KINETIC MODEL OF HIGH TEMPERATURE PHYSICOCHEMICAL PROCESSES

M. ZinigradCollege of Judea and Samaria, Science Park, Ariel, 44837 Israel

zinigrad@research.yosh.ac.il

Abstract The quality of metallic materials depends on their composition

and structure and these are determined by various physicochemical and technological factors.

We have developed unique method of mathematical modeling of phase interaction at high temperatures. This method allows us to build models taking into account: thermodynamic characteristics of the processes, influence of the initial composition and temperature on the equilibrium state of the reactions, kinetics of heterogeneous processes, influence of the temperature, composition, hydrodynamic and thermal factors on the velocity of the chemical and diffusion processes.

The model can be implemented in optimization of various technological processes in welding, surfacing, casting as well as in manufacturing of steels and non-ferrous alloys, materials refining, alloying with special additives, removing of non-metallic inclusions.

IntroductionOne of the most important and complicated problems in the

modern industry is to obtain materials with required composition, structure and properties. For example, deep refining is a difficult task by itself, but the problem of obtaining the material with the required specific level of refining is much more complicated. It will take a lot of time and will require a lot of expanses to solve this problem empirically and the result will be far from the optimal solution.

The effective method used to solve such problems is computer modeling. The use of mathematical models decreases the amount of time and the amount of labor needed for an investigation, as well as makes it possible to carry out experiments which cannot be performed or can be performed only with great difficulty on a real object.

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The development of computer technology and its accessibility have made it possible to solve problems for which there were previously no known methods of solution or these methods were so tedious that they proved to be unsuitable for practical application.

It has become possible to mathematically model complex physicochemical processes in metallurgical systems [1-22] both in reference to processes involving the smelting and refining of steels and alloys [1- 4, 6, 10, 11, 14 - 19, 46, 47] and for the analysis welding technologies [5, 7-9, 20 -23, 37, 48-60].

The complexity of these processes stems from the simultaneous occurrence of a considerable number of physical and chemical processes involving liquid, solid, and gas phases, as well as the high temperatures, the complex character of the hydrodynamic and heat fluxes, and the nonstationary nature of the processes. This complexity is manifested in the large number of parameters determining the course of the processes and the fact that the variation of a few parameters causes the variation of many others. Therefore, such complex objects are studied by constructing models, i.e., simplifying systems, which reflect the most significant aspects of the object under consideration.

Physical modeling, i.e., the representation of experimental data in the form of dependences of dimensionless variables (similarity criteria), which are composed of various physical quantities and linear dimensions, is convenient for comparatively simple systems. Interesting results were obtained, for example, from the physical modeling of slag foaming [24]. “Cold” and “hot” models, the bubble sizes, and such important characteristics as the viscosity and the adhesive force of oxide melts were investigated. In such cases, the focus is generally placed on physical or physicochemical characteristics of the phases which are subject to direct measurement.

Such a method is poorly suited to complex systems and processes described by systems of equations. In the case of mathematical modeling, the process is studied on a mathematical model using a computer, rather than on a physical object. The input parameters of the mathematical model are fed into the computer, and the computer supplies the output parameters calculated in the process. The first stage in the mathematical modeling of physicochemical systems is generally the construction of thermodynamic models [10, 11, 14-17, 25-36]. This

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stage is very important both for ascertaining the fundamental possibility of the combined occurrence of particular chemical processes and for listing the most important thermodynamic characteristics. The investigation of the activities of the components in binary and more complex, i.e., ternary, systems and the creation of a database of thermodynamic characteristics was the subject, for example, of [10, 35]. If the rates of the chemical reactions are sufficiently high, the composition of the reactant mixture at the outlet of the chemical reactor should be fairly close to the equilibrium composition and can be found by thermodynamic methods. There are several approaches to the creation of thermodynamic models. They include the employment of polymer theory to model complex multicomponent systems [15], modeling for the purpose of constructing phase diagrams [11,25,28, 31,32, 36], the construction of statistical-thermodynamic models [13, 14, 17-19, 26, 27, 29], the determination of the enthalpy and other thermal characteristics [16, 30], the modeling of melting processes and structure-building processes [36]. Very interesting and important results were obtained from the investigation of the microstructure and properties of deposited weld metals [48, 49, 50, 52-54] and susceptibility to cracking and extent of hot cracking [51]. The results of these investigations will be very useful for us - for determination of the influence of weld metal composition (which will be obtained from our model computations) on the structure and properties.

