Kinetic model of high temperature physicochemical processes

M. Zinigrad

College of Judea and Samaria, Science Park, Ariel, 44837 Israel

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.

Introduction

One 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.

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 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 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.

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;

- 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]0k 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), VEn 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); r 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 VEn < 0) of the element to the neighboring phases as a result of a chemical reaction; the third term (the integrals with respect to 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 analysis

Let 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)

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: