Numerical Investigations of Equilibrium

Numerical Investigations of Equilibrium Concentrations for Oxide Mixtures in Thermal Treatment Oleg Chizhko Moscow Environmental Center, Post-Box 88, 369000 Cherkessk, Russia, [email protected] Abstract. According to DSC-measures (differential scanning calorimetry) of warm capacities we study the thermodynamic properties of Tl-species, produced with thermal treatment of multi-oxide systems of combustion slag. They are the aluminates of thallium, complex crystals and the simple minerals (anorthite, melilith, pyroxens, which include the silicates such as mullite and andalusite). The practical application for these materials is the anti-corrosion cover for constructional elements of burning chamber. A comparison with the Al2O3-layers gives us a chance to propose the Tl2O3-containing composites because of their higher electro-chemical potentials for building of the geological mixtures against an aggression of hot gases. We suppose that the protective films should be created with a participation of the fly and bottom ach ingredients, but not with a permanent degradation of aluminum alloys. The second advantage is the conversion of mechanical properties for the metal acid substances with improved packing symmetry and higher viscosity characteristics of coatings from these components. We should not refuse a light toxic effect of thallium aluminates for prevention of micro-biological corruption too. In the investigations the heterogeneous solid-gas equilibriums were regarded with input of fugacities in terms of calculation for the chemical potentials. We introduce the ab initio computations of enthalpies for gaseous phase; kinetic mechanisms of formation for structures with concentration gradients; implementation of the calphad technique for the modeling of plasma sprays, molten state and condencing matter; distribution of software for metal rest substances. MeRS-data is the database operating and information system, produced and designed with aim of estimation for equilibrium concentrations and treatment for experimental results, their depositing and presentation in the form of finite element graphics, tables and three dimensional diagrams.

I. INTRODUCTION AND ACTUALITY OF INVESTIGATION The main target of the scientific work is the idea of iso-morphous incoming for the thallium oxide substances into the Al-rich minerals according to the periodic table Tl 3+Al3+. With opportunity of high temperature production for anti-corrosion coatings we introduce: 1. the XRD (x-ray difractometry) structural analogy of condensed bodies from the aluminium and thallium oxides; 2. the higher reaction ability of Tl2O3 in compare with Al2O3, which initiates the electro-chemical potential of formation for the TlAl3O6 composites of improved packing symmetry of their cubic crystal cells; 3. higher viscosity characteristics of covers from these species; 4. defense against aggression of waste combustion gases; 5. light toxic effect of thallium aluminates for prevention of micro-biological corruption. The main tasks of the paper are 1. correct formulation of start tasks and mathematical accuracy of computations; 2. permanent dependence of solution methods on input thermodynamic dates for mixed oxide crystals, spinels, complex chemical compounds and main minerals; 3. consistence of theoretical constructs and their convergence with experimental results; 4. implementation of numerical schemas for description of local chemical potentials, which take part during the building of waste systems for the thermal processes; 5. independent model approach for regarding of solid-gas equilibriums in terms of estimation of Global minimums for separate chemical contents; 6. new software for calculation of phase diagrams.

II. THERMAL TREATMENT OF FURNACE SLAG SYSTEMS Experiments of thermal treatment for a fuel from waste were carried on grates of the incineration plant TAMARA in the Karlsruhe Institute of Technology. They proved a building quality of bottom ashes, consisted from the main minerals: pyroxenes, melilith, wollastonite. At the higher concentrations of aluminum oxide the phase content was characterized with anorthite and andalusite. A stability of the slag against ground water depended on a binding of toxic substances in silicate matrix because of a formation of the complex composites from aluminates.

Homogenization of the molten products was prevailed at 2000°C with aim of its purification in a periphery, where the active additives diffused according to difference of the chemical potential between a solid phase and oxygen atmosphere. Berthod [1] introduced a softening of surface layers for steels as a result of the reaction ability of carbides during a contact with aggressive environments. Kjellqvist [2] described the same effect of lowering for the sulphur and aluminum content in an alloyed steel by the addition of a slag. On the other hand there were welding and sintering processes of formation of glassed films for local conglomerations of oxide compounds with metallic educts in the case of burning with lower coefficients of air excess. Analogous enriching of Tl in pyroxenes and anorthite was confirmed by [3] as a consequence of spontaneous geological activities.

