Exploring Alloy Composition Dynamics: Thermodynamic Analysis of Fe-Al-Si-Cr System in Homogeneous Liquid State

Abstract

This study employs thermodynamic-diagram analysis to investigate component ratios within the Fe-Al-Si-Cr system, focusing on the behavior of homogeneous liquid states. Through comprehensive modeling, a phase diagram is constructed, elucidating the interplay of iron, aluminum, silicon, and chromium components. This study identifies stable elementary tetrahedra within the system, providing insights into phase compositions and distribution. Key findings reveal the significance of tetrahedral geometry in understanding and optimizing alloy compositions, particularly in the context of complex chromium alloys. This research underscores the utility of thermodynamic analysis in advancing our understanding of alloy systems and facilitating the optimization of production processes.

1. Introduction

An analysis of the global raw material base for metallurgy indicates a decline in the availability of high-quality ores, coupled with increasing demands for superior metal quality [1]. This necessitates the development of materials with new, complex properties. To enhance metal quality and introduce advanced technologies, comprehensive physicochemical studies are essential. These studies underscore the growing importance of physical chemistry as a foundation for producing metals with specific, desired properties [2].

The motivation for studying the Fe-Al-Si-Cr system lies in its significant potential for developing advanced high-temperature materials with improved mechanical properties, corrosion resistance, and oxidation resistance, which are crucial for various industrial applications such as aerospace, automotive, and energy sectors [3,4].

Conducting thermodynamic studies on multicomponent systems is inherently complex, often requiring extensive mathematical calculations [5,6]. These studies are directly related to defining the thermodynamic functions of numerous independent reactions. However, the thermodynamic-diagram analysis of complex systems, developed at the Zh. Abishev Chemical-Metallurgical Institute, offers a simpler and more accurate alternative compared to classical thermodynamic studies of metallurgical processes [7].

In line with our objectives, we have performed a thermodynamic assessment of the feasibility of melting a complex aluminum–silicon–chromium alloy with a given composition. We have plotted the phase diagram of the Fe-Al-Si-Cr metal system in the homogeneous liquid state and identified the elementary tetrahedra.

This phase diagram can predict the phase composition of aluminum–silicon–chromium alloys at varying element concentrations. It effectively identifies the optimal composition ranges for forming high-quality aluminum–silicon–chromium alloys. This predictive capability is a significant advantage of the thermodynamic-diagram analysis of multicomponent systems.

Plotting phase diagrams for the Fe-Cr-Si-Al metal system across the full temperature range of phase transitions requires accurate thermodynamic characteristics of the involved compounds [8,9,10]. These characteristics are crucial for the comprehensive understanding of the four-component system. In addition to the standard values of enthalpy and entropy of formation, it is essential to determine the temperature dependences of heat capacities, enthalpies, and entropies of these compounds to calculate their thermodynamic properties over a wide temperature range.

2. Materials and Methods

2.1. Thermodynamic-Diagram Analysis

A thermodynamic-diagram analysis was employed to study the phase compositions of the complex aluminum–silicon–chromium alloys. The compositions of individual grades were recalculated into the four primary elements of the Cr-Fe-Al-Si system. The phase compositions of the alloys were determined by calculating the normative number of secondary phases within the corresponding tetrahedra.

2.2. Phase Diagram Plotting

The phase diagram of the Fe-Al-Si-Cr system in the homogeneous liquid state was plotted based on the recalculated compositions. The elementary tetrahedra and their volumes were identified to map the phase space.

2.3. Mathematical Modeling

The analytical expressions for the secondary phases in each tetrahedron were derived using the Heath’s transformation equation:

aiCr + biFe + ciAl + diSi

The coefficients were used to compute the amounts of the formed secondary phases.

3. Results and Discussion

The values of heat capacity −${\mathrm{C}}_{\mathrm{p}}^0\left(\mathrm{T}\right)$, the enthalpy of melting −∆H⁰_mel. and the entropy of melting −∆S⁰_mel. of compounds Cr₂Al, Fe₂Al₅, FeSi, Cr₃Si, Cr₅Si₃, CrSi₂, formed in the Fe-Cr-Si-Al metal system, have been calculated. Their reference data have not been found.

The precise thermodynamic methods can be applied to define the melting heat. For instance, it can be calculated using the Clapeyron–Clausius equation [11]. Therefore, data on the dependence of melting point on pressure should be requested. However, the data are rarely available to a researcher.

The authors of [12,13,14] have recommend to calculate ΔH_mel. of compounds from the phase diagrams of systems using the Schröder’s equation in the ideal solutions [15]. Unfortunately, ideal solutions and melts are rare. Thus, the Schröder’s equation should be applied with great caution for the practical calculations.

The known methods to estimate the melting heat are few in number, and they do not provide a reliable assessment. The paper of [16] has pointed out that there is no precise and universal correlation for the melting heat. The paper of [17] has described that ΔH_mel. cannot be estimated with high accuracy.

Referring to Turkdogan and Pearson [18,19], a value of the melting entropy (ΔS_mel.) correlates with T_mel. and number of atoms in the compound molecule (m). The recommended formula to estimate ΔH_mel. is as follows:

(ΔH_mel./T_mel.) × m = K

(1)

where, K is determined by a melting point of the compound, i.e., K has the same numerical value for compounds with the same melting point.

However, the data in [20] have described that this correlation does not provide the satisfactory accuracy of assessment of the melting heat, i.e., a value of (ΔH_mel./T_mel.) × m changes within a wide range for different compounds with the same melting point.

It should be stated that the reference data on the melting heat of molecular compounds contains large gaps. Data on many compounds are contradictory. As a result, it makes difficult to find the universal correlations.

