The data used come from several publications. The phase equilibrium diagram of the SiO₂-Na₂O-CaO system established by Rankin, Wright, Greig and Clifton provides the liquidus of silica (1723˚C) and lime (2570˚C) [2] [3] . The phase diagram of the SiO₂-Na₂O-CaO system from the FACT database provides the liquidus of soda ash (1132˚C) [4] [5] . The data is collated by considering the vitrification tests of said ternary mixture. A table summarizing the data is drawn up. This table is used to determine the matrices for the modeling phase. A starting matrix whose columns are the constituents of the ternary system. A column matrix of the melting temperatures of the respective mixtures is also constructed.

We are looking for a mathematical expression of the correlations between compositions and melting temperature. The mixture models described in Chapter 4 are determined for this phase data. These range from the first-order model to the third-order synergistic model. The mathematical temperature functions of these models are listed below [9] - [11] .

First-order model

${\hat{T}}_1\left(˚\text{C}\right)={\displaystyle \underset{i=1}{\overset{3}{\sum }}{b}_i{x}_i}$

Second-order model

${\hat{T}}_2\left(˚\text{C}\right)={\displaystyle \underset{i=1}{\overset{m}{\sum }}{b}_i{x}_i}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{b}_{ij}{x}_i{x}_j}}$

Complete third-order model

${\hat{T}}_{3,C}\left(˚\text{C}\right)={\displaystyle \underset{i=1}{\overset{m}{\sum }}{b}_i{x}_i}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{b}_{ij}{x}_i{x}_j}}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{b}_{ij}{x}_i{x}_j\left(x_i-x_j\right)}}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{\displaystyle \sum b_{ijk}{x}_i{x}_j{x}_k}}}$

Synergistic third-order model

${\hat{T}}_{3,S}\left(˚\text{C}\right)={\displaystyle \underset{i=1}{\overset{m}{\sum }}{b}_i{x}_i}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{b}_{ij}{x}_i{x}_j}}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{\displaystyle \sum b_{ijk}{x}_i{x}_j{x}_k}}}$

For these models, the parameters have the following characteristics.

$b_i,b_{ij},b_{ijk},T\left(˚\text{C}\right)\in ℝ$

$x_i,x_j,x_k\in \left\{\%{\text{SiO}}_2;\%{\text{Na}}_{\text{2}}\text{O};\%\text{CaO}\right\}\text{et}{x}_i,x_j,x_k\in \left[0;1\right]$

The coefficients $b_i,b_{ij},b_{ijk}$ of the component proportions $x_i$, $x_i{x}_j$ and $x_i{x}_j{x}_k$ respectively represent their influence on the melting temperature [9] - [11] .

Empirical analysis has shown that the best results are obtained by allocating 20% - 30% of the initial data points for the tests and using the remaining 70% - 80% for the training set [12] - [14] . Knowing that we have 46 data points, this means that we’ll have around 9 to 14 points for the tests and around 32 to 37 points for the training set. This distribution will enable us to test the performance of our model on a separate dataset from the one on which it was trained, which is essential for assessing its ability to generalize to new data. We will then be able to adjust our model according to the performance observed on the test set to optimize its performance on future datasets.

The statistical study is carried out to assess the quality of the model through the coefficient of determination (correlation). This coefficient is defined by the expression below

$R^2=\frac{SSR}{SST}=1-\frac{SSE}{SST}$

With

Total sum of squares:

$SST=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(T_i^{*}\right)}^2=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(T_i-\bar{T}\right)}^2$

Sum of squares due to error

$SSE=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(T_i-{\hat{T}}_i\right)}^2$

sum of squares due to regression

$SSR=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left({\hat{T}}_i-\bar{T}\right)}^2$

The closer R² is to 1, the “closer” the fitted model is to the observed responses. A classic threshold is to look for models for R² ≥ 0.9500. This procedure enables us to assess the accuracy of the mathematical model. This procedure enables us to assess the accuracy of the mathematical model .