When there are no or only few theoretical data on the process being modeled, the mathematical description can take the form of a system of empirical equations obtained from a statistical study of the real process. As correlation between the input and output parameters of the object is established as a result of such a study [14, 33]. Naturally, the employment of statistical models is restricted by the width of the range of variation of the parameters studied.

In recent years mathematical modeling has been applied not only to the investigation of theoretical aspects of physicochemical processes, but also to the analysis of real technologies.

The areas of the prediction and optimization of the composition and properties of materials obtained in different technological processes are especially promising [2, 3, 4-9, 36, 57, 58]. Some of the results were obtained from the modeling of the process of the formation of a weld

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pool [5], from modeling of weld metal transformations [58,59], and from the modeling of processes involving the segregation of nonmetallic inclusions in steel [4], the interaction of particles during welding [9], and diffusion-controlled kinetics [2, 3, 6, 7, 54-57].

Important results were obtained from the studies of the physical and chemical parameters of welding processes [54] and development of kinetic model of alloy transfer [55-57].

By determining the chemical composition of the weld metal researchers have developed the kinetic model [56,57]. Basing on this model the authors described the transfer of alloying elements between the slag and the metal during flux-shielded welding. Although the model takes into consideration the practical weld process parameters such as voltage, current, travel speed, and weld preparation geometry. The model was tested experimentally [57] for transfer Mn, Si, Cr, P, Ni, Cu, and Mo.

In our opinion the problem of modeling complex objects with consideration of the kinetics of the chemical processes occurring them is more complicated. This applies both to diffusion processes [56] and especially to the analysis of the kinetics of complicated heterogeneous reactions [12].

A more complete, adequate description of real chemical processes requires the construction of a mathematical model which takes into account the diffusion of all the components in the complex multicomponent system, the kinetics and mechanism of the individual chemical reactions, the special features of their simultaneous occurrence, and the influence of heat transfer and the hydrodynamics, as well as the influence of the engineering parameters and other factors. There is presently a large amount of experimental and theoretical data, which make it possible to solve such problems.

The present research is also intended to be devoted to the development of mathematical models of such a type on the basis of a new method for the kinetic analysis of reactions in multicomponent systems.

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Research objectives Metal products with required properties is usually obtained with

the use of various initial materials. These materials are employed in all branches of modern industry for metallurgy, castings, welding, hardening the surfaces of items, and corrosion protection, as well as for restoring worn items.

The main problem which the technologies solve is the production of metal or alloy with a required composition and assigned properties. At the present time, these problems are generally solved empirically, i.e., either by means of technological experiments or by the statistical treatment of existing experimental data.

Such an approach requires great expenditures of time and resources and the consumption of considerable amounts of expensive materials. In addition, the results of such investigations have a random character and are far from optimal.

A fundamentally different approach will be employed in the present research: the mathematical model of the physicochemical processes developed and the computer program written on its basis will make it possible to “run” a large number of variants within a short time without considerable expenses and to select the optimal variant, which provides products with the required composition and properties. Such a result cannot be obtained, in principle, even after the performance of hundreds of technological experiments.

The mathematical model of the physicochemical processes involved in the interaction of the metallic, oxide, and gas phases will take into account the following:

- the mass and energy balance equations, which are written with consideration of the hydrodynamics of the phases and characterize the distribution of the temperature, concentrations of the components, and various properties;

- equations describing heat and mass transfer, equilibrium, and the kinetics of reactions;

- relations between individual parameters of the processes;

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- restrictions on the values of the parameters of the process, for example, the permissible fluctuations of the concentrations of certain substances, the maximal permissible temperature of the process, etc.

General approach The object of modeling for the analysis of the physicochemical

processes taking place during all of high temperature technologies is a system which includes the following phases: a metal, an oxide melt, a gas, and solid phases, in which various chemical and physical processes take place.