Morachevsky et al. [21] have analyzed the experimental data on the melting heat of the molecular compounds. Their vapor pressure at a triple point has not exceeded 10⁴ Pa.

The non-polar and low polar molecular compounds (ψ < 0.05) have a universal ratio between the value of ΔS_mel. and a melting point, presented as follows:

ΔS_mel. = ΔH_mel./T_mel.

(2)

Thus, it leads to the formula to calculate a melting heat [13,14]:

ΔH_mel. = 6.147 × Tmel.^1.333

(3)

Since the compounds of Cr₂Al, Fe₂Al₅, FeSi, Cr₃Si, Cr₅Si₃a and CrSi₂ are low polar, thus, the Formula (3) is used to calculate values of ΔH_mel.:

ΔH_mel., Cr₂Al = 6.147 × 15281.333 = 76.727 kJ/mol

ΔH_mel., Fe₂Al₅ = 6.147 × 14271.333 = 100.084 kJ/mol

ΔH_mel., FeSi = 6.147 × 14221.333 = 70.103 kJ/mol

ΔH_mel., Cr₃Si = 6.147 × 14851.333 = 131.284 kJ/mol

ΔH_mel., Cr₅Si₃ = 6.147 × 12381.333 = 126.364 kJ/mol

ΔH_mel., CrSi₂ = 6.147 × 13031.333 = 102.959 kJ/mol

Then the Formula (2) is used to calculate ΔS_mel.:

ΔS_mel., Cr₂Al = 76727/1183 = 64.86 J/(mol∙K)

ΔS_mel., Fe₂Al₅ = 100084/1144 = 69.31 J/(mol∙K)

ΔS_mel., FeSi = 70103/1678 = 41.8 J/(mol∙K)

ΔS_mel., Cr₃Si = 131284/1170 = 74.17 J/(mol∙K)

ΔS_mel., Cr₅Si₃ = 126364/1720 = 73.47 J/(mol∙K)

ΔS_mel., CrSi₂ = 102959/1475 = 69.8 J/(mol∙K)

The literature has not yet described methods for the approximate calculation of the heat capacity of liquids for the inorganic compounds. Only the Neumann–Kopp approach has been found from the original literature [22,23]. This approach has been successfully justified for the intermetallic compounds. This approach has been based on the additive contribution of different atoms to the molecular heat capacity of a compound [24,25]:

$${\mathrm{C}}_{\mathrm{p}}^{ж}={\displaystyle \sum _{\mathrm{i}}{\mathrm{C}}_{\mathrm{p}}^{ж}\left(\mathrm{i}\right)}·{\mathrm{n}}_{\mathrm{i}},$$

(4)

where ${\mathrm{C}}_{\mathrm{p}}^{ж}\left(\mathrm{i}\right)$ is an atomic component (increment) of the heat capacity of an element, J/(mol∙K); n_i—number of atoms; i—an element in a molecule of the compound.

The thermodynamic constants of compounds of the Fe-Si-Al-Cr system calculated by Formula (4) are presented in

Table 1. The calculated thermodynamic constants of compounds formed in the Fe-Si-Al-Cr system.

Compounds	∆Hmel., kJ/mol	∆Smel., J/(mol∙K)	${\mathit{C}}_{\mathit{p}}^0(\mathit{T})$, J/(mol∙K)	Tmel., K
Cr2Al	−33.0	75.31	77.58	298
	76.727	64.86	27.32	1183
Fe2Al5	−200.83	206.37	197.76	298
	100.08	69.31	254.49	1444
FeSi	−76.57	46.024	48.53	298
	70.103	41.8	87.51	1678
Cr3Si	−138.07	86.61	90.41	298
	131.284	74.17	28.15	1770
Cr5Si3	−326.57	182.5	291.77	298
	126.364	73.47	269.35	1720
CrSi2	−100.42	55.65	63.59	298
	102.959	69.8	27.75	1475

.

Thus, based on the semi-empirical methods to calculate the thermodynamic characteristics of substances, the values of melting enthalpy and melting entropy of compounds of Cr₂Al, Fe₂Al₅, FeSi, Cr₃Si, Cr₅Si₃ and CrSi₂ have been determined for the following tetrahedration of the Fe-Al-Si-Cr system at a melting temperature.

In order to plot the phase diagram of the Fe-Al-Si-Cr metal system at temperatures of phase transitions of the compounds, a complete calculation of the thermodynamic constants of reactions (in these systems) has been performed using the “Gibbs-MISiS” software package (National University of Science and Technology “MISIS”, Moscow, Russia, 2011). The results have been described in [26].

The Gibbs software package developed by scientists developed at Zh.Abishev Chemical-Metallurgical Institute determines the change in enthalpy, entropy, heat capacity and Gibbs energy of any reaction in the heterogeneous and homogeneous liquid-phase states at different temperatures including all phase transitions (the allotropic transformations, melting, etc.) for all system components and calculates the equilibrium constant of the reaction. The algorithm for calculation of the reaction value ${∆\mathrm{G}}_{\mathrm{T}}^0$ based on the Gibbs software package can be represented by formulas of (5)–(8):

(1)

The Gibbs energy values $∆{\mathrm{G}}_{\mathrm{f},298.15}^0$ of these compounds have been calculated by the following formula:

$${∆\mathrm{G}}_{\mathrm{f},298.15}^0={∆\mathrm{H}}_{\mathrm{f},298.15}^0-\mathrm{T}·{\mathrm{S}}_{298.15}^0$$

(5)

(2)

Calculation of the reaction of the enthalpy change at a given temperature:

$$∆{\mathrm{H}}_{\mathrm{T}}^0=∆{\mathrm{H}}_{298.15}^0+\underset{298.15}{\overset{{\mathrm{T}}_{пл}}{\int }}{∆\mathrm{C}}_{\mathrm{p}}\mathrm{dT}+{∆\mathrm{H}}_{\mathrm{mel}}^0$$

