theory. In the free volume theory, the total volume (v) of molten slag is the sum of the occupied volume (v₀) and free volume (v_f). The occupied volume is the sum of the volume of the molecule and the volume related to vibration; the rest is the free volume. Furthermore, the occupied volume is dependent on temperature. However, the temperature dependence of the viscosity of the melt is governed by changes in the free volume since the temperature dependence of the free volume is considerably greater. Based on these ideas, Doolittle’s viscosity formula is expressed by Eq. (1).

  

$$\eta =\text{Aexp}\left(\text{B}\frac{v_0}{v_f}\right),$$(1)

where constants A and B are determined by the system. It is known that the viscosities of organic small molecules and polymer liquids are well expressed by Eq. (1). Here, the occupied volume was calculated from the density and thermal expansion coefficient of the glass solid and glass transition temperature. First, we improved the handmade setup of the density measurement of high-temperature melts through the Archimedean double-bob method.¹⁴⁾ Thereafter, the measurement conditions of a standard slag sample with relatively high accuracy were established. The temperature dependence of the density of binary alkali silicate glass solids and melts was investigated over a wide temperature range. Alkali silicate glasses are well known for their microscopic structures^20,21) in the basic compositional system. Finally, using the measured density, volume expansion coefficient, and glass transition temperature for slag solids and melts, we developed an equation for predicting melt density based on Doolittle’s free volume theory, and the measured and predicted values were compared.

2. Experimental Procedure

2.1. Melt Density

The density of binary alkaline silicate slag melt with a viscosity ranging from 10⁻² to 10 Pa·s was measured using the Archimedean double-bob method. Figure 1 shows that a Pt weight of volume V connected to a wire with radius r was immersed in a melt with density ρ. The difference between the present and a previous study¹⁴⁾ is that the radiant heat from the electric furnace to the balance was blocked as much as possible with refractory materials and ice water for cooling. The buoyancy (W) received by the weight is expressed by Eq. (2), including the surface tension term on the wire, effect of air convection, and vapor deposition of the substance.

  

$$W=V\cdot \rho -2\pi r\cdot \gamma \cdot \text{cos}\theta +\text{A,}$$(2)

where W (10⁻³ kg) is the buoyancy force that the weight receives from the liquid (melt), and V (10⁻⁶ m³) is the volume of the weight in the liquid (melt) (10³ kg·m⁻³). ρ is the density of the melt, r (10⁻³ m) is the diameter of the wire, and γ (10⁻³ N·m⁻¹) is the surface tension of the liquid (melt), θ (°) is the contact angle between the hanging thread and liquid surface, and A (10⁻³ kg) is a constant related to the convection of air and vapor matters on the hanging thread. In Eq. (2), two weights of different volumes are used to offset the effects of the surface tension, vapor deposition, and updraft of the melt on the wire.

  

$$W_{\text{L}}=V_{\text{L}}\cdot \rho -2\pi r\cdot \gamma \cdot \text{cos}\theta +\text{A,}$$(3)

  

$$W_{\text{S}}=V_{\text{S}}\cdot \rho -2\pi r\cdot \gamma \cdot \text{cos}\theta +\text{A,}$$(4)

where L and S mean large and small, respectively, and W_L is the buoyancy force on the large bob. W_S is the buoyancy force on the small bob, V_L is the volume of the large bob, and V_S is the volume of the small bob. The difference between W_L and W_S when two bobs were immersed in the melt under the same conditions was obtained. The density of the liquid was obtained from the difference in the buoyancy and the volume difference of the weight using Eq. (5).

  

$$\rho =\frac{W_{\text{L}}-W_{\text{S}}}{V_{\text{L}}-V_{\text{S}}}.$$(5)

Fig. 1. Schematic of density measurement setup using the Archimedean double-bob method. (Online version in color.)

Considering that the volume of the Pt bob changes because of thermal expansion at high temperatures, the correction of the volume after the thermal expansion of the Pt bob is expressed by Eq. (6).

  

$$\rho =\frac{W_{\text{L}}-W_{\text{S}}}{\left(1+3\alpha \mathrm{\Delta }T\right)V_{\text{L}}-V_{\text{S}}},$$(6)

where α is the coefficient of the linear thermal expansion of Pt (1.0102 × 10⁻⁵ K⁻¹),²²⁾ and ΔT is the temperature difference between the room and density measurements. When determining the density of the melt, the volume difference (V_L − V_S) of the Pt bob must be determined beforehand. Using the weight of both Pt bobs and the density of Pt,²³⁾ the volume difference (ΔV) of the Pt bobs was obtained using Eq. (7).

