The method for the mathematical description of phase composition diagrams has been developed. It is based on the balance of distribution of the initial components of the slag or metal over the secondary compounds (phases) formed from them. The equations relating the phase composition to the chemical one are obtained, and a computer program is created on their basis. After setting the chemical composition of the slag from the console and starting it, the computer outputs the phase composition in mass percent, the number of the polytope, its volume, initial and final equations. The method has no restrictions on the number of components considered in the system under study and allows describing five or more component systems that cannot be correctly displayed in three-dimensional space on a plane. Examples of using the model are given. Calculations using the obtained model established that deterioration in the production of carbonaceous ferrochromium from Kazakh ores is due to the movement of slags within the SiO 2 -Al 2 O 3 -MgO triple diagram from the region of magnesia spinel (MgOAl 2 O 3 ) to the region of predominant forsterite crystallization (2MgOSiO 2 ) during transition to use of magnesia ores. A predictive estimation of the phase composition of steel-smelting slags with high basicity is given. It was found out that, on average, they contain high-temperature compounds such as periclase (MgO), dicalcium silicate (Ca 2 SiO 4 ), calcium monoaluminate (CaAl 2 O 4 ), and magnesia spinel (MgAl 2 O 4 ) as phase components. All of them are high-temperature compounds, which determine refractoriness of the slag. Presence of Ca 2 SiO 4 in the slag in a large amount (74 %), capable of polymorphic transformation with a volume change by 12 %, can cause decomposition of slags with formation of dispersed dust, which is hazardous to the environment. The calculation results are confirmed by X-ray phase analysis of the slag. The proposed mathematical model also allows solving inverse problems, i.e. to find the composition of the charge to obtain a product with the desired phase composition. It is recommended to load charge containing 68.48 % CaO, 25.10 % SiO 2 and 6.43 % Al 2 O 3 into the kiln to obtain high quality white Portland cement clinker with the following composition: 45 % of 3CaOSiO 2 , 38 % of 2CaOSiO 2 and 17 % of 3CaOAl 2 O 3 .

The diagram of the phase composition of the Fe -Si -Al system and its isothermal sections within the temperature range of 300-3000 K are plotted. Thermodynamic data of