II.1. Thermodynamic and mathematical modelling of oxide composites In a form of the cubic spline let us write the Gibbs energy, which excludes standard building enthalpy  ik H 0  and is divided on temperature

T    0  1T   2T 2   3T 3   4 ln T  .

(1)

It interpolates experimental results of DSC measurements of heat capacities for condensed compositions with help of coefficients  0 , 1 ,  2 ,  3 ,  4 . According to the expression (1) we calculate activities of the defined above metal rest substances (MeRS), arising and disappearing at the stable conditions of a constant elementary content, pressure and temperature. Selection of an i-substance from the inserted database is succeeded with the qualitative analyze of elements in chemical formulas. The number K of all possible reactions depends on the quantity C of components, which participate a common process of thermal transformation, and we use it during the estimation of stoichiometric coefficients. The equations for potential differences of formation of the i-compound during conglomeration of oxide mixtures

i     ik G n, p, T  RT   ik Фn, p, T  R   ik H 0  RT

(2)

describe the closed isobar-isothermal system of the crystal bodies at a global minimum. An opportunity to describe solid-gas interactions describes the case of combustion for energy fuels on the basis of aluminum with thallium technological ingredients, which takes place at the boundary between the both phases and after the condensation of ionized Tl-Al-O agglomerates at the surface, defended with anti-corrosion coating. These inhomogeneous relations, characterized with existential gradient (diffusion) and temperature degradation (cooling) in duration of time, we simulate with help of fugacities. They are the source of a power, influencing the substance at the heterogeneous equilibrium of electrical essences and mineralogical existences. Thus, the complex chemical potentials for a contribution  ik of k-reactions are

ik  i    `ln  f ik p  . k   k 1 It inputs the moving engine f ik  ik xi p K

(3) of the inter-phase transformation together with the partial pressures

pik  xi p of gas molecules above a solid solution. And the formula for computation of an isothermal activity is

ai 

ik

(4)

C

 ik i 1

According to the regarded formalisms we introduce difference schemas for the boundary conditions of elementary balances [4] in a system of the i-linear differential equations Ah ci b j  ai ; 0  b j   j ; ci 0   0; ci  j  i ;  j ,i  I  0,   , (5)

 

where

 j is the molar break of an j-element.

 

The key value ci  ai

h

li

is obtained with Ritz’s variation method in approximation of the discretization

l





error 10-7. The mathematical model (5) is stable at the interval 0,  j , if we use all calculated parameters of the inconstant width of the step hl. Thus, at each l-iteration on the fixed temperature level the concentration of the substance is equal to the multiplication of its activity and hli. On the figure 1 the estimation of the system Al2O3-Tl2O3 are shown at the temperature 1200 K of thermal treatment in the dependence of the aluminum oxide quantity so that the total molar sum of the oxide content is equal 1.

FIGURE 1. Two dimentional phase diagram for thallium aluminates

II.2. Interpolation and assessment of experimental results for heat capacities of thallium aluminates We search a solution of the differential equation AhT   f T , T  , 298 ,15   AT0 ,

(6)

which was evaluated with the approximation (4) at the interval T  I T0 , T0  H  . Taylor's operator Ah of morder is implemented for precision of the interpolation quality and improves the splining results of modified Euler's, Heun's, Runge-Kutta's terms of 1,2 and 4 orders. А298,15 - value of the function at 298,15 K. The realization of the I  T0 , T0  H  and the condition start task (6) exists because (1) is continuous at

f T , T   f T , ' T   L T   ' T  converges [5]. In the table 1 the convergence of discretization

consistence of the methods

 nm 1   n

m 1

  nm1



hn



enm 1  max Tn    nm 1 T  is presented together with the 0 n  N



 f Tn 1 , nm1 T  , n  0,, N ,

in correspondence with the assessment

(7)

enm 1  enm1  hnm 1L enm1  hnm 1  nm 1 .

TABLE 1. Local and global errors of discretization

Method Step width h 0,01 0,001 0,0001

Euler |τh| 8,3666234391 0,6296028597 0,0607887368

Heun |τh| 0,6163134392 0,0048414538 4,7448783604E-5

Runge-Kutta |τh| 0,0706482156 4,7279124686E-6 4,4970249746E-10

|еh| 0,2707517365 0,2115998264 0,2096778906

In the following we are going to statute a strategy, according which the step width of our calculations is selected so that the error tolerance L > 0 confirms the condition

max enm1  L .