Mass and energy balance equations will be written for each of the phases. The material balance for any chemical element E in a given phase (kg E/s) (for example in the metallic phase) can be written in the general case as:

(1)

where mk is the rate of entry of the substance into the respective phase from one of the K incoming fluxes (kg/s), ml the rate of departure of the substance from the phase with one of the L outgoing fluxes, [E]0

k and [E] l are the mass concentrations of the element in the input and output fluxes (%), [E] is the mass concentration of the element within the phase in a given element of its volume dV (%), ME is the molecular weight of the element (kg/mol), VE

n is the rate of transfer of the element from the respective phase to one of the N phases interacting with the it on an element of the interface dAn (mol/m2·s); is the density of the phase within an element of volume dV (kg/m3).

Thus, the sum on the left-hand side of the equality is total rate of entry of the element into the respective phase with the incoming fluxes. The first sum on the right-hand side is the total rate of departure of the element with the outgoing fluxes; the second sum (the sum of the integrals over the surface An) is the total rate of departure or entry (in the case in which VE

n < 0) of the element to the neighboring phases as a result of a chemical reaction; the third term (the integrals with respect to

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the volume) is the rate of accumulation of the element within the phase as the concentration varies. Integral dependences appear in the equation, since the rates of the chemical reactions can differ [3] at different points on the interface due to the nonuniformity of the phase with respect to the temperature, composition and convection conditions, and the concentrations can vary with time with varying rates.

The mathematical model of the processes involved in the physicochemical interaction of the phases is based on the method of the kinetic analysis of the interaction of multicomponent metallic and oxide melts previously developed with participation of the author of the present work [43]. It will be used to solve the most complex problems in modeling, viz., consideration of the rates of transfer VE of all the elements through the phase boundaries, as well as consideration of the mutual influence of all the chemical reactions taking place on these boundaries. On the basis of this method it is possible to take into account the complex interactions between all components, i.e. their interactions with each other.

Method of kinetic analysisLet us perform a kinetic analysis of the reactions between metal

and slag which occur simultaneously in the diffusion mode. The mutual influence of the reactions and the diffusion of ah the reagents in the metal and slag are taken into account. The developed method of kinetic analysis is convenient for calculation and does not involve any assumptions of the form in which the elements in the metal and slag exist.

Its theoretical basis are two statements, namely: 1) in the diffusion mode, in all reactions, the ratio of concentrations at the metal -- slag interface for each reaction is close to that of equilibrium; 2) the rate of transfer of each reagent in the metal and slag towards or away from their interface is proportional to the difference of the concentrations in the phase volume and at the metal-slag interface.

Let us consider the method in a case of practical importance which is the oxidation of iron impurities by slag:

, (2)

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where [Ei] - Mn, Si, Cr, P, Ti, etc. Let us define the rate of the reaction (2) as Vi, mol/m2 · s, and limit ourselves to an analysis of the interaction of the homogeneous metal and slag when the reactions (2) occur at the their interface. In the diffusion mode, the ratio of concentrations near the metal - slag interface is given by the following expression:

. (3)

Hence,

,

(4)

where X is a mole fraction of reagents; Ki is a constant, including the equilibrium constant Kpi, activity coefficients γi, and coefficients of the recalculation of concentrations for the weight concentrations; [Fe]' and (FeO)' are the equilibrium mass concentrations of iron in the metal phase and iron oxide in the slag; [Ei]' and (EinOm)' are the equilibrium mass concentrations of i-element in the metal phase and their oxide in the slag.

In a steady-state mode, the reagent concentration at the interface will be equal to [43]:

(5)

where Vj, mol/(cm2 ·s) – the rate of mass-transfer of any j-reagent to metal-slag interface; = β·Dj

1/2·Cj is the limit diffusion flow; Cj is the concentration of any j-reagent into interaction phases, mol/cm3; Dj is the diffusion coefficient of any j-reagent, m2·s-1; β is the mass-transfer coefficient, s-1/2. In expression (5) the 'minus' sign corresponds to the initial reagents, and the 'plus' sign to the reaction products. Replacing the

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concentrations in the equations (5) by the weight concentrations and substituting them into the expression (4), we will have:

. (6)

It follows from the stoichiometry of the reaction (2) that:

(7)

Also, it should be taken into account that Vi is a proportion of the total diffusion flow of iron monoxide (VFeO) consumed in the oxidation of the impurity. Therefore,

(8)

Let us express VFeO and VEi through x using the equations (6) and (7):

, (9)

at n =1

; (10)

at n =2

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, (11)

where

.

