## 3 Results  
From the solidification curves and applying higher order derivatives, the rigidity point temperatures (RPT) were determined with the different studied methods and the comparison with the two methods presented in the literature while considering the gaps of both methods. In the following equations, from number 1 to 5, the linear regression equations obtained using the different methods studied for the calculation of the RPT in function of the alloy composition are shown. **Method Equation (1)**  
\[ \begin{split}&\text{RPT(${}^{\circ}$C)}=564.48+0.65.\text{Si}-8.48.\text{Mg}-2.37.\text{Fe}+6.09.\text{Cu}-0.26.\text{Ni}+3.56.\text{Cr}-4.46. \text{Mn}\\ &-9.01.\text{Ti}-4.89.\text{Zn}+1.05.\text{Pb}-7.79.\text{Sn}-0.00 2.\text{Sr}\end{split}  \tag{1}\]  
**Method Equation (2)**  
\[ \begin{split}&\text{RPT(${}^{\circ}$C)}=564.99+0.59.\text{Si}-8.28.\text{Mg}-2.35.\text{Fe}+5.90.\text{Cu}-0.32.\text{Ni}+3.42.\text{Cr}-4.47. \text{Mn}\\ &-8.84.\text{Ti}-5.36.\text{Zn}+1.37.\text{Pb}-7.84.\text{Sn}+1. 97.\text{Sr};\text{r}^{2}=0.77\end{split}  \tag{2}\]  
**Method Equation (3)**  
\[ \begin{split}&\text{RPT(${}^{\circ}$C)}=567.74+0.58.\text{Si}-8.24.\text{Mg}-2.89.\text{Fe}+7.41.\text{Cu}-2.04.\text{Ni}+3.90.\text{Cr}-4.25. \text{Mn}\\ &-7.70.\text{Ti}-6.24.\text{Zn}+0.37.\text{Pb}+5.13.\text{Sn}-5.20.\text{Sr}\end{split}  \tag{3}\]  
**Method Equation (4)**  
\[ \begin{split}&\text{RPT(${}^{\circ}$C)}=560.92+0.76.\text{Si}-10.6 2.\text{Mg}-1.42.\text{Fe}+3.06.\text{Cu}-1.95.\text{Ni}+0.69.\text{Cr}-2.01. \text{Mn}\\ &-3.74.\text{Ti}-0.16.\text{Zn}-2.09.\text{Pb}-15.03.\text{Sn}+1.42.\text{Sr}.\end{split}  \tag{4}\]  
**Method Equation (5)**  
\[ \begin{split}&\text{RPT(${}^{\circ}$C)}=565.96+0.55.\text{Si}-7.97.\text{Mg}-2.76.\text{Fe}+6.50.\text{Cu}-0.01.\text{Ni}+3.83.\text{Cr}-4.58. \text{Mn}\\ &-8.20.\text{Ti}-5.75.\text{Zn}+1.16.\text{Pb}+0.29.\text{Sn}-1.03.\text{Sr}\end{split}  \tag{5}\]  
Equations (1)-(5) show that Mg, Cu, Cr, Mn, Ti, Zn, and Sn have in general a decreasing effect on RPT values. Si, Cr and Cu are the only elements that promote an increase of RPT values. Alloying elements Mg, Fe, Ni, Mn, Ti, and Zn decreases RPT values, and the rest of the alloying elements have positive or negative values in the function of the employed method. Pb, Sn, Sr do not show a specific influence as in some methods their influence on RPT is positive and in other methods negative. The rigidity temperature values obtained employing the different studied methods are displayed in the following Figure 7 in order to compare the differences and tendencies of the obtained values. Methods 1, 2, 3, and 5 have very similar temperature values, while Method 4 shows lower RPT values. Furthermore, it can be appreciated that the influence of the alloy elements is clear on the values obtained with similar differences regardless of the method selected. Small changes in the composition of the alloy promote important changes in the RPT values. There is up to approximately 10 \({}^{\circ}\)C of difference in the RPT values with the same method depending on the real alloy composition. Table 2 shows the accuracy of the RPT values obtained for each method, with the linear regression coefficient (r\({}^{2}\)) and the standard error (S\({}_{\text{ey}}\)). Method 1 and Method 2 showed similar coefficient of determination values compared to Method 3 and Method 4, with values between 0.77 and 0.78. Method 5 using the plotting of the derivatives vs. temperature parameter is the one with more precision, with 0.82. The use of higher order derivatives which made easier the determination of the RPT and the plotting of the dT/dt curve vs. T, that is independent of the sand cup size provides better results in Method 5. A value of the coefficient of determination between 0.5 and 0.8 is classified as a regular correlation and a value from 0.8 to 0.9 isclassified as a good correlation. We can observe that Method 5 is the only one characterized as a good correlation, with the rest of the methods very close in obtaining this classification.