(8)

Tn  I

Thus, for a practical assessment of application of a numerical method it’s necessary to use the sharpest resolution of Taylor’s term. In our investigations we implement the common regular of halfing for the step. An important part of this strategy is the computation of consistence H /2 H  nm11/ 2   nm12  OH m  2   n 1  n 1

H

m 2



 O H m 2

.

(9)

The comparison of (8) and (9) gives us the final formula

hnm 1  L

H m 1  nm 1

(10)

III. ELECTRO-CHEMICAL OPTIIZATION OF THERMODYNAMIC MODELS For the CaO–Al2O3, Al2O3– SiO2 and CaO–Al2O3– SiO2 systems the quasi-chemical approach of substances (CaO, AlO1.5, SiO2) was introduced by [21]. [22] applied the sublattice model of the (Al3+, Ca2+)p(O-2)q ionic liquid for calcium and aluminum oxides. The polymerization in the CaO-SiO2 melts was developed with help of the (Ca2+)p and (SiO4-4, SiO2, O-2)q species [6]. If Al2O3 presented in these mixtures, so it was simulated with the formula (Al3+)1(AlO2-1)3. Another solution constructions for calculation of bottom ash compounds were investigated by [7]. [8] used the triangular stoichiometry of Al2O3 in compare with tetrahedral of [9]. They described the reaction of Al3+ with O and formation of AlO2-2 species. The modified associate species model as well as the four sublattice approach have been applied to the mullite phase in the work [10]. It reported about mullite with the composition 3Al2O3·2SiO2, which is the only stable compound up to 2163 K, where it melts congruently. In correspondence with [11] an eutectic point of cristobalite was equal 1868 K. The calcium silicates Ca2SiO4, Ca3Si2O7, CaSiO3 were in the molten state at 2154, 1464, 1540(1437) °C. [12] described the main minerals of the combustion slag for municipal solid waste. They were pyroxenes and melilith. Anorthite was an ingredient of the bottom ash, characterized with a high Al2O3 content of 16,43% and the liquidus temperature 1252 °C. The DSC measures of the transformation from molten to solid state presented the dates of 1310 °C for CaAl2SiO6 and 1350 °C for Ca2AlSi2O7,5. The experimental special points along the liquidus surface of [6] gave us 1555 °C for anorthite and 1595 °C for gehlenite. The aluminates Ca3Al2O6, CaAl2O4, CaAl4O7, CaAl12O19 are listed at 1812, 1875, 2035, 2103K in the form of an invariant equilibria with the liquid phase in the CaO–Al2O3 system [13]. According to Hillert’s model of the ionic two-sublattice compounds we regard the binary Al2O3-Tl2O3 system and use the formula (Tl3+)p(AlO2-1)q with aim to describe the electro-chemical process of their condensation and formation of crystal structure. The coefficients P=1 and Q=3 satisfy the condition of electro neutrality in correspondence with Mao [13]. Thus, the Gibbs-Helmholtz energy of the solid solutions is Gm = P yTl(3+) ∙°GTl(3+) + Q yAlO2(-1)∙RT ln kAlO2(-1) + EGm

(11)

y is mole brake of a substance, kAlO2(-1) – reaction constant for building of AlO2-1. Excess energy EGm introduces interactions between the sublattice units and we calculate it with help of finite element models for the crystal structures of thallium aluminate. On the figures 2a and 2b the temperatures of congruent melting for CaO-Al2O3-SiO2 compounds are drawn with help of Matlab 6 Software in the form of finite element objects. They plot triangular parts of the Gibbs surfaces for condensation of the complex substances from the systems Al2O3-Tl2O3-SiO2 and CaO-Al2O3-Tl2O3. A crystallization of Tl-containing minerals was interpolated according to its isomorphic incoming Tl3+Al3+ in correspondence with the periodic table of chemical elements. Three dimentional (3D) format of these pictures reproduces all features of transformation of mixture oxide agglomerates from liquid to solid state on the single graphic with possibility to change a point of viewing in compare with slices, which are complicated on the 2D-

planes. Thermodynamic databases and mathematical models are implemented in the Object Pascal Program «MeRSdates» and realized with the software package.

FIGURE 2. Solius-liquidus surfaces for the systems: a) Al2O3-Tl2O3-SiO