Having substituted VFeO and VEi, in the expression (8), we will have an equation with one unknown - x. Having found x from the expressions (10) and (11) we can find VEi . Note, that the x value is related unambiguously to the oxidation potential (equilibrium pressure of oxygen) at the metal – slag interface. Indeed, from the equilibrium of the reaction:

,

it follows that:

. (12)

It should be noted, that where there are significant differences in the metal and slag compositions in the volume and at their interface, the equation (9) should allow for changes of the Ki values, which are, first of all, related to the change of the activity coefficients. Here, the method of successive approximations can be recommended, namely: first perform calculations for the values of Ki which correspond to the volume fractions of the phases and then, calculate from the equations (5) the content of the components near their interface (C*). Then repeat the calculation with more precisely defined values of Ki which correspond to the values of C* which are found.

It should also be noted that the results of the calculation will not depend on which, of all the possible reactions, have been considered. It is sufficient for them to be algebraically independent, include all the

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required components and contain one common reagent. This is related to the essence of the method itself which allows not only the rates of the reactions proper to be determined, but the flow of the reagents toward or away from the interface, i.e. of the rates of their transfer through this interface. The question of what are the adsorption-chemical processes that make such a transfer possible cannot be resolved by the diffusion mode of the reactions.

Examples of calculationLet us first consider a relatively simple case of simultaneous

oxidation of two elements by slag i.e. the simultaneous oxidation of manganese and silicon:

, (13)

. (14)

From equations (8)-(10), neglecting the retardation by diffusion in the metal, we will have:

, (15)

where x = (FeO)*. The following was assumed for 1600 o C: DFeO = =DMnO = 10-5 cm2/s, DMn= DSi = 10-4 cm2/s, DSiO2 = 0.10 DFeO , KMn = 0.33, KSi = 0.08 [12, 43-45].

Having substituted these values into the equation (15), we will have:

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For a melt of iron with l% Si, 1% Mn, and oxide melt with 25% SiO2 , 5% MnO, 15% FeO, 50% CaO, 5% MgO, we have found the

following values for the above ( ) oxidation rates:

IndividualSimultaneous

Oxidationoxidation

8.1 -2.9

8.7 10.2

The negative value indicates that, in the presence of silicon,

reduction and not oxidation of manganese, is taking place. This result agrees with experimental data [60], which show that in the same conditions, one can observe a decrease of the silicon concentration and increase of the manganese concentration in the metal phase. For comparison, reactions rates in individual oxidation mode have been calculated (see the table). The result shows that with separate oxidation,

the values of and are relatively close. It does not agree with

experiments.

Iron refinement by slagLet us consider this more complicated case, i.e. the simultaneous

transfer of six elements (Fe, C, Si, Mn, O, S) through the metal/slag interface. The diffusion mode is that which is the more probable with

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small concentrations of impurities in the metal. Let us assume, therefore, the following compositions of metal in wt%: 0.05 C, 0.1 Si, 0.1 Mn, 0.01 O, 0.02 S and slag: 40 CaO, 12 MgO, 8 MnO, 9 SiO2, 0.1 S, up to 31 FeO, these proportions being characteristic of the melting of pure iron .

The phase interaction here can be related by the following reactions:

; (16)

; (17)

; (18)

; (19)

. (20)

According to the expression (8), we have:

. (21)

Therefore, the equation (21) should be complimented by the expressions for VC, VO , VS. Neglecting the influence of CaO and Fe diffusion because of their high concentrations and CO diffusion in gas and since pCO 1 atm in the bubbles at the metal/slag interface, we will have, from equation (10):

; (22)

; (23)

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. (24)

In the calculations, in addition to the above data, the following values of constants for 1600oC were used: D[C] =13.3 ·10-5, D[S] = 6.2 ·10-

5; D[O] = 13.3 · 10-5; D[S]= 10-5, cm2/s; KC = [C]·(FeO) = 0.014; KS

=[S]/[(FeO)·(S)]=0.0025, KO = [O]/(FeO)= 0.00176 [12, 43-45]. Assuming that slag with 9% SiO2 is sufficiently close to a perfect ion solution, the ion activities were replaced by ion fractions and the latter were recalculated into the weight concentrations. Where there are significant changes in the FeO content (0 – 30%), such a recalculation yields a non-linear dependence of [O] on (FeO) i.e. variability of the K value. The averaged values KC ,KS and KO are assumed for calculations which differ from the actual ones by not more than 6%. After appropriate substitutions we have derived the following system of equations which can be regarded as a mathematical model of metal and slag interaction at 1600oC:

