Isothermal sections of the phase composition diagram of the Fe – Si – Cr – B system in the temperature range of 300 – 3 000 K with a step of 200 K are constructed. The diagram is shown at a temperature of 1 900 K, which is typical for silicochromium at the outlet from the furnace. A mathematical model of the diagram has been created in the form of a system of linear equations connecting the chemical composition of the metal with its phase. A computer program has been developed for the numerical calculation of the number of formed phases. Silicochromium with 0,3 – 0,5 % boron is located in the crystallization field CrSi2 – Si – SiB4 – FeSi

Akberdin, A. A.

Kim, A. S.

Lu, N. Yu.

Orlov, A. S.

Sultangaziyev, R. B.

English

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305METALURGIJA 61 (2022) 2, 305-308A. S. KIM, A. A. AKBERDIN, N. YU. LU, R. B. SULTANGAZIYEV, A. S. ORLOVDIAGRAM OF THE EQUILIBRIUM PHASE COMPOSITION OF THE Fe – Cr – Si – B SYSTEMReceived – Primljeno: 2021-08-19Accepted – Prihvaćeno: 2021-11-15Original Scientific Paper – Izvorni znanstveni radA. S. Kim, A. A. Akberdin, N. Yu. Lu., A. S. Orlov Chemical-Metallurgi-cal Institute, R. B. Sultangaziyev  Karaganda Technical University, Kara-ganda, Kazakhstan, e-mail: sulrus83@mail.ruIsothermal sections of the phase composition diagram of the Fe – Si – Cr – B system in the temperature range of 300 – 3 000 K with a step of 200 K are constructed. The diagram is shown at a temperature of 1 900 K, which is typical for silicochromium at the outlet from the furnace. A mathematical model of the diagram has been created in the form of a system of linear equations connecting the chemical composition of the metal with its phase. A computer pro-gram has been developed for the numerical calculation of the number of formed phases. Silicochromium with 0,3 – 0,5 % boron is located in the crystallization field CrSi2 – Si – SiB4 – FeSi.Keywords: Fe‐Si‐Cr‐B system, phase, diagram, equations, mathematical modelINTRODUCTIONChromium ores and products made from them are in high demand on the world market. This is due to the pos-sibility of obtaining with their use a wide range of high-quality steels, cast irons, refractories, dyes, bleaches, protective coatings. The undisputed leadership in terms of the consumption of chrome ores belongs to metallur-gy. Globally, about 80 % of mined chrome ore is used in metallurgy, 10 % for the production of refractories and the remaining 10 % in the chemical industry. Such a high proportion of the use of chromium in metallurgy is ex-plained by the excellent performance characteristics of chromium-containing steels and alloys. They are ob-tained using chromium ferroalloys, among which silico-chromium and refined ferrochrome deserve special at-tention, used for smelting stainless, heat-resistant, corro-sion-resistant and other critical steel grades. The two-stage method of silicochromium production used in factories has a number of unsolved problems. Often, there is a disturbance in the operation of the furnaces due to the formation of a viscous and refractory slag, which impedes the free exit of metal from the tap hole and leads to the formation of crust on the bottom. Another disadvantage of the two-stage method is the formation of a large number of sublimates under the roof and clog-ging of gas ducts with them, which makes it necessary to periodically stop the furnaces to remove the deposits. In the production of re-fined ferrochrome grades, silico-chromium is used as a reducing agent. This process also has a number of unresolved problems, where, along with a high consumption of charge materials and energy, the resulting slags decompose into a finely dispersed pow-ISSN 0543-5846METABK 61(2) 305-308 (2022)UDC / UDK 546.72.76.27/168.4:536.425.717:519.9:51/608.4=111der that poses a threat to the environment. This work is devoted to the construction and mathematical descrip-tion of the phase