Mixtures of the ternary SiO₂-Na₂O-CaO system were prepared and melted to validate the best model obtained in terms of the correlation coefficient. These mixtures were melted at the temperatures predicted by the model. Micrography of the glass samples obtained is used to validate the model by observing the melting state of the test mixtures. The silica raw material is collected locally in the Maféré area, Côte d’Ivoire .

In view of the equipment available in our laboratory, we have chosen melting temperatures below 1250˚C. This temperature limit is set according to our furnace, whose maximum temperature is 1280˚C. Using this temperature, we generate ternary system mixtures .

Melt tests are carried out with a powdered siliceous raw material with a granularity of between 80 μm and 100 μm ( and ). This particle size is similar to that used by Marlind Daud and Mahadi Abu in their work on soda-lime glass production .

Sodium and calcium are supplied to the mixtures in a solid state (powder) through their carbonates. Grynberg’s work on the ternary system SiO₂-Na₂O-CaO showed the possibility of using these carbonates for the production of soda-lime glass [7] [17] . and below illustrate the sodium and calcium carbonate labels used in our experiment.

The 200 grams mixtures are introduced into crucibles for melting (). The temperature rise from ambient to target temperature takes about an hour.

Several mixtures with varying compositions of silica, soda ash and lime have been identified. summarizes these mixtures and their respective melting temperatures. Compositions are considered in the interval [0; 1] and temperatures are in degrees Celsius.

Source*: [2] - [5] [7] [8] [16] [17] .

The majority of SiO₂-Na₂O-CaO mixtures studied in the literature contain at most 80% silica [4] [5] [7] [18] . below illustrates the silica proportion as a function of temperature of the blends used for modeling. The silica proportion of ternary mixtures ranges from 0.39 to 0.8 for laboratory tests and industrial production. The limit of 80% silica in blends is linked to the high melting temperature of silica (1723˚C). In , the letters C and B are used to identify pure bodies and binary mixtures (mixtures of two components) respectively. In these figures, the areas circled with the letter T within these circles represent ternary mixtures. In ternary mixtures, the proportion of flux (sodium oxide) typically ranges from 5% to 25% (), while the stabilizer (calcium oxide) content varies between 4% and 50% ().

The data in above have been used to establish the mathematical expressions below $T\left(˚\text{C}\right)=f\left(\%{\text{SiO}}_2;\%{\text{Na}}_{\text{2}}\text{O};\%\text{CaO}\right)$. These are linear expressions of the components of the ternary system. Models from first-order mixtures to third-order synergistic models were determined. The accuracy of the models obtained was assessed.

The first-order polynomial model adapted to the study of mixtures, for 3 components, is given by the expression below.

${\hat{T}}_1\left(˚\text{C}\right)={\displaystyle \underset{i=1}{\overset{3}{\sum }}{b}_i{x}_i}$

$b_i,T\left(˚\text{C}\right)\in ℝ$

$x_i\in \left\{\%{\text{SiO}}_2;\%{\text{Na}}_{\text{2}}\text{O};\%\text{CaO}\right\}\text{and}{x}_i\in \left[0;1\right]$

In detail

${\hat{T}}_1\left(˚\text{C}\right)=b_{{\text{SiO}}_{\text{2}}}{x}_{{\text{SiO}}_{\text{2}}}+b_{{\text{Na}}_{\text{2}}\text{O}}{x}_{{\text{Na}}_{\text{2}}\text{O}}+b_{\text{CaO}}{x}_{\text{CaO}}$

${\hat{T}}_1\left(˚\text{C}\right)=1305.69\cdot x_{{\text{SiO}}_{\text{2}}}+994.45\cdot x_{{\text{Na}}_{\text{2}}\text{O}}+1710.48\cdot x_{\text{CaO}}$

The quality of the polynomial ${\hat{T}}_1\left(˚\text{C}\right)$ is given by its coefficient of determination. This coefficient of the first-order polynomial is very low compared with the reference (R² = 0.2082 < 0.9500).