;

(25)

; (26)

; (27)

; (28)

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; (29)

; (30)

(31)

Equations (25)-(31) allow calculation of the rate of transition of the elements through the metal/slag interface with the assigned composition of slag, metal and the mass-transfer coefficients. The following equation should be used in order to find the dependence of the concentrations upon time:

,

where Q is the phase volume which contains the element j, S is the metal/slag interface. It is more convenient to integrate by the Euler method, assuming sufficiently small time intervals ( ) at which the rates can be regarded as constant:

.

This equation allows the , value and the concentration to be found by the start of the next interval. Figure gives the dependencies of Vj on the iron monoxide concentration in the slag, computed from equations (25)-(31) for the above compositions. For comparison, the same plot shows in the form of dashed lines, the rates in a case of individual oxidation of the impurities which were also found from the

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equations (25)-(31) with zero concentrations of other impurities in the metal and slag.

ConclusionThe mathematical model of high temperature physicochemical

processes on the basis of the kinetic analysis method had been developed.The model takes into consideration:- stage-by-stage implementation of the process;- continuous renovation of interacting phases;- simultaneous running of all reactions and their mutual influence;- physico-chemical properties of the interaction phases;- interconnection between technological process parameters and

kinetics of reactions.The proposed method can be applied to the development of new

compositions of special alloys and welding materials (electric coatings, flux cored wires, welding fluxes). The practical implementation of this approach is considered.

Figure. The dependencies of Vj on the iron monoxide concentration in the slag

References

167

1. Matsumiya T., Nogami A., Fukuda Y.,Applicability of molecular dynamics to analyses refining slags, ISIJ Int., 1995, 33,1, 217-223.

2. Boronenkov V., Zhadkevich M., Statnikov B., Salamatov A., Zalomov N.,Mathematical modeling of the physical-chemical processes of the evaporation and degassing in electron-beam remelting of alloys, Vide, Couches Minces, 1992, 261, 74-76.

3. Shanchurov S., Boronenkov V.,The determination of mass transfer parameters between the metal and slag by physical modeling methods and in the real process, Vide, Couches Minces, 1992, 261, 77-79.

4. Matsumiya T., Mathematical analyses of segregation and chemical compositional changes of nonmetallic inclusions during solidification of steels, Mater. Trans., JIM, 1992, 33, 9, 783-794.

5. Ducharme R., Kapadia P., Dowden J., Williams K, Steen W. ,An integrated mathematical model for the welding of thick sheets of metal with a continuous CO2 laser, Laser Inst. Am., 1994, 77, 97-105.

6. Boronenkov V., Zhadkevich M., Shanchurov S., Yanishevskaya A., Mathematical model of chemical processes in centrifugal electroslag casting, Metally, 1993, 5, 35-42.

7. Davydov Yu., Boronenkov V., Salamatov A., Prediction of the weld formation of variable composition based on modeling of metallurgical processes, Autom. Svarka, 1992, 7-8, 23-26.

8. Norrish J., Gray D., Computer simulation and off-line programming in integrated welding systems, Weld. Met. Fabr.,1992, 60, 3, 119-122.

9. Kozlovsky S., Modeling of the interaction of parts in the contact area in spot welding , Isv.VUZ. Mashinostr.,1990, 9, 89-94.

10. Pelton A., Modeling the thermodynamic properties of slags, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992, 122.

11. Yamada W., Matsumiya T., Calculation of phase diagrams for oxide slags by thermodynamic models, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992, 133.

168

12. Boronenkov V., Shalimov M., Shanchurov S., Method for analysis of the kinetics of simultaneously occurring electrode reactions under nonsteady-conditions, Rasplavy,1994, 5, 12-17.

13. Zalomov N., Boronenkov V., Calculation of activities and ionic composition of multicomponent silicate melts, Rasplavy, 1991,3, 39-42.

14. Gaye H., Lehman J., Matsumiya T., Yamada W., A statistical thermodynamics model of slags: application to systems containing S, F, P2 O5 and Cr-oxides, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992, 126.