(6)

(3)

Determination of the reaction of the entropy change at a given temperature:

$$∆{\mathrm{S}}_{\mathrm{T}}^0=∆{\mathrm{S}}_{298.15}^0+\underset{298.15}{\overset{{\mathrm{T}}_{\mathrm{п}\mathrm{л}}}{\int }}{∆\mathrm{C}}_{\mathrm{p}}{\displaystyle \frac{\mathrm{dT}}{\mathrm{T}}}+{∆\mathrm{S}}_{\mathrm{m}\mathrm{e}\mathrm{l}.}^0$$

(7)

(4)

Calculation of the reaction of the Gibbs energy change at a given temperature—${∆\mathrm{G}}_{\mathrm{T}}^0$ using the Gibbs–Helmholtz formula.

After substituting formulas of (6) and (7) into formula (5), an expression has been presented:

$$∆{\mathrm{G}}_{\mathrm{T}}^0=∆{\mathrm{H}}_{298.15}^0+\underset{298.15}{\overset{{\mathrm{T}}_{пл}}{\int }}{∆\mathrm{C}}_{\mathrm{p}}\mathrm{dT}+{∆\mathrm{H}}_{\mathrm{m}\mathrm{e}\mathrm{l}}^0-\mathrm{T}·\left({∆\mathrm{S}}_{298.15}^0+\underset{298.15}{\overset{{\mathrm{T}}_{пл}}{\int }}{∆\mathrm{C}}_{\mathrm{p}}{\displaystyle \frac{\mathrm{dT}}{\mathrm{T}}}+{∆\mathrm{S}}_{\mathrm{mel}.}^0\right)$$

(8)

where:

-

${∆\mathrm{H}}_{298.15}^0$—a standard value of the reaction of enthalpy, J/mol;

-

${∆\mathrm{S}}_{298.15}^0$—a standard value of the reaction of entropy, J/(mol∙K);

-

${∆\mathrm{C}}_{\mathrm{p}}$—value of the reaction of the heat capacity, J/(mol∙K);

-

T_mel.—a current temperature or temperature of the phase transition (melt, evaporation, etc.) of the component, respectively, K;

-

T—a temperature where the system is in the homogeneous liquid-phase state, K;

-

ΔH⁰_mel.—enthalpy of phase transition or melt of a component, respectively, J/mol;

-

∆S⁰_mel.—entropy of phase transition or melt of a component, respectively, J/(mol∙K).

In order to plot a phase diagram of tetrahedration of the Fe-Al-Si-Cr system in the homogeneous liquid state by the thermodynamic-diagram analysis, the Si-Al-Fe, Si-Cr-Fe, Fe-Al-Cr and Si-Al-Cr boundary subsystems have been decomposed into elementary independent triangles. The principle of Gibbs free energy minimization has been used [27]. If the Gibbs energy change was negative during the reaction between the phases, then the reaction products as the coexisting phases have been connected with a straight line in the diagram. The sequence of this operation leads to a phase equilibrium diagram [28]. First of all, the metal compounds with the varying complexity forming the studied system should be described.

The approved coordinates (based on mass fraction × 1000) of the congruent and incongruent compounds of the Cr-Fe-Al-Si system used in the further study of their crystallization regions are presented in

Table 2. The congruent and incongruent metallic compounds in the Cr-Fe-Al-Si system and their coordinates on the quadruple concentration simplex (tetrahedron).

Compounds	Coordinates Based on Mass Fraction × 1000
	Cr	Fe	Al	Si
Cr	1000	0	0	0
Fe	0	1000	0	0
Al	0	0	1000	0
Si	0	0	0	1000
FeAl	0	675	325	0
Fe2Al5	0	453	547	0
FeAl3	0	409	591	0
Fe2Si	0	800	0	200
Fe5Si3	0	769	0	231
FeSi	0	667	0	333
FeSi2	0	500	0	500
CrAl7	216	0	784	0
Cr2Al11	259	0	741	0
CrAl4	325	0	675	0
Cr4Al9	461	0	539	0
Cr5Al8	546	0	454	0
Cr2Al	790	0	210	0
Cr3Si	848	0	0	152
Cr5Si3	755	0	0	245
CrSi	650	0	0	350
CrSi2	481	0	0	519

. The data of this table focus attention on 21 simple and complex compounds.

Tetrahedration of the Si-Al-Fe subsystem. The Gibbs energy values (∆G⁰₂₅₀₀) of the reactions calculated with the Gibbs software package for the Si-Al-Fe subsystem are demonstrated in

Table 3. Thermodynamics of reactions in the Si-Al-Fe subsystem (T = 2500 K).

Equation of Reaction	∆G02500, (kJ/mol)
3Al + FeSi2 = FeAl3 + 2Si	112.61
FeAl3 + FeSi2 = 3Al + 2FeSi	−40.52
2FeAl3 + 3FeSi = 6Al + Fe5Si3	−93.33
FeAl3 + Fe5Si3 = 3Al + 3Fe2Si	−39.12

.

The reaction of 3Al + FeSi₂ = FeAl₃ + 2Si has been observed in the Si-Al-Fe subsystem. the Gibbs energy was positive at a temperature of 2500 K. It was equal ΔG⁰₂₅₀₀ = 112.61 kJ/mol, i.e., the reaction has shifted to formation of the initial compounds. The plotted phase diagram based on the data obtained in the Si-Al-Fe subsystem at a temperature of 2500 K is illustrated in Figure 1.