  

$$\mathrm{\Delta }V=V_{\text{L}}-V_{\text{S}}=\left(\frac{m_{\text{L}}}{{\rho }_{\text{Pt}}}\right)-\left(\frac{m_{\text{S}}}{{\rho }_{\text{Pt}}}\right),$$(7)

where m_L is the weight of a large Pt bob, m_S is the weight of a small Pt bob, and ρ_Pt is the density of Pt (21.45 g·cm⁻³). As shown in Fig. 1, the buoyancy force on the Pt bob was read down to 0.1 mg using an automatic precision balance (CP224S, Sartorius). A wire made of 0.25-mm Pt–Rh13 wt% was used to connect the precision balance and Pt weight. The container used for the molten slag was a Pt–20%Rh crucible, and the weight for immersion was made of Pt. A MoSi₂ heating element was used in the setup. The maximum measured temperature with the heat resistance of the aforementioned wire-suspended Pt weight was 1823 K.

To prepare a synthetic slag sample for the measurements, SiO₂ (High Purity Chemical Research Institute, 99.9% purity), Li₂CO₃ (High Purity Chemical Research Institute, 99.9%), Na₂CO₃ (Kanto Chemical Co., Ltd., 99.8%), and K₂CO₃ powder (Kanto Chemical Co., Ltd., 99.5%) were used. Table 1 presents the sample composition. The sample names were abbreviated by the content and type of alkali oxides. For example, the 30 mol% Li₂O–70 mol% SiO₂ sample was denoted by “30LS.” The raw materials were weighed and mixed to obtain the molten slag sample (100 g), which was melted for 2 h in the atmosphere at 1873 K using a Pt–20%Rh crucible. The molten sample was rapidly cooled and vitrified by pouring it onto a stainless-steel mold. The crushed glass sample was remelted in the atmosphere at 1873 K for 2 h, and the density was measured during a temperature-lowering process from 1823 to 1673 K.

Table 1. Slag batch compositions.

Sample	Composition (mol%)
	Li2O	Na2O	K2O	SiO2
30LS	30	–	–	70
35LS	35	–	–	65
25NS	–	25	–	75
30NS	–	30	–	70
35NS	–	35	–	65
40NS	–	40	–	60
25KS	–	–	25	75
30KS	–	–	30	70

The buoyancy measurements were started after holding 30 min at each temperature for melt homogenization. Buoyancy readings were continuously performed at each temperature for 30 min, and the average value was used to calculate the density as a buoyancy measurement. The composition and temperature range of the slag for density measurement were determined using the thermodynamic equilibrium calculation software & thermodynamic database, FactSage. A composition of 40Na₂O–60SiO₂ (mol%), which had a low liquidus temperature and can be measured over a wide temperature range, was selected as the standard. For the molten slag, the reproducibility of the measurement and the comparison between experimental and reported^9,24) values were evaluated within the range of 1673–1773 K. The measurement temperature of the density of each slag melt (Table 1) was determined from the liquid phase region in the diagram of the R₂O–SiO₂ (R = Li, Na, K) binary system up to a maximum of 1823 K.

2.2. Solid Density, Linear Thermal Expansion, and Glass Transition Temperature

Starting materials and sample compositions similar to those used for the determination of the melt density were selected. After weighing and mixing the raw materials (30 g) of the glass samples, they were melted for 2 h in the atmosphere at 1873 K using a Pt crucible. The molten sample was poured onto a carbon mold heated to 673 K and annealed for 1 h at its glass transition temperature plus 20 K and slowly cooled to room temperature at a cooling rate of 1 K·min⁻¹. Finally, a glass sample was obtained. The density of the glass solids at room temperature was measured via the Archimedes method using kerosene as an immersion liquid (accuracy = 0.01 g·cm⁻³).²⁵⁾ The coefficient of linear expansion of glass, α_S, was evaluated using a thermomechanical analyzer (TMA8310, Rigaku). Samples (3.5 × 3.5 × 16 mm³) were heated from room temperature to 873 K at a heating rate of 10 K·min⁻¹. The glass transition temperature (T_g) was calculated from the bend points with different inclinations in the positive thermal expansion curve.²⁵⁾ In addition, α_S was calculated from the slope of the expansion curve. The coefficient of volume expansion of the glass solid, β_S, was calculated as 3α_S ≈ β_S.²⁶⁾

3. Results and Discussion

3.1. Accuracy of Melt Density Measurements

Figure 2 shows the buoyancy measurement results as the raw data of the 40Na₂O–60SiO₂ (mol%) melt. Figure 3 shows the melt densities calculated from the average of each buoyancy value. Here, the difference between the values of the two measurements was 0.2%. Using the apparatus described in our previous report,¹⁴⁾ the difference between two measurements of the slag melt was 0.6%. Here, the reproducibility was improved using the modified apparatus due to the arrangements of ice water cooling and of refractory between the furnace and the electrical balance (Fig. 1). The difference between the density and reported values at each measured temperature was up to 0.23%. These comparisons confirmed that the self-made apparatus (Fig. 1) could measure the melt density with high accuracy and reproducibility.