composition diagram of the Fe-Si-Cr-B system. It can be used not only to improve the technol-ogy of smelting chromium ferroalloys. The basis for the production of various grades of ferrosilicon is the Fe-Si system [1], aluminum-thermal ferroboron – Fe-B [2], and silicothermal ferroboron-Fe-Si-B [3]. It is also used to create amorphous alloys [4]. WAYS OF RESEARCHPhase analysis can be performed using graphically depicted diagrams. If for three-component systems this is done relatively easily, then for multicomponent systems, for example, dimensions four, five or more, this is a dif-ficult and in many cases unsolvable problem. Its solution can be facilitated by creating mathematical models of diagrams, since they allow you to operate not only in three-dimensional space, where diagrams are usually dis-played graphically, but also in multidimensional one.The studied four-component system Fe-Si-Cr-B in the graphic image is a tetrahedron, including one, two and three-component partial systems. Their list is given in Table 1.The construction of a diagram of the phase composi-tion of a four-component system began with particular three-component systems (Table 1), identifying binary Table 1 List and number of partial systemsNumberSystemscomponents systems1 4 Fe; Cr; Si; B2 6 Fe – Cr; Fe – B; Fe – Si; B – Cr; B – Si; Cr – Si3 4 Fe – B – Cr; Fe – B – Si; Fe – Cr – Si; B – Cr – Si4 1 Fe – Si – Cr – B306  METALURGIJA 61 (2022) 2, 305-308A. S. KIM et al.: DIAGRAM OF THE EQUILIBRIUM PHASE COMPOSITION OF THE Fe – Cr – Si – B SYSTEMFigure 1  Diagrams of the phase composition of the system (a) Fe – B – Cr, (b) Fe – B – Si, (c) Fe – Cr – Si, (d) B – Cr – Si at 1 900 Kand triple phases in them by triangulation using the “Triangle” program of the “Terra” complex [5, 6]. The database of the complex was supplemented with infor-mation from reference publications [7]. Isothermal sec-tions of all three-component systems were built in the temperature range of 300 - 3 000 K with a step of 200 K. Below are the data for the characteristic temperature at the outlet of silicon ferroalloys 1 900K (Figure 1).Table 2 lists all the elementary triangles found after triangulation.The presence of phase composition diagrams of par-ticular three-component systems made it possible to break down the four-component system Fe-Si-Cr-B into elemen-tary tetrahedrons. We applied the method of “closing” tri-angles on a tetrahedron by comparing their compositions [8 - 10]. This made it possible to establish that the Fe – Cr – Si – B system has 14 tetrahedra of coexisting phases:1. Cr – Cr3Si – CrB – Fe; 2. Fe – FeB – CrB – Fe5SiB2; 3. CrB – Cr3Si – Fe5SiB2 – Fe;4. Fe – FeSi – Fe5SiB2 – Cr3Si;5. FeSi – Fe5SiB2 – CrB – Cr3Si;6. FeSi – FeB – Fe5SiB2 – CrB; 7. CrB – CrB2 – FeB – FeSi; 8. FeB – FeSi – CrB2 – SiB6; 9. FeB – B – SiB6 – CrB2; 10. Cr3Si – CrSi2 – CrB – FeSi; 11. CrB – CrB2 – FeSi – CrSi2; 12. CrB2 – CrSi2 – SiB6 – FeSi; 13. SiB6 – SiB4 – CrSi2 – FeSi; 14. CrSi2 – Si – SiB4 – FeSi.A graphic representation of the diagram of the phase composition of the Fe-Cr-Si-B system in elementary tetrahedra of coexisting phases is given in Figure 2.The diagram can be used to find the phase composi-tion of metals by geometric constructions, but it is dif-ficult to apply this method in a tetrahedron. Therefore, Figure 2  Diagram of the phase composition of the Fe – Cr – Si – B systemTable 2  Elementary triangles of the Fe – Si – Cr – B system at 1 900 KTriple systemsFe – B – Cr Fe – B – Si1. Fe – CrB – Cr2. Fe – CrB – FeB3. FeB – CrB – CrB24. FeB – B – CrB25. Fe – FeSi – Fe5SiB26. FeSi – FeB – Fe5SiB27. FeB – SiB6 – B8. SiB6 – FeB – FeSi9. SiB6 – S iB4  – FeSi10. SiB4 – FeSi – SiFe – Cr – Si B – Cr – Si11. Fe – Cr – Cr3Si12. Fe – Cr3Si – FeSi13. FeSi – Cr3S i – CrSi214. FeSi – CrSi2 – Si15. Cr – CrB – Cr3Si 16. CrB – Cr3Si – CrSi2  17. CrB – CrB2 – CrSi218. CrB2 – B – SiB6 19. CrB2 – SiB6  – CrSi220. SiB6  –  SiB4  –  CrSi221. CrSi2  – SiB4  – Sia)b)c)d)307A. S. KIM et al.: DIAGRAM OF THE EQUILIBRIUM PHASE COMPOSITION OF THE Fe – Cr – Si – B SYSTEMMETALURGIJA 61 (2022) 2, 305-308for each of the 14 elementary tetrahedra, we created mathematical models using the balance method devel-oped by us [11]. In this method, the balance equations for the distribution of the initial alloy elements over the formed phases are first compiled, the subsequent solu-tion of which allows finding expressions for the quanti-tative calculation of the phases. RESULTS DISCUSSION Below, for example, are the initial balance and final design equations for tetrahedron No. 1. In them, Fe0, Cr0, Si0 and B0 denote the content of iron, chromium, silicon and boron in the studied metal according to chemical analysis.Balance initial equations:Сr0 =1∙Cr – 0,85∙Cr3Si – 0,83∙ CrBSi0 = 0,15∙Cr3SiB0 = 0,17∙ CrBFe0 = 1∙FeCalculated final equations:Cr = 1∙ Cr0 – 5,666∙Si0– 4,882∙B0 Cr3Si = 6,666∙Si0CrB = 5,882∙ B0Fe = 1∙ Fe0The coefficients obtained in this way for Fe0, Cr0, Si0 and B0 in the calculated equations for all tetrahedra were entered in Table 3. Using the obtained equations, a computer program was created, which, when setting the chemical composition of the metal (Fe0, Cr0, Si0 and B0) Table 3 Coefficients of equations for calculating the phase composition Ref.Coeff .Coeffi  cient values in tetrahedra1 2 3 4 5 6 7CrCr3SiCrBFeFeFeBCrBFe5SiB2CrBCr3SiFe5SiB2FeFeFeSiFe5SiB2Cr3SiFeSiFe5SiB2CrBCr3SiFeSiFeBFe5SiB2CrBCrBCrB2FeBFeSiV 0,025 0,01 0,02 0,2 0,028 0,015 0,037Cr0a1 1 1,075 0,517 1,176 -0,763 -0,773 2,307a2 0 -1,28 0,671 0 0,593 -2,717 -1,307a3 0 -1,257 -0,519 1,414 3,286 0a4 0 1,205 1,069 0,343 -0,244 1,204 0S0b1 -5,666 -6,031 -2,934 0 4,328 4,337 -1,99b2 6,666 -5,469 2,865 0 -3,36 2,595 2,36b3 0 12,5 7,127 2,941 -1,351 -5,932 -2,31b4 0 0 -6,058 -1,941 1,383 0 2,94F0c1 0 1 0 0 -0,714 -0,719 1,026c2 0 0 0 0 1,731 -1,337 -1,216c3 0 0 0 0 0,695 3,056 1,19c4 1 0 1 1 -0,712 0 0B0d1 -4,882 -5,25 3,353 0 3,728 3,775 -5,384d2 0 6,25 -3,274 14,285 -2,895 13,269 6,384d3 5,882 0 6,14 -3,361 -6,908 -16,044 0d4 0 0 -5,219 -9,924 7,075 0 08 9 10 11 12 13 14Ref.Coeff .FeFeSiCrB2SiB6FeBBSiB6CrB2Cr3SiCrSi2CrBFeSiCrBCrB2FeSiCrSi2CrB2CrSi2SiB6FeSiSiB6SiB4CrSi2FeSiCrSi2SiSiB4FeSiV 0,167 0,176 0,041 0,044 0,196 0,028 0,193Cr0a1 -0,366 0 1,405 2,307 1,221 7,342 2,083a2 0,466 -0,428 -0,405 -1,307 0,302 -8,425 -1,083a3 1,428 0 0 0 -0,523 2,083 0a4 -0,528 1,428 0 0 0 0 0S0b1 -1,993 0 -1,297 -2,13 -1,127 -6,777 0b2 2,538 -2,333 2,297 1,207 1,644 7,777 1b3 0,455 3,333 0 1,923 0,483 0 0b4 0 0 0 0 0 0 0F0c1 1,027 1,19 0,668 1,096 0,58 3,491 0c2 0,207 -0,19 -1,183 -0,621 -0,847 -4,006 -0,515c3 0 0 0 -0,99 -0,248 0 0c4 -0,234 0 1,515 1,515 1,515 1,515 1,515B0d1 0,855 0 -6,861 -5,384 0,483 4,333 0d2 -1,088 1 1,979 6,384 -0,704 -3,333 -0,639d3 0 0 5,882 0 1,221 0 1,639d4 1,233 0 0 0 0 0 0308  METALURGIJA 61 (2022) 2, 305-308A. S. KIM et al.: DIAGRAM OF THE EQUILIBRIUM PHASE COMPOSITION OF THE Fe – Cr – Si – B SYSTEMdisplays its phase composition in mass percent on the monitor. An example of using the program is shown in Table 4, which presents the phase compositions of con-ventional and boron-containing ferroalloys calculated using the obtained mathematical models.All alloys are located in tetrahedron No. 14 CrSi2 – Si – SiB4 – FeSi. The main phase in silicochromium of the FeSiCr48 brand is CrSi2 and FeSi, boron is a part of SiB4. Found the volumes (V) of each elementary tetra-hedron (table 3). In the phase space of the Fe – Si – Cr – B diagram, the largest volume is occupied by the ele-mentary tetrahedron №14 CrSi2 – Si – SiB4 – FeSi, and the smallest  –  №2 Fe – FeB – CrB – Fe5SiB2. The op-eration of the melting furnace in the phase regions of a small volume cannot be recommended due to the pos-sible transition of the metal composition into undesira-ble fields with inevitable fluctuations in the composi-tion of the charged charge under production conditions.Determined