As a result of triangulation of the Si-Al-Fe system (Figure 1) including all complex compounds, five regions have been formed. The incongruent compounds have not been found in regions of Si-Al-FeSi₂ (1), FeSi₂-Al-FeSi (2), FeSi-Al-Fe₅Si₃ (3), Fe₅Si₃-Al-Fe₂Si (4). The diagram region of Fe₂Si-Al-Fe has one congruent (Fe₂Al₅) and two incongruent compounds (FeAl₃, FeAl). These compounds divide this region into four sub-regions: Fe₂Si-Al-FeAl₃ (5I), Fe₂Si-FeAl₃-Fe₂Al₅ (5II), Fe₂Si-Fe₂Al₅-FeAl (5III) and Fe₂Si-FeAl-Fe (5IV).

Tetrahedration of the Si-Cr-Fe subsystem. Values of changes in the Gibbs energy of the reactions for the Si-Cr-Fe subsystem are presented in

Table 4. Thermodynamics of reactions in the Si-Cr-Fe subsystem (T = 2500 K).

Equation of Reaction	∆G02500, (kJ/mol)
FeSi + CrSi2 = FeSi2 + CrSi	−45.77
2FeSi + 5CrSi = 2FeSi2 + Cr5Si3	−32.06
4FeSi + 3Cr5Si3 = 4FeSi2 + 5Cr3Si	−70.95
FeSi + Cr3Si = FeSi2 + 3Cr	14.61
Fe5Si3 + 2Cr3Si = 5FeSi + 6Cr	39.79
5Fe2Si + Cr3Si = 2Fe5Si3 + 3Cr	7.09
2Fe + Cr3Si = Fe2Si + 3Cr	41.24

.

Based on the obtained thermodynamic data, the phase diagram of the Si-Cr-Fe subsystem has been plotted up to a temperature of 2500 K (Figure 2).

Eight regions have been formed during the triangulation of the ternary Si-Cr-Fe system (Figure 2). However, the Si-CrSi-FeSi₂ region has one congruent compound—CrSi₂ which resulted in formation of two sub-regions: Si-CrSi₂-FeSi₂ (1I) and CrSi-FeSi₂-CrSi₂ (1II). The incongruent compounds have not been observed in regions of CrSi-Cr₅Si₃-FeSi₂ (2), Cr₅Si₃-Cr₃Si-FeSi₂ (3), Fe₂Si-Cr₃Si-FeSi (4), FeSi-Cr₃Si-Fe₅Si₃ (5), Fe₅Si₃-Cr₃Si-Fe₂Si (6), Fe₂Si-Cr₃Si-Fe (7) and Cr₃Si-Cr-Fe (8) [14].

Tetrahedration of the Fe-Al-Cr subsystem. The values of changes in the Gibbs energy of the reactions for the Fe-Al-Cr subsystem are demonstrated in

Table 5. Thermodynamics of reactions in the Fe-Al-Cr subsystem (T = 2500 K).

Equation of Reaction	∆G02500, (kJ/mol)
Fe + Cr2Al = FeAl + 2Cr	−71.62
Fe2Al5 + 6Cr = 2FeAl + 3Cr2Al	338.09
2FeAl3 + 3Cr = Fe2Al5 + Cr2Al	−282.52
11FeAl3 + 2.5Cr2Al = 5.5Fe2Al5 + Cr5Al8	−1614.65
26FeAl3 + 4Cr5Al8 = 13Fe2Al5 + 5Cr4Al9	−3489.90
14FeAl3 + Cr4Al9 = 7Fe2Al5 + 4CrAl4	−2194.17
3FeAl3 + CrAl4 = 1.5Fe2Al5 + 0.5Cr2Al11	−388.55
3FeAl3 + 0.5Cr2Al11 = 1.5Fe2Al5 + CrAl7	−440.55

.

Based on the obtained thermodynamic data, the phase diagram of the Fe-Al-Cr subsystem at a temperature of 2500 K has been constructed (Figure 3). The phase diagram of the Fe-Al-Cr subsystem is a set of binary compounds which have been found in its binary Cr-Fe, Fe-Al, Cr-Al systems.

The triangulation of the ternary Fe-Al-Cr system (Figure 3) has formed nine regions. The incongruent compounds have not been found in the regions of Fe-FeAl-Cr (1), Fe₂Al₅-Cr-FeAl (2), Fe₂Al₅-Cr₂Al-Cr (3), Fe₂Al₅-Cr₅Al₈-Cr₂Al (4), Fe₂Al₅-Cr₄Al₉-Cr₅Al₈ (5), Fe₂Al₅-CrAl₄-Cr₄Al₉ (6), Fe₂Al₅-Cr₂Al₁₁-CrAl₄ (7) and Fe₂Al₅-CrAl₇-Cr₂Al₁₁ (8). The Fe₂Al₅-Al-CrAl₇ region has one incongruent compound—FeAl₃. This incongruent compound divides this region into two sub-regions, respectively: Fe₂Al₅-FeAl₃-CrAl₇ (9I) and FeAl₃-Al-CrAl₇ (9II).

Tetrahedration of the Si-Al-Cr subsystem. Values of changes in the Gibbs energy of reactions for the Si-Al-Cr subsystem are presented in

Table 6. Thermodynamics of reactions in the Si-Al-Cr subsystem (T = 2500 K).

Equation of Reaction	∆G02500, (kJ/mol)
2Si + CrAl7 = CrSi2 + 7Al	−190.22
CrSi2 + CrAl7 = 2CrSi + 7Al	−214.37
3CrSi + 2CrAl7 = Cr5Si3 + 14Al	−339.78
0.5Cr5Si3 + 2CrAl7 = 1.5Cr3Si + 14Al	−359.51

. Based on the obtained thermodynamic data, the phase diagram of the Si-Al-Cr subsystem at a temperature of 2500 K has been plotted (Figure 4).

The triangulation of the ternary Si-Al-Cr system (Figure 4) has formed five regions.