The second-order polynomial model adapted to the study of mixtures, for 3 components, is given by ${\hat{T}}_2\left(˚\text{C}\right)$.

${\hat{T}}_2\left(˚\text{C}\right)={\displaystyle \underset{i=1}{\overset{3}{\sum }}{b}_i{x}_i}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{b}_{ij}{x}_i{x}_j}}$

$b_i,b_{ij},T\left(˚\text{C}\right)\in ℝ$

$x_i,x_j\in \left\{\%{\text{SiO}}_2;\%{\text{Na}}_{\text{2}}\text{O};\%\text{CaO}\right\}\text{et}{x}_i,x_j\in \left[0;1\right]$

In detail

$\begin{array}{c}{\hat{T}}_2\left(˚\text{C}\right)=b_{{\text{SiO}}_{\text{2}}}{x}_{{\text{SiO}}_{\text{2}}}+b_{{\text{Na}}_{\text{2}}\text{O}}{x}_{{\text{Na}}_{\text{2}}\text{O}}+b_{\text{CaO}}{x}_{\text{CaO}}+b_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}}{x}_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}}\\ +b_{{\text{SiO}}_{\text{2}}\text{.CaO}}{x}_{{\text{SiO}}_{\text{2}}\text{.CaO}}+b_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}}{x}_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}}\end{array}$

$\begin{array}{c}{\hat{T}}_2\left(˚\text{C}\right)=1662.1\cdot x_{{\text{SiO}}_{\text{2}}}+1141.45\cdot x_{{\text{Na}}_{\text{2}}\text{O}}+2644.36\cdot x_{\text{CaO}}\\ +125.392\cdot x_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}}-2729.23\cdot x_{{\text{SiO}}_{\text{2}}\text{.CaO}}-3070.31\cdot x_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}}\end{array}$

The quality of the polynomial ${\hat{T}}_2\left(˚\text{C}\right)$ is given by its coefficient of determination. This coefficient of the second-order polynomial is lower than the reference (R² = 0.8805 < 0.9500).

The third-order polynomial model adapted to the study of mixtures, for m = 3 components, is given by ${\hat{T}}_{3,C}\left(˚\text{C}\right)$.

${\hat{T}}_{3,C}\left(˚\text{C}\right)={\displaystyle \underset{i=1}{\overset{3}{\sum }}{b}_i{x}_i}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{b}_{ij}{x}_i{x}_j}}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{a}_{ij}{x}_i{x}_j\left(x_i-x_j\right)}}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{\displaystyle \sum b_{ijk}{x}_i{x}_j{x}_k}}}$

$b_i,b_{ij},a_{ij},b_{ijk},T\left(˚\text{C}\right)\in ℝ$

$x_i,x_j,x_k\in \left\{\%{\text{SiO}}_2;\%{\text{Na}}_{\text{2}}\text{O};\%\text{CaO}\right\}\text{and}{x}_i,x_j,x_k\in \left[0;1\right]$

In detail

$\begin{array}{c}{\hat{T}}_{3,C}\left(°C\right)=b_{{\text{SiO}}_{\text{2}}}{x}_{{\text{SiO}}_{\text{2}}}+b_{{\text{Na}}_{\text{2}}\text{O}}{x}_{{\text{Na}}_{\text{2}}\text{O}}+b_{\text{CaO}}{x}_{\text{CaO}}+b_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}}{x}_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}}\\ +b_{{\text{SiO}}_{\text{2}}\text{.CaO}}{x}_{{\text{SiO}}_{\text{2}}\text{.CaO}}+b_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}}{x}_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}}+b_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}\text{.CaO}}{x}_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}\text{.CaO}}\\ +b_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}\left({\text{SiO}}_{\text{2}}{\text{-Na}}_{\text{2}}\text{O}\right)}{x}_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}\left({\text{SiO}}_{\text{2}}{\text{-Na}}_{\text{2}}\text{O}\right)}+b_{{\text{SiO}}_{\text{2}}\text{.CaO}\left({\text{SiO}}_{\text{2}}\text{-CaO}\right)}{x}_{{\text{SiO}}_{\text{2}}\text{.CaO}\left({\text{SiO}}_{\text{2}}\text{-CaO}\right)}\\ +b_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}\left({\text{Na}}_{\text{2}}\text{O-CaO}\right)}{x}_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}\left({\text{Na}}_{\text{2}}\text{O-CaO}\right)}\end{array}$