the probability of the existence of phas-es W [9] (Table 5).It can be seen that the phase FeSi, is the most wide-spread, and Cr is the least widespread.For practical purposes, it is important that the ob-tained mathematical models make it possible to solve both direct and inverse problems of phase formation. Calculation equations are used to determine the phase composition of a metal from a known chemical compo-sition (direct problem), and the above-derived balance equations are used to determine the required chemical composition of the metal (or charge into the melt) to obtain a metal of a given phase composition. CONCLUSIONThus, a diagram of the phase composition of the 4-component metal system Fe-Si-Cr - B has been con-structed. It contains 13 phases: Fe, Si, Cr, B, Cr3Si, CrSi2, CrB, CrB2, FeB, FeSi, Fe5SiB2, SiB6 and SiB4. The sizes of the fields formed and the prevalence of in-dividual phases in them have been determined. The most common phases in the system under study are FeSi, CrSi2, CrB, CrB2, SiB4.A mathematical model of the phase diagram has been created. It allows you to determine the phase com-position of the material by the chemical composition. Using the model, it is also possible to determine the type and number of components of the initial charge to obtain a product of the required phase composition.AcknowledgementThis research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09259038).REFERENCES[1] Zubov V. L., Gasik M. I. Electrometallurgy ferrosilicon. Dnepropetrovsk: System technologies (2002) 704[2] Lyakishev I. P. , Pliner Yu. L., Lap-by S. I. Boron steels and alloys. - M.: Metallurgy. 1986. -192 p.[3] Zhuchkov V. I., Zayakin O. V., Leontiev L. I., Sychev A. V., Kel I. N. Physicochemical characteristics, production and application of complex boron-containing ferroalloys. Izvestiya Ferrous Metallurgy,60(2017) 5, 348-354 [4] Dong W., Wu X., Yan M. Enhancement of Thermal Stabil-ity and Bending Ductility of Fe-Si-B Amorphous Ribbons with Minor Y Addition / Metals and materials internation-al, 26(2020) 4, 20-23.[5] Trusov B. G. TERRA software system for modeling phase and chemical equilibria at high temperatures. - In the book: III Int. symposium «Combustion and Plasma Chemistry». 24 - 26 August 2005. Almaty, Kazakhstan, (2005), 52-57.[6] Udalov Yu. P. Application of software complexes of computa-tional and geometric thermodynamics in the design of techno-logical processes of inorganic substances. SPb. (2012) 147[7] State diagrams of binary metal systems: Handbook: In 3 volumes: Vol. 1 / Pod obshch. ed. N. P. Lyakisheva, M.: Mashinostroenie (1996) 992[8] Berezhnoy A. S. Multicomponent oxide systems. - Kiev: Dumka, 1970, 572[9] Gabdullin T. G., Takenov T. D., Baysanov S. O., Buketov E. A. Physicochemical properties of manganese slags. - Alma-Ata: Science. Kaz.det., 1984.[10] Kelamanov, B., Samuratov, Y., Zhumagaliyev, Y., Akuov, A., Sariev, O. Titanium and chrome oxides system thermo-dynamic diagram analysis Metalurgija, 59(2020)1, 101-104[11] Akberdin. A. A. Balance method for calculating the phase composition of multicomponent systems // Complex use of mineral raw materials (1995) 3, 92-93Note:  The responsible translator for English language is Nataliya Drag, Karaganda, KazakhstanTable 4 Compositions of ferroalloysNo AlloyAlloy composition / wt. %Chemical PhaseFe Сr Si B CrSi2 Si SiB4 FeSi1 FeSiCr48 20 33 47 0 68,7 1 0 30,32 FeSiCr 48B0,3 19,7 33 47 0,3 68,7 1,1 0,5 29,73 FeSiCr 48B0,5 19,7 33 47 0,5 68,7 1 0,9 29,44 FeSiCr 48B0,7 19,7 33 47 0,7 68,7 0,9 1,2 29,2Table 5 Probability of the existence of phasesNo. Phase W. % No. Phase W. %1 Fe 0,0605 8 CrSi2 0,12552 Si 0,04825 9 CrB 0,0553 Cr 0,00625 10 CrB2 0,1554 B 0,044 11 SiB6 0,141755 FeSi 0,19225 12 SiB4 0,055256 FeB 0,0595 13 Fe5SiB2 0,023257 Cr3Si 0,0335

Diagram of the equilibrium phase composition of the Fe – Cr – Si – B system

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