The incongruent compounds have not been observed in regions of Si-Al-CrSi₂ (1), CrSi₂-Al-CrSi (2), CrSi-Al Cr₅Si₃ (3) and Cr₅Si₃-Al-Cr₃Si (4). Two congruent (Cr₂Al and CrAl₇) and four incongruent compounds (Cr₂Al₁₁, CrAl₄, Cr₄Al₉, Cr₅Al₈) have been found in the diagram region of Fe3Si-Al-Cr which resulted in formation of seven sub-regions: Cr₃Si-Al-CrAl₇ (5I), Cr₃Si-CrAl₇-Cr₂Al₁₁ (5II), Cr₃Si-Cr₂Al₁₁-CrAl₄ (5III), Cr₃Si-CrAl₄-Cr₄Al₉ (5IV), Cr₃Si-Cr₄Al₉-Cr₅Al₈ (5V), Cr₃Si-Cr₅Al₈-Cr₂Al (5VI) and Cr₃Si-Cr₂Al-Cr (5VII).

Figure 5 illustrates a general view of the analyzed Cr-Fe-Al-Si system in the homogeneous liquid state including the congruent and incongruent compounds.

Thus, the above-mentioned data and results on calculations have confirmed the reliability of tetrahedration of the phase diagram of the metal Cr-Fe-Al-Si system in the liquid state. Thereafter, the phase compositions of the metal products can be predicted in the sub-solidus state during the melting of different grades of the complex aluminum–silicon–chromium alloy [29,30].

The quasi-systems of the general Fe-Cr-Si-Al system and their mathematical models.

The above-stated data on the boundary ternary systems derived from the sub-solidus structure of the four elements of three-component subsystems are sufficient to divide the concentration space of tetrahedron of the Fe-Cr-Si-Al system into elementary ones.

Distances between pairs of the coexisting compounds are picks of quasi-systems with oblique axes of A1 (x1; y1; z1; u1 …), A2 (x2; y2; z2; u2 …). They have been needed to plot the composition–property diagrams in the corresponding sections. They have been calculated by Formula (9) [15,16]:

l2 = (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 + (u2 − u1)2 + … + (x2 − x1) × (y2 − y1) + (x2 − x1) × (z2 − z1) + (x2 − x1) × (u2 − u1) + … + (y2 − y1) × (z2 − z1) + (y2 − y1) × (u2 − u1) … + (z2 − z1) × (u2 − u1) + …

(9)

The four-component Fe-Cr-Si-Al metal system has been composed of four ternary metal systems of Si-Al-Fe, Si-Cr-Fe, Fe-Al-Cr and Si-Al-Cr.

Based on the results of tetrahedration of the above four ternary systems, the elementary tetrahedrons of the general Fe-Cr-Si-Al system are easier to determine by searching through the related (differ in one component) triangles of subsystems. Then, the resulting tetrahedron of the studied quadruple system is derived by summing up these triangles. Therefore, the exclusion of one of the components of this four-vertex can be reduced to a triangle of the subsystem of the general system to find the next adjacent four-vertex. It is performed schematically as described in

Table 7. Determination of the resulting tetrahedra of the Fe-Cr-Si-Al system by triangulation data of its boundary systems.

System	Initial Triangles
Boundary	1	6	16
Si-AlFe	FeSi2-Al-Si	Fe5Si3-Al-FeSi	–
Si-Cr-Fe	FeSi2-CrSi2-Si	Fe5Si3-Cr3Si-FeSi	–
Fe-Al-Cr	–	–	Fe2Al5-Cr4Al9-Cr5Al8
Si-Al-Cr	Si-Al-CrSi2	–	Cr3Si-Cr4Al9-Cr5Al8
General	The resulting tetrahedra
Fe-Cr-Si-Al	Si-FeSi2-Al-CrSi2	Fe5Si3-FeSi-Al-Cr3Si	Fe2Al5-Cr4Al9-Cr5Al8-Cr3Si

.

As a result, the phase diagram of the general Cr-Fe-Al-Si system (Figure 5) has been derived by searching through all related triangles of the four ternary subsystems using a similar method.

Thus, based on the tetrahedration of the quadruple metal Fe-Cr-Si-Al system for the liquid-phase state (T = 2500 K) it has been established that the system consists of 21 elementary independent tetrahedral. They model the compositions of different grades of aluminum–silicon–chromium in recovery process of elements from ash of the high-ash coal. The resulting elementary quadruple systems and their volumes are summarized in

Table 8. List of tetrahedra of the metal Cr-Fe-Al-Si system.

Tetrahedra No.	Tetrahedra	Elementary Volumes
1	Si-FeSi2-Al-CrSi2	0.2405
2	CrSi2-FeSi2-CrSi-Al	0.0845
3	CrSi-Cr5Si3-Al-FeSi2	0.0525
4	Cr5Si3-Cr3Si-Al-FeSi2	0.0465
5	FeSi-FeSi2-Al-Cr3Si	0.1416
6	Fe5Si3-FeSi-Al-Cr3Si	0.0865
7	Fe2Si-Fe5Si3-Al-Cr3Si	0.0263
8	FeAl3-Al-Cr3Si-Fe2Si	0.0693
9	FeAl3-Fe2Al5-Cr3Si-Fe2Si	0.0074
10	Fe2Al5-FeAl-Cr3Si-Fe2Si	0.0376
11	FeAl-Fe-Cr3Si-Fe2Si	0.0551
12	Fe-Cr-Cr3Si-FeAl	0.0494
13	FeAl-Fe2Al5-Cr-Cr3Si	0.0337
14	Fe2Al5-Cr-Cr3Si-Cr2Al	0.0144
15	Fe2Al5-Cr2Al-Cr5Al8-Cr3Si	0.0168
16	Fe2Al5-Cr4Al9-Cr5Al8-Cr3Si	0.0058
17	Fe2Al5-CrAl4-Cr4Al9-Cr3Si	0.0093
18	Fe2Al5-Cr2Al11-CrAl4-Cr3Si	0.0045
19	Fe2Al5-CrAl7-Cr2Al11-Cr3Si	0.0029
20	FeAl3-Fe2Al5-CrAl7-Cr3Si	0.0014
21	Al-FeAl3-CrAl7-Cr3Si	0.0134
Amount		0.9994

[26].