15. Zalomov N., Boronenkov V., Lyudmilin A., Application of the polymer theory to multicomponent silicate and alumino-silicate melts, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992,144.

16. Iida T., Kita Y., Morita Z., Estimation of the Enthalpy of Evaporation of molten slags and fluxes based on a harmonic oscillator model, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992, 147.

17. Matsumiya T., Nogami A., Fukuda Y., Applicability of molecular dynamics simulation to analysis of slags, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992,128.

18. Shakmatkin B., Vedishcheva N., Shultz M., Simulation of thermodynamic properties of slag melts, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992, 120.

19. Zaitsev A., Mogutnov B., The theory of associated solutions in thermodynamics of metallurgical slags, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992, 121.

20. Grigorenko V., Kiselev O., Chernyshov G., Mathematical model and its practical evaluation for weld formation, Svar. Proizvod., 1994, 2, 30-32.

21. Tsybulkin G., Mathematical models in adaptive control of arc welding, Autom. Svarka, 1994, 1,24-27.

169

22. Cerjak H., Easterling K. E., Mathematical modeling of weld phenomena, 1993, London, 369p.

23. Vitek J., Zacharia T., David S., Rappaz M., Boather H., Modeling of single-crystal laser-weld microstructures, Laser Mater Process.,Proc. Sypm. TMS, USA, 1994, 213-220.

24. Ogawa Y., Huin D., Gaye., Tokumitsu N., Physical model of slag foaming, ISIJ Int., 1993, 33, 1, 224-232.

25. Blander M., Pelton A.,Eriksson G., Analyses and predictions of thermodynamic properties and phase diagrams of silicates, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992, 117.

26. Jahanshahi S., Wright S., Aspects of the regular solutions model and its application to metallurgical slags, Proceedings of the 4th Molten Slags and Fluxes International Conference, Sendai (Japan), 1992, 118.

27. Banya Sh., Hino M., Nagasaka T., Estimation of hydroxyl capacity of molten silicates by quadratic formalism based on the regular solution model, Tetsu to Hagane, 1993, 79, 1, 26-33.

28. Pelton A., Talley P., Shama R., Thermodynamic evaluation of phase equilibria in the calcium chloride-magnesium chloride-calcium fluoride-magnesium fluoride system, J. Phase Equilib., 1992,13, 4, 384-390.

29. Matsumiya T., Nogami A., Sawada H., Monte carlo simulations of intermetallic compounds, Adv. Mater. Processes,1995, 147, 2, 51-53.

30. Iita T., Kita Y.,Okano H., Katayama I., Tanaka T., An equation for the vapor pressure of liquid metals and calculation of their enthalpies of evaporation, High Temp. Mater. Processes, 1992, 10, 4, 199-207.

31. Zuo Y., Chang Y., Calculation of phase diagram and solidification paths of aluminium-rich aluminium-magnesium-copper ternary alloys, Light Met, 1993, 935-942.

32. Wu P., Eriksson G., Pelton A., Blander M., Prediction of the thermodynamic properties and phase diagrams of silicate systems-evaluation of the iron (II) Oxide-magnesia-silica system, ISIJ Int.,1993, 33, 1, 26-35.

170

33. Blander M., Bloom I., A statistical mechanical theory for molten silicate solutions, Proc. Electrochem. Soc., 1994, 94-13, 1-7.

34. Van Niekerk W., Dippenaar R., Thermodynamic aspects of sodium oxide and calcium fluoride containing lime-based slags used for the desulfurization of hot metal, ISIJ In., 1993, 33, 1, 59-65.

35. Xiao Y., Holappa L., Determination of activities in slags containing chromium oxides, ISIJ Int., 1993,33, 1, 66-74.

36. Computer assisted materials design and process simulation, Proceedings of International Conference COMMP-93, 1993, Tokyo, Japan.

37. Dowden J., Ducharme R., Kapadia P., Clucas A., A mathematical model for the penetration depth in welding with continuous CO2 lasers, Laser Inst. Am., 1994, 79, 451-460.

38. Panphilova L., Zinigrad M., Barmin L., Effect of surface concentration of oxygen in Me-S melts on the kinetics of its transfer through a sulfur-oxide melt interface, J. Phys.Chem., 1978, 5, 10, 2491-2494.