$\begin{array}{c}{\hat{T}}_{3,C}\left(˚\text{C}\right)=1726.15\cdot x_{{\text{SiO}}_{\text{2}}}+1132.2\cdot x_{{\text{Na}}_{\text{2}}\text{O}}+2572.46\cdot x_{\text{CaO}}\\ -136.163\cdot x_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}}-2479.67\cdot x_{{\text{SiO}}_{\text{2}}\text{.CaO}}-10511\cdot x_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}}\\ +25264\cdot x_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}\text{.CaO}}-230.175\cdot x_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}\left({\text{SiO}}_{\text{2}}{\text{-Na}}_{\text{2}}\text{O}\right)}\\ -4790.55\cdot x_{{\text{SiO}}_{\text{2}}\text{.CaO}\left({\text{SiO}}_{\text{2}}\text{-CaO}\right)}+16880.8\cdot x_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}\left({\text{Na}}_{\text{2}}\text{O-CaO}\right)}\end{array}$

The coefficient of determination of the polynomial ${\hat{T}}_{3,C}\left(˚\text{C}\right)$ is $R^2=0.9908$. This value of the coefficient is greater than the reference $R^2=0.9908>0.9500$).

The third-order synergetic polynomial model adapted to the study of mixtures, for m = 3 components, is given by ${\hat{T}}_{3,S}\left(˚\text{C}\right)$.

${\hat{T}}_{3,S}\left(˚\text{C}\right)={\displaystyle \underset{i=1}{\overset{3}{\sum }}{b}_i{x}_i}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{b}_{ij}{x}_i{x}_j}}+{\displaystyle \sum {\displaystyle \underset{i<j}{\sum }{\displaystyle \sum b_{ijk}{x}_i{x}_j{x}_k}}}$

$b_i,b_{ij},b_{ijk},T\left(˚\text{C}\right)\in ℝ$

$x_i,x_j,x_k\in \left\{\%{\text{SiO}}_2;\%{\text{Na}}_{\text{2}}\text{O};\%\text{CaO}\right\}\text{and}{x}_i,x_j,x_k\in \left[0;1\right]$

In detail

$\begin{array}{c}{\hat{T}}_{3,S}\left(˚\text{C}\right)=b_{{\text{SiO}}_{\text{2}}}{x}_{{\text{SiO}}_{\text{2}}}+b_{{\text{Na}}_{\text{2}}\text{O}}{x}_{{\text{Na}}_{\text{2}}\text{O}}+b_{\text{CaO}}{x}_{\text{CaO}}+b_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}}{x}_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}}\\ +b_{{\text{SiO}}_{\text{2}}\text{.CaO}}{x}_{{\text{SiO}}_{\text{2}}\text{.CaO}}+b_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}}{x}_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}}+b_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}\text{.CaO}}{x}_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}\text{.CaO}}\end{array}$

$\begin{array}{c}{\hat{T}}_{3,S}\left(°C\right)=1678.52\cdot x_{{\text{SiO}}_{\text{2}}}+1136.42\cdot x_{{\text{Na}}_{\text{2}}\text{O}}+2637.32\cdot x_{\text{CaO}}+193.998\cdot x_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}}\\ -2697.82\cdot x_{{\text{SiO}}_{\text{2}}\text{.CaO}}-309.294\cdot x_{{\text{Na}}_{\text{2}}\text{O}\text{.CaO}}-6826.57\cdot x_{{\text{SiO}}_{\text{2}}{\text{.Na}}_{\text{2}}\text{O}\text{.CaO}}\end{array}$

The quality of the polynomial ${\hat{T}}_{3,S}\left(˚\text{C}\right)$ is given by its coefficient of determination. This coefficient of the second-order polynomial is lower than the reference (R² = 0.8820 < 0.9500).

