A Statistical Approach for Estimation of Fusion Behavior of Alumino-Thermic Ferrochrome Slags

Abstract

Effects of simultaneous variations of Al₂O₃/CaO ratio, and MgO and CaF₂ contents, on the fusion behavior of synthetic alumino-thermic ferrochrome slags are systematically measured using a hot-stage-microscope in accordance with German standard 51730. Empirical equations are developed for the prediction of the softening temperature (ST), liquidus/hemispherical temperature (HT), and flow temperature (FT), in terms of aforementioned variables using Factorial Design Technique. To verify the predicted equations, further groups of synthetic slags are prepared in the laboratory using pure oxides, and the characteristic temperatures of these slags are measured in the same manner using hot-stage microscope. It is observed that within the range of variables examined, equations developed render the best fit for HT, the next best fit for FT, and somewhat good fit for ST.

Appendix

Calculation of Variance for ST

$$ \begin{gathered}{{\text{Response}},\; \rho = f\left( {R_{a} , M_{b} , C_{c} } \right)\;{\text{where}}\; a = b = c = 1, 2} \hfill \\ {{\text{and}}\;\rho = \rho_{R1} \;{\text{for}}\;a = 1,\;b = c = 1, 2} \hfill \\ {\rho = \rho_{R2} \;{\text{for}}\;a = 2, b = c = 1, 2} \hfill \\ {\rho = \rho_{M1} \;{\text{for}}\;b = 1, a = c = 1, 2} \hfill \\ {\rho = \rho_{M2} \;{\text{for}}\;b = 2, a = c = 1, 2} \hfill \\ {\rho = \rho_{C1} \;{\text{for}}\;c = 1, a = b = 1, 2} \hfill \\ \rho = \rho_{C2} \;{\text{for}}\;c = 2, a = b = 1, 2 \hfill \\ {{\text{Similarly}},\;\rho = \rho_{R1M1} \;{\text{for}}\;a = 1, b = 1, c = 1, 2} \hfill \\ {\rho = \rho_{R1M2} \;{\text{for}}\;a = 1, b = 2, c = 1, 2 } \hfill \\ {\rho = \rho_{R2M1} \;{\text{for}}\;a = 2, b = 1, c = 1, 2 } \hfill \\ {\rho = \rho_{R2M2} \;{\text{for}}\;a = 2, b = 2, c = 1, 2 } \hfill \\ {\rho = \rho_{M1C1} \;{\text{for}}\;b = 1, c = 1, a = 1, 2 } \hfill \\ {\rho = \rho_{M1C2} \;{\text{for}}\;b = 1, c = 2, a = 1, 2 } \hfill \\ {\rho = \rho_{M2C1} \;{\text{for}}\;b = 2, c = 1, a = 1, 2 } \hfill \\ {\rho = \rho_{M2C2} \;{\text{for}}\;b = 2, c = 2, a = 1, 2 } \hfill \\ {\rho = \rho_{C1R1} \;{\text{for}}\;a = 1, c = 1, b = 1, 2 } \hfill \\ {\rho = \rho_{C2R1} \;{\text{for}}\;a = 1, c = 2, b = 1, 2 } \hfill \\ {\rho = \rho_{C1R2} \;{\text{for}}\;a = 2, c = 1, b = 1, 2 } \hfill \\ {{\text{and}}\; \rho = \rho_{C2R2} \;{\text{for}}\;a = 2, c = 2, b = 1, 2 } \end{gathered} $$

$$ {\text{Correction}}\;{\text{factor}}\; ( {\text{CF) }} = \frac{{(\sum \rho )^{2} }}{8} = 24321825.13\;({\text{see}}\;{\text{Table}}\;{\text{V}}) $$

$$ {\text{Total variance}} = (\Upsigma \rho^{2} ) - CF = 2501.87 $$

$$ {\text{Variance due to}}\;R = \frac{{(\Upsigma \rho_{R1} )^{2} + (\Upsigma \rho_{R2} )^{2} }}{4} - CF = 36.12 $$

$$ {\text{Variance due to }}M = \frac{{(\Upsigma \rho_{M1} )^{2} + (\Upsigma \rho_{M2} )^{2} }}{4} - CF = 1035.12 $$

$$ {\text{Variance due to}}\;C = \frac{{(\Upsigma \rho_{C1} )^{2} + (\Upsigma \rho_{C2} )^{2} }}{4} - CF = 903.13 $$

$$\text{Interaction variance due to} \; R \; \text{and} \; M = \frac{{\left( {\Upsigma \rho_{R1M1} } \right)^{2} + \left( {\Upsigma \rho_{R1M2} } \right)^{2} + \left( {\Upsigma \rho_{R2M1} } \right)^{2} + \left( {\Upsigma \rho_{R2M2} } \right)^{2} }}{2} - \text{Variance due to} \; R - \text{Variance due to}\; M - CF = 325.12$$

$$\text{Interaction variance due to} \;M \;\text{and} \; C = \frac{{\left( {\Upsigma \rho_{M1C1} } \right)^{2} + \left( {\Upsigma \rho_{M1C2} } \right)^{2} + \left( {\Upsigma \rho_{M2C1} } \right)^{2} + \left( {\Upsigma \rho_{M2C2} } \right)^{2} }}{2} - \text{Variance due to} \; M - \text{Variance due to} \; C - CF = 28.13$$

$$ \text{Interaction variance due to} \; C \; \text{and} \;R = \frac{{\left( {\Upsigma \rho_{C1R1} } \right)^{2} + \left( {\Upsigma \rho_{C1R2} } \right)^{2} + \left( {\Upsigma \rho_{C2R1} } \right)^{2} + \left( {\Upsigma \rho_{C2R2} } \right)^{2} }}{2} - \text{Variance due to} \; C - \text{Variance due to} \; R - CF = 171.13$$

$$\text{Residual variance} = \text{Total variance} - \text{Sum of above six variances} = 3.12$$

$$ \text{Variance ratio or Fischer ratio} \left( F \right) = \frac{\text{Variance}}{\text{Residual variance}} $$

In order for a factor to be significant, F should be greater than the value given by Snedecor’s table with respect to the degree of freedom at 90, 95, or 99 pct of confidence.

To calculate relative significance of the three factors R, M, and C, ST [ai] is calculated by averaging the response with signs of R, M, and C.

$$ {\text{So}},\;ST\left[ {ai} \right]\;{\text{for}}\;R = + 2. 1 2 5 $$

$$ ST\left[ {ai} \right]\;{\text{for}}\;M = - 1 1. 3 7 5 $$

$$ ST\left[ {ai} \right]\;{\text{for}}\;C = - 10. 6 2 5 $$

$$ {\text{Softening Temperature}}\;(\rho_{ST} )= a_{0} + a_{1} R + a_{2} M + a_{3} C, $$

where a ₀ is the average response values of ST, and (a ₁, a ₂, a ₃) are the ST [ai] values of R, M, and C, respectively.

Hence, ${\text{Softening Temperature}}\;(\rho_{ST} ) { } = \, 1743.625 + 2.125R - 11.375M - 10.625C$