39. Flyagin A. Zinigrad M., Barmin L., Kinetics of ion exchange between an iron-carbon-aluminium melt and an oxide electrolyte, Electrochem., 1979, 5, 12, 1858-1861.

40. Panphilova L., Zinigrad M., Barmin L., Quick stage kinetics of oxygen ion discharge on the boundary of sulfide melts and liquid oxides, Electrochem., 1981, 17, 9, 1346-1349.

41. Zinigrad M., Phephelov A., Barmin L., Shalimov M., Kinetics of the interaction of a boron containing metal melt with an oxide electrolyte, Electrochem., 1986, 22, 1, 74-76.

42. Zinigrad M., Okolzdajev A., Flyagin A., Limiting stage of anodic oxidation of tungsten at the boundary of metallic and oxide melts, Rasplavy, 1988, 2, 3, 46-51.

43. Boronenkov V., Shanchurov S., Zinigrad M., Kinetics of the interaction of multicomponent metal with slag under diffusion conditions, Izvestiya Ac. Nauk USSR. Metal, 1979, 6, 21-27.

44. Boronenkov V., Zinigrad M., Shalimov M., Mathematical modeling of metal and slag processes interaction in a ladle Izv.VUZ. Tcher. metallurg., 1983, 1, 36-4.

45. Zinigrad M., Simulation of metal and slag interaction for optimization and development of technological processes,

171

Proceedings of the 4th Molten Slags and Fluxes International Conference, 1992, 125.

46. Hang GuangWei,Fend Di, Ye Wujun, Luo Heli, Mathematical model on ductile-brittle transition of metallic materials, Proceedings of the International Conference on Modelling and Simulation in Metallurgical Engineering and Materials Science, Beijing (China), 1996, 744-748.

47. Kim Y.S., Yoon J.K., A model study on the simulation dephosphorization reaction in the combined blowing converter, Taehan Kumsok Hakhoechi , 1992, 30(3), 256-61.

48. Grong O., Kluken A.O., Nylund H.K., Hjelen J. Andersen I. Mechanisms of cicular ferrite formation in low-alloy steel weld metals, SINTEF Rep.,1992

49. Grong O., Kluken A.O., Microstructure and properties of steel weld metals,Key Eng.Mater.,1992, 47-93.

50. Olson David L., Influence of welding flux on the pyrometallurgical; physical and mechanical behavior of weld metal, Gov.Rep. Announce. Index (U.S.), 1986, 86(21). Abstr.No.647,023.

51. Olson David L., Lui Stephen, Edwards Gl. R.,Role of solidification on HSLA steel weld metal chemistry, Weldability Mater.,Proc.Mater.Weldability Symp.1990,183-9.

52. Olson David L., Matlock D.K., The role of composition and microstructure gradients on weld metal properties and behavior, Energy, Res. Abstr., 1987, 12(16), Absstr. No.33161.

53. Olson David L., Matlock D.K., Compositional gradient effects on weldment properties, Weldability Mater., Proc. Mater. Weldability Symp., 1990, 49-55.

54. Eager Thomas W., The physics and chemistry of welding processe, Adv. Weld. Sci. Technol., Proc. Int. Conf.Trnds. Weld.Res., 1986, 281-8.

55. Mitra U., Eagar T.W., Slag-metal reactions during welding: Part I. Evaluation and reassessment of existing theories, Metall.Trans. (B), 1991, vol.22, No 1, 65-71.

56. Mitra U., Eagar T.W., Slag-metal reactions during welding: Part II. Theory, Metall.Trans. (B), 1991, vol.22, No 1, 73-81.

172

57. Mitra U., Eagar T.W., Slag-metal reactions during welding: Part III. Verification of the theory, Metall.Trans. (B), 1991, vol.22, No 1,83-100.

58. Olson D. L., Lui S., Edwards G. R., Physical metallurgical concerns in the modeling of weld metal transformations, Math. Modell. Weld. Phenom., 1993, 89.

59. Liao F.C., Lui S., Olson D.L., Effect on titanium nitride precipitates on the weldability of nitrogen enhanced Ti-V microalloyed steels, 1994 , 31,511-22.

60. Erokhin A.A., The Fundamentals of Fusion Welding, Mashinostroenie, Moscow, 1973.

173