Fig. 2. Experimental data of buoyancy of 40 mol% Na₂O–60 mol%SiO₂ slag melt obtained via the Archimedean double-bob method. (Online version in color.)

Fig. 3. Variation in density with temperature for the 40 mol% Na₂O–60 mol%SiO₂ slag melt. The error bars with each color means the density range due to the fluctuation of measured buoyancy values, as shown in Fig. 2. (Online version in color.)

3.2. Density of Alkali Silicate Melts

Figure 4 shows variations in the melt density with temperature, along with the temperature coefficient of density, $\frac{d\rho }{dT}$. Table 2 lists the measured density values. In all the melts, the densities linearly increased with a decrease in temperature. The temperature coefficient was −1.37 and −1.32 × 10⁻⁵ g·cm⁻³·K⁻¹, −2.30 and −1.56 × 10⁻⁵ g·cm⁻³·K⁻¹, and −2.15 and −1.62 × 10⁻⁵ g·cm⁻³·K⁻¹ for the LS, NS, and KS slags, respectively. The absolute value of the temperature coefficient of the 30KS slag was the largest; that of the 30LS slag was the smallest. The order of $\frac{d\rho }{dT}$ corresponded to the cation radius of the alkali oxide component.²⁷⁾ This was presumed to be related to the anharmonic lattice vibrations caused by the difference in Coulombic attraction between ions based on the difference in the distance between ions.¹⁷⁾ The absolute value of $\frac{d\rho }{dT}$ increased with an increase in the alkali oxide content. This was related to the fact that the network structure of Si–O–Si covalent bonds was cleaved by alkali oxides for network modifiers and changed to small silicate anion forms.^28,29)

Fig. 4. Variations in density with temperature for binary alkali silicate slag melts. (Online version in color.)

Table 2. Measured densities of alkali silicate slag melts.

Temperature (K)	30LS	35LS	25NS	30NS	35NS	40NS	25KS	30KS
1673	2.143	2.124	2.216	2.216	2.220	2.201	2.180	2.167
1723	2.136	2.119	2.212	2.214	2.212	2.187	2.169	2.158
1773	2.127	2.109	2.203	2.199	2.203	2.179	2.164	2.144
1823	2.124	2.105	2.193	2.191	2.185	2.165	2.155	2.135

3.3. Molar Volume of Alkali Silicate Slags

Table 3 lists the values of the measured density (ρ_S), coefficient of linear thermal expansion (α_S), calculated coefficient of body expansion (β_S), and glass transition temperature (T_g) of the alkaline silicate glass solids at room temperature. Figure 5 shows the measurement results of the density of the alkaline silicate glass solids and melts, and the molar volume (V_m(T)) is shown as a function of temperature. V_m(T) was calculated using Eq. (8).

  

$$V_m\left(T\right)=\frac{{\sum }_i{M}_i{x}_i}{\rho \left(T\right)},$$(8)

where M_i and x_i are the molecular weight and molar fraction of each component, respectively. ρ_S and ρ_L are the densities of the alkaline silicate glass solids and melts, respectively; V_S(T) is the molar volume of glass solids, and V_L(T) is the molar volume of the melt. Table 4 presents the calculated results of the V_S(T) and V_L(T). The V_S(T) was calculated using Eq. (9), assuming that the volume linearly changed from room temperature to the T_g.

  

$$V_{\text{S}}\left(T\right)=\frac{{\sum }_i{M}_i{x}_i}{{\rho }_{\text{S}}}\left(1+{\beta }_{\text{S}}\mathrm{\Delta }T\right).$$(9)

Table 3. Measured density, glass transition temperature, and linear and volume thermal expansion coefficients of alkali silicate glasses.