Earlier, the results of the thermodynamic-diagram analysis were confirmed by studying the phase compositions of the complex aluminum–silicon–chrome alloy. The aluminum–silicon–chrome alloy was smelted in a single-stage carbothermic process in an open-type ore-thermal furnace, and the alloy samples were then sent for phase composition determination. In the aforementioned study, the thermodynamic-diagram analysis for the Cr-Fe-Al-Si metallic system was conducted using only congruent compounds, and accordingly, the system consists of 11 tetrahedra [31].

The accuracy of division of the general system into the quasi-systems is controlled by the equality to unity of the amount of the relative volumes of quasi-systems to avoid losses of the transient polytopes. The above data of the table demonstrate that the amount of the relative volumes of the elementary tetrahedra is practically equal to unity (0.9994). Thus, it confirms the accuracy of the made tetrahedration.

The data of phase diagrams can be processed analytically without visualizing the system. Thus, the equations can be obtained to study its physical and chemical properties [32,33,34,35]. The simple and accessible method for manual calculation to derive the transformation equations expressing any secondary system using the primary components of the general system has been described in the well-known papers of [36,37]. The criterion for position of a given composition of the melts into one of the quasi-systems is positive values of the n- amount of the secondary components of a certain polytope calculated by the Heath’s equation [38].

Based on the above mentioned, the coefficients calculated for each secondary component of the 21 quasi-systems of the basic tetrahedron have been derived by the procedure described in [39], and they are presented in

Table 9. List of elementary tetrahedra, their volumes and coefficients of equations to calculate the equilibrium ratios of secondary components of the Cr-Fe-Al-Si system.

Initial Components	Coefficients	Polytopes, Their Volumes and Transformation Coefficients
		1	2	3	4	5	6	7	8	9	10	11
		Si FeSi2 Al CrSi2	CrSi CrSi2 Al FeSi2	CrSi Cr5Si3 Al FeSi2	Cr5Si3 Cr3Si Al FeSi2	FeSi Cr3Si Al FeSi2	FeSi Cr3Si Al Fe5Si3	Fe2Si Cr3Si Al Fe5Si3	FeAl3 Al Cr3Si Fe2Si	FeAl3 Fe2Al5 Cr3Si Fe2Si	FeAl Fe2Al5 Cr3Si Fe2Si	FeAl Fe Cr3Si Fe2Si
Volumes		0.2405	0.0845	0.0525	0.0465	0.141616	0.086496	0.026288	0.069366	0.007462	0.037651	0.05512
Cr	a1	−1.079	−2.84615	7.19048	−1.6344	−2.994	7.53922	−24.8065	1.75301	−8.91337	1.76662	0
	a2	0	3.84615	−6.1905	2.63441	0	0	0	−1.03603	9.63036	−1.04963	0.71698
	a3	0	0	0	0	0	0	0	1.17925	1.17925	1.17925	1.17925
	a4	2.079	0	0	0	3.99401	−6.5392	25.8065	−0.89623	−0.89623	−0.89623	−0.8962
Fe	b1	−1	2.84615	−7.1905	−9.1183	2.99401	−2.2647	7.45161	2.44499	−12.4318	2.46396	0
	b2	2	−3.84615	6.19048	8.11828	0	0	0	−1.44499	13.4318	−1.46396	1
	b3	0	0	0	0	0	0	0	0	0	0	0
	b4	0	2	2	2	−1.994	3.26471	−6.45161	0	0	0	0
Al	c1	0	0	0	0	0	0	0	0	10.2955	−2.04054	3.07692
	c2	0	0	0	0	0	0	0	1	−9.29545	3.04054	−2.0769
	c3	1	1	1	1	1	1	1	0	0	0	0
	c4	0	0	0	0	0	0	0	0	0	0	0
Si	d1	1	3.07101	−2.3333	9.11828	0.53666	−1.3514	4.44644	−9.77995	49.7272	−9.85586	0
	d2	0	−2.07101	3.33333	−8.1183	1.17925	1.17925	1.17925	5.77995	−53.7273	5.85586	−4
	d3	0	0	0	0	0	0	0	0	0	0	0
	d4	0	0	0	0	−0.7159	1.17212	−4.62568	5	5	5	5
Initial Components	Coefficients	Polytopes, Their Volumes and Transformation Coefficients
		12	13	14	15	16	17	18	19	20	21
		Fe Cr Cr3Si FeAl	FeAl Fe2Al5 Cr Cr3Si	Fe2Al5 Cr Cr3Si Cr2Al	Fe2Al5 Cr5Al8 Cr3Si Cr2Al	Fe2Al5 Cr4Al9 Cr5Al8 Cr4Al9	Fe2Al5 CrAl4 Cr3Si Cr4Al9	Fe2Al5 CrAl4 Cr3Si Cr2Al11	Fe2Al5 CrAl7 Cr3Si Cr2Al11	FeAl3 Fe2Al5 CrAl7 Cr3Si	FeAl3 Al CrAl7 Cr3Si
Volumes		0.0494	0.033744	0.01446	0.016801	0.005853	0.009364	0.004544	0.002961	0.001445	0.013428
Cr	a1	0	−2.0405	0	0	0	0	0	0	−37.3687	0
	a2	1	3.04054	1	−0.86066	6.34118	−3.96323	11.2273	−17.2326	33.7391	−3.6296
	a3	0	0	0	0	0	0	0	0	4.62963	4.62963
	a4	0	0	0	1.86066	−5.3412	4.96323	−10.2273	18.2326	0	0
Fe	b1	1	0	2.20751	2.20751	2.20751	2.20751	2.20751	2.20751	−12.4318	2.44499
	b2	0	0	4.54252	−3.90955	6.54894	−4.09309	4.73854	−7.27311	13.4318	−1.445
	b3	0	−5.579	0	0	0	0	0	0	0	0
	b4	0	6.57895	−5.75	2.70204	−7.7565	2.88558	−5.94605	6.06561	0	0
Al	c1	−2.0769	0	0	0	0	0	0	0	10.2955	0
	c2	0	0	−3.7619	3.2377	−5.4235	3.38971	−3.92424	6.02325	−9.29545	1
	c3	0	1	0	0	0	0	0	0	0	0
	c4	3.07692	0	4.7619	−2.2377	6.42353	−2.38971	4.92424	−5.02326	0	0
Si	d1	0	2.46396	0	0	0	0	0	0	208.478	0
	d2	−5.579	−1.464	−5.579	4.80155	−35.377	22.1107	−62.6364	96.1395	−188.228	20.2495
	d3	6.57895	0	6.57895	6.57895	6.57895	6.57895	6.57895	6.57895	−25.8285	−25.828
	d4	0	0	0	−10.3805	29.7982	−27.6896	57.0574	−101.719	6.57895	6.57895