From the above, the best model is the complete third-order model. Using the mathematical expression of the complete third-order model, the melting temperature $T^{\prime }\left(˚\text{C}\right)$ of each mixture is calculated. Thus, a 100% silica (SiO₂) mixture has its liquidus at $T^{\prime }\left(˚\text{C}\right)=1708.78$ according to the model versus $T\left(˚\text{C}\right)=1723$ according to Grynberg . Similarly, a mixture with 100% Sodium oxide (Na₂O) has its liquidus at $T^{\prime }\left(˚\text{C}\right)=1131.57$ according to the model versus $T\left(˚\text{C}\right)=1132$ according to Zhan Zhang [4] [5] . Also, a 100% Calcium oxide (CaO) mixture has its liquidus at $T^{\prime }\left(˚\text{C}\right)=2570.36$ according to the model versus $T\left(˚\text{C}\right)=2570$ according to Clifton . A statistical study is then carried out to assess the accuracy of the model.

The model was used to calculate liquidus values from the starting mixtures. Using the calculated values and those taken from the literature, we carry out a statistical study. below summarizes the data used to study the accuracy of the full third-order model.

The correlation between the predicted (estimated) melting temperature and the experimental melting temperature of each mixture is shown in . The coefficient of determination of the polynomial ${\hat{T}}_{3,C}\left(˚\text{C}\right)$ is $R^2=0.9908$. This value of the coefficient is greater than the reference ( $R^2=0.9908>0.9500$) and represents the highest result obtained. The formula provided in Section 2.4 and the appendix outlines the calculation procedure

The third-order mathematical model established above was calibrated. An experimental phase dedicated to verifying the model’s predictions. Test mixes are melted in the mechanics and materials science laboratory using two Nabertherm furnaces. The compositions of the test mixes are generated using the model and their melting temperature. below summarizes the test mixes.

Table 3. Compositions of test mixtures.

Using the complete third-order model, an illustration of the ternary diagram of the SiO₂-Na₂O-CaO system is proposed.

and are illustrations of the ternary diagram of the SiO₂-Na₂O-CaO system obtained using the full third-order model.

Continuous lines are shown in this diagram. They represent liquidus of mixtures whose compositions coincide with these. Red indicates very high temperatures of around 2570˚C.

However, blue indicates a temperature of 1132˚C. So, the color gradient observed on the diagram is similar to the temperature of the liquidus of the mixtures.

The test samples resulted in the formation of a glass paste. Due to the high content of sodium carbonate (flux) in the composition of the samples, there was good melting of the mixtures .

The casting of each glass pastes enabled samples of soda-lime glass to be obtained easily. above illustrates the casting of the glass pastes obtained.

Once the cast glass melt has cooled, the samples obtained are described. The glass samples of the test mixtures are micrographed to provide a better view of the melting state. below shows the degree of melting of the blends.

Overall, the predicted liquidus resulted in a significant degree of melting, making glass melt casting feasible. However, all produced glass samples contained trapped un-melted particles. Despite this, the findings justify maintaining the complete third-order model, based on its accuracy (R² = 0.9908), even though it remains insufficient. Further studies are being carried out to develop a more predictive model. Series of tests need to be carried out for an adjustment of the coefficients $b_i,b_{ij},{\delta }_{ij},b_{ijk}\in ℝ$ in order to improve the accuracy of said model. The presence of unmelted particles can be linked to several factors. Firstly, some quartz grains in the mixture may exceed 200 micrometers in size, as indicated by Emmanuelle Gouillart . In addition, the temperature may be insufficient to achieve complete melting of each batch of samples, according to J. Barton’s work . From another perspective, unmelted particles may persist due to the mixture’s composition, particularly the proportion of flux. These hypotheses help explain the presence of unmelted particles.