Sample	ρs (g·cm−3)	Tg (K)	αs (×10−5 K−1)	βs (×10−5 K−1)
30LS	2.336	748	0.977	2.932
35LS	2.352	741	1.125	3.375
25NS	2.441	752	1.199	3.598
30NS	2.476	753	1.397	4.191
35NS	2.500	729	1.544	4.632
40NS	2.533	724	1.663	4.990
25KS	2.422	781	1.382	4.146
30KS	2.452	793	1.702	5.107

Fig. 5. Variations in molar volume with temperature for binary alkali silicate slag solids and melts. (Online version in color.)

Table 4. Equations for the temperature dependences of the solid and molten volumes of the alkali silicate slags.

Sample	VS(T) (cm3·mol−1)	VL(T) (cm3·mol−1)
30LS	0.000708T + 21.810	0.001754T + 21.336
35LS	0.000784T + 21.012	0.002001T + 20.468
25NS	0.000987T + 24.745	0.002215T + 24.186
30NS	0.001103T + 24.426	0.002542T + 23.758
35NS	0.001243T + 24.248	0.002663T + 23.619
40NS	0.001268T + 23.969	0.003238T + 23.096
25KS	0.001212T + 28.285	0.002808T + 27.522
30KS	0.001620T + 28.580	0.003381T + 27.701

The V_L(T) was calculated using Eq. (10), assuming that the volume change from the melt density, ρ_L, at a certain temperature to the T_g occurred linearly.

  

$$V_{\text{L}}\left(T\right)=\frac{{\sum }_i{M}_i{x}_i}{{\rho }_{\text{L}}}.$$(10)

In each melt, the molar volume increased with an increase in temperature. This corresponded to a decrease in the density with temperature.

3.4. Prediction of Density of Alkaline Silicate Glass Melts by Multiregression Analysis

3.4.1. Application of Doolittle’s Free Volume Theory

According to Doolittle’s free volume theory,^18,19) the volume, v(T), of glass is expressed by the sum of the occupied volume, v₀(T), and free volume, v_f(T) (Eq. (11)).

  

$$v\left(T\right)=v_0\left(T\right)+v_f\left(T\right).$$(11)

The v₀(T) was estimated to be the sum of the volume of the molecule and the volume related to the lattice vibration and equal to the molar volume of the glass solid, V_S(T), using Eq. (12).

  

$$v_0\left(T\right)=V_{\text{S}}\left(T\right).$$(12)

The v_f(T) was calculated using Eq. (13).

  

$$v_f\left(T\right)=V_{\text{L}}\left(T\right)-V_{\text{S}}\left(T\right).$$(13)

Equations (11) and (12) were substituted into Eq. (1), and a prediction equation expressing the relationship between the viscosity and molar volume was formed.

  

$$\begin{array}{l}\eta \left(T\right)=Aexp\left(B\frac{V_{\text{S}}\left(T\right)}{V_{\text{L}}\left(T\right)-V_{\text{S}}\left(T\right)}\right),\\ \text{ln}\mathrm{\hspace{0.17em} }\eta \left(T\right)=lnA+B\left(\frac{V_{\text{S}}\left(T\right)}{V_{\text{L}}\left(T\right)-V_{\text{S}}\left(T\right)}\right).\end{array}$$(14)

By organizing Eq. (14) with the Arrhenius-type equation, the intercept (A) and slope (B) of the straight line were obtained. Figure 6 shows the relationship between ln η(T) and V_S(T)/(V_L(T)−V_S(T)) for 40NS slag melt as an example. The viscosity η(T) was quoted from the database (INTERGLAD ver. 8).³⁰⁾

Fig. 6. The relationship between ln η(T) and V_S(T)/(V_L(T)−V_S(T)) for 40NS slag melt as an example. (Online version in color.)

The relationship between the values of ln A and B and the glass composition was investigated, and the regression of the composition of the alkaline silicate glass was obtained via the least squares method. Based on the results, the molar volume of the alkaline silicate glass melts, V_L(T), was predicted; the melt density (ρ_L) was calculated using Eq. (10), and the melt density prediction equation was created. For example, the calculation process of density ρ_cal for 40NS was described in the following. The obtained values of V_S(T) = 25.75 and lnA = −5.940, and B = 0.439 at ln η(T) = 0 were substituted into Eq. (14) and then V_L(T) was calculated as 24.40. The density ρ_cal was calculated as 2.220 from Eq. (10) with the substituted values of 24.40 for V_L(T) and 60.8424 for ${\sum }_i{M}_i{x}_i$.

Using the obtained molar volumes, V_S(T) and V_L(T), and the viscosity η(T) quoted from the database (INTERGLAD ver. 8),³⁰⁾ the glass composition and calculated values of ln A and B were obtained (Table 5). During the regression analysis, (mol% Li₂O, hereinafter referred to as %Li₂O), (%Na₂O), (%K₂O), and (%SiO₂) were used as explanatory variables. In Eqs. (15) and (16), the compositions of ln A and B used to perform regression analysis on the alkaline silicate slag composition from the values in Table 5 are shown as a function.

  

$$\begin{array}{l}ln\mathrm{\hspace{0.17em} }\text{A}=-0.13028\left({\text{\%Li}}_2\text{O}\right)-0.15236\left({\text{\%Na}}_2\text{O}\right)\\ \hspace{1em}\hspace{1em}-0.10557\left({\text{\%K}}_2\text{O}\right)-0.00427\left({\text{\%SiO}}_2\right).\end{array}$$(15)

  

$$\begin{array}{l}\text{B}=0.005915\left({\text{\%Li}}_2\text{O}\right)+0.008670\left({\text{\%Na}}_2\text{O}\right)\\ \hspace{1em}+0.008806\left({\text{\%K}}_2\text{O}\right)+0.001034\left({\text{\%SiO}}_2\right).\end{array}$$(16)

Table 5. Calculation results of ln A and B via the least-squares method for the temperature dependence of melt density.

Sample	ln A	B
30LS	−4.597	0.244
35LS	−4.801	0.267
25NS	−4.431	0.286
30NS	−4.937	0.329
35NS	−5.813	0.345
40NS	−5.940	0.439
25KS	−2.590	0.279
30KS	−4.224	0.323
30LS	−3.860	0.264
25KS	−2.443	0.319

In Eqs. (15) and (16), it is known that coefficients A and B of the alkali oxide, which is a modified oxide,³¹⁾ were larger than those of the network component, SiO₂.

3.4.2. Prediction of Melt Density

The densities calculated using the prediction equation, ρ_cal., were compared with the measured densities, ρ_mea. at 1673 K. Figure 7 shows the difference between the calculated and experimental values. In addition to the measured values used in the regression analysis, the measured values reported in the literature^9,24,32) were added. The dotted line in Fig. 7 represents the point where the predicted and measured values coincide. The maximum difference between both values at 1673 K was 0.025 g·cm⁻³, which was approximately a 1.1% difference from the measured value. Considering that the difference between the predicted and measured values was an average of 0.5%, Eqs. (15) and (16) for the melt density predicted from the composition and temperature were reasonably accurate.

Fig. 7. Relationship between the calculated and measured densities of binary alkali silicate slag melts at 1673 K. (Online version in color.)

Figure 8 shows the density of the alkaline silicate slag melt measured in this study, ρ_mea., and a dashed line connecting the density measurement of the glass and the density at the T_g. Furthermore, the melt density predicted over the range of 1273–1823 K in the melt state is shown. ρ_cal. is indicated by the colored solid lines. Within the range of 1273–1823 K in the melt state, the difference between the measured and predicted values in this study was 0.4% on average.

Fig. 8. Temperature dependences of the measured and calculated densities of the alkali silicate slag melts. The black dotted lines show the connections between measured densities and predicted densities at T_g by least squares method. The colored solid lines indicate predicted densities ρ_cal. based on Doolittle’s free volume theory. The temperature range of predicted melt density depends on quoted melt viscosity.³⁰⁾ (Online version in color.)

4. Conclusions

We evaluated the accuracy of measuring the density of molten alkaline silicate slag with a composition of 40 mol%Na₂O–60 mol%SiO₂ at high temperatures via the Archimedean double-bob method. In addition, we established a measurement method for a handmade setup based on the comparison and reproducibility between the measured and reported values. The measurements of the melt density of eight binary alkaline silicate glasses linearly increased with a decrease in temperature. The temperature coefficient of density ranged from −21.5 to −13.2 × 10⁻⁵ g·cm⁻³·K⁻¹, and the order of the coefficient corresponded to the order of the ionic radii of the alkali oxide components. The reported values of melt viscosity were combined with the measurement results of the melt density, glass solids, and volume expansion coefficients. Thus, Doolittle’s free volume theory was applied to create a prediction equation for melt density with a good correlation between the predicted and measured values. Although the composition system was limited, melt densities at any temperature within the range of 1273–1823 K were predicted with an error range of 0.4% based on the composition, temperature, molar volume, and viscosity of the alkaline silicate slags.

Conflict of Interest

Authors are requested to declare any conflicts of interest related to the conduct of this research.

No conflicts exist for this research.

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