.

Hereafter, in order to apply results on the thermodynamic-diagram analysis to the compositions of different grades of aluminum–silicon–chromium, the elementary tetrahedra with their compositions have been found. Then, based on the normative distribution of the primary phases between the compounds (secondary phases) located at the peaks of a given tetrahedron, the metallurgical assessment of the melts has been made.

For this purpose, the chemical compositions of the individual grades of aluminum–silicon–chromium and the complex aluminum–chromium–silicon alloy have been recalculated to the four elements of the Cr-Fe-Al-Si system. Their compositions are presented in

Table 10. The chemical composition of complex alloys based on the metal Cr-Fe-Al-Si system.

Material	The Chemical Composition, %
	Cr	Si	Al	Fe	C	P	S	Ti	Ca
Complex aluminum–silicon–chromium alloy
AS65Cr15	15.71	53.4	10.47	13.25	0.54	0.037	0.07	0.58	1.40
AS65Cr20	20.33	48.48	13.21	15.09	0.75	0.028	0.047	0.66	1.10
AS60Cr25	25.5	49.41	10.79	11.25	0.40	0.046	0.033	0.31	0.86
AS60Cr30	28.95	47.15	10.12	11.25	0.56	0.05	0.04	0.42	0.94
Complex aluminum–chromium–silicon (ACS) alloy
ACS No.1 alloy	33.4	23.57	13.07	24.29	0.69	0.10	0.006	-	0.06
ACS No.2 alloy	41.88	27.42	10.73	16.62	0.58	0.08	0.005	-	0.16
ACS No.3 alloy	61.61	14.40	0.17	15.39	0.35	-	0.004	-	0.32

. Then based on Table 9, their positions in the factor space of this system have been determined by calculation of the normative number of the secondary phases in the corresponding tetrahedra (

Table 11. The standard phase compositions of complex alloys based on the Cr-Fe-Al-Si system.

Material	The Standard Phase Composition, %										Tetrahedron	Volume
	Si	FeSi2	Al	CrSi2	CrSi	Cr5Si3	FeAl	Fe	Cr3Si	Fe2Si
	Complex aluminum–silicon–chromium alloy
AS65Cr15	25.0	28.54	11.28	35.18	-	-	-	-	-	-	Si-FeSi2-Al-CrSi2 (No.1)	0.2405
AS65Cr20	11.78	31.08	13.6	43.53	-	-	-	-	-	-	Si-FeSi2-Al-CrSi2 (No.1)	0.2405
AS60Cr25	10.99	23.2	11.13	54.68	-	-	-	-	-	-	Si-FeSi2-Al-CrSi2 (No.1)	0.2405
AS60Cr30	17.69	23.08	10.38	48.85	-	-	-	-	-	-	Si-FeSi2-Al-CrSi2 (No.1)	0.2405
	Complex aluminum–chromium–silicon (ACS) alloy
ACS No.1 alloy	-	51.5	13.85	-	11.15	23.50	-	-	-	-	CrSi-Cr5Si3-Al-FeSi2 (No.3)	0.0525
ACS No.2 alloy	-	34.4	11.1	41.74	12.75	-	-	-	-	-	CrSi2-FeSi2-CrSi-Al (No.2)	0.0845
ACS No.3 alloy	-	-	-	-	-	-	0.55	1.75	79.34	18.35	FeAl-Fe-Cr3Si-Fe2Si (No.11)	0.05512

).

The phase composition in each of the tetrahedra presented in Table 8 can be described by substituting of the corresponding coefficients from Table 10 into the Equation (10) [39]:

Xi = aiCr + biFe + ciAl + diSi

(10)

It is the Heath’s transformation equation, where Xi—the amount of the formed secondary phase; ai, bi, ci and di—the transformation coefficients; Cr, Fe, Al and Si—the amounts of the primary metal components in the metal.

The results of the calculations have found that all grades of the complex aluminum–silicon–chromium alloy by a chemical composition were located in tetrahedron No.1 (Table 11) and contained 10.99–28.95% Si, 35.18–54.68% CrSi₂, 10.38–13.6% Al and 23.08–31.08% FeSi₂. The petrographic analysis has really found these phases in the composition of aluminum–silicon–chromium. The found tetrahedron was volumetric (Vi = 0.2405) phase triangle of the metal Cr-Fe-Al-Si system. Therefore, the large volume of the tetrahedron has provided good conditions for the melt of aluminum–silicon–chromium. As a result, it was possible to freely adjust the compositions of the charge to obtain the required grade composition of the alloy [40].

The analytical expressions of the secondary phases of tetrahedron No.1 of Si-FeSi₂-Al-CrSi₂ are presented below:

Si = a1Cr + b1Fe + c1Al + d1Si = −1.07900∙Cr − 1.0∙Fe + 1.0∙Si
CrSi₂ = a2Cr + b2Fe + c2Al + d2Si = 2.07900∙Cr
Al = a3Cr + b3Fe + c3Al + d3Si = 1.0∙Al
FeSi₂ = a4Cr + b4Fe + c4Al + d4Si = 2.0∙Fe

(11)

Compositions of the aluminum–silicon–chromium alloy have changed their positions to tetrahedral of No.2, No.3, No.11 (CrSi₂-FeSi₂-CrSi-Al, CrSi-Cr₅Si₃-Al-FeSi₂, FeAl-Fe-Cr₃Si-Fe₂Si) when the chromium content increased, and aluminum content decreased. These tetrahedra have taken from 5 to 8% of the factor space of the general system.

The normative amounts of the secondary phases in tetrahedra of No.2, No.3 and No.11 are calculated by the Formulas (12)–(14):

CrSi = a1Cr + b1Fe + c1Al + d1Si = −2.84615∙Cr + 2.84615∙Fe + 3.07101∙Si
CrSi₂ = a2Cr + b2Fe + c2Al + d2Si = 3.84615∙Cr − 3.84615∙Fe − 2.07101∙Si
Al = a3Cr + b3Fe + c3Al + d3Si = 1.0∙Al
FeSi₂ = a4Cr + b4Fe + c4Al + d4Si = 2.0∙Fe

(12)

CrSi = a1Cr + b1Fe + c1Al + d1Si = 7.19048∙Cr − 7.19048∙Fe − 2.33333∙Si
Cr₅Si₃ = a2Cr + b2Fe + c2Al + d2Si = −6.19048∙Cr + 6.19048∙Fe + 3.33333∙Si
Al = a3Cr + b3Fe + c3Al + d3Si = 1.0∙Al
FeSi₂ = a4Cr + b4Fe + c4Al + d4Si = 2.0∙Fe Fe

(13)

FeAl = a1Cr + b1Fe + c1Al + d1Si = 3.07692∙Al
Fe = a2Cr + b2Fe + c2Al + d2Si = 0.71698∙Cr + 1.0∙Fe − 2.07692∙Al − 4.0∙Si
Cr₃Si = a3Cr + b3Fe + c3Al + d3Si = 1.17925∙Cr
Fe₂Si = a4Cr + b4Fe + c4Al + d4Si = −0.89623∙Cr + 5.0∙Si Fe

(14)

Figure 6 schematically illustrates the phase diagrams of the complex aluminum–silicon–chromium alloy (a) and aluminum–chromium–silicon (ACS) alloy (b, c, d). They have the following average chemical composition after recalculation into four components (

Table 12. Components of complex chromium alloys based on the metal Cr-Fe-Al-Si system.

Name	Component, %
	Cr	Si	Al	Fe
Aluminum–silicon–chromium grade: AS65Cr15	16.92	57.53	11.28	14.27
Aluminum–silicon–chromium grade: AS65Cr20	20.94	49.92	13.6	15.54
Aluminum–silicon–chromium grade: AS60Cr25	26.3	50,97	11.13	11.6
Aluminum–silicon–chromium grade: AS60Cr30	29.7	48.38	10.38	11.54
ACS No.1 alloy	35.41	24.99	13.85	25.75
ACS No.2 alloy	43.33	28.37	11.10	17.20
ACS No.3 alloy	67.28	15.73	0.18	16.81

).

Thus, the region of compositions of the complex chromium alloys, in particular aluminum–silicon–chromium, is characterized by the set of the above mentioned tetrahedra.

By volume they take up 43.26% of the factor space of the general Cr-Fe-Al-Si system. The volumes of tetrahedra of No.2 and No.11 can be disregarded, i.e., only alloy compositions with the high chromium content and the low aluminum content are able to penetrate them. However, tetrahedral of No.1 and No.3 are of direct interest, i.e., the amount of their relative volumes equals 29.3% of the general system volume (Table 8).

4. Conclusions

Based on the theoretical studies conducted by the thermodynamic-diagram analysis, the following results have been obtained:

• A phase diagram of the metal Fe-Al-Si-Cr system modeling the compositions of the multicomponent chromium alloys including the complex aluminum–silicon–chromium alloy has been plotted. It has been found that the system consisted of 21 stable elementary tetrahedra. The amount of the relative volumes of the elementary tetrahedra was practically equal to unity (0.999999), and thus, it has confirmed the accuracy of the made tetrahedration;
• The analytical expressions for each tetrahedron have been derived. The created mathematical model can find the phase composition of the complex aluminum–silicon–chromium alloy in combination with other properties to optimize the technological process of its production. Based on the calculations, it has been stated that the phase compositions of aluminum–silicon–chromium are characterized by tetrahedron No.1 (Si-FeSi₂-Al-CrSi₂). The found tetrahedron is the most volumetric (Vi = 0.2405) phase triangle of the metal Cr-Fe-Al-Si system. Therefore, the large volume of the tetrahedron has provided good conditions for the melt of aluminum–silicon–chromium. As a result, it was possible to freely adjust the composition of the charge to obtain the required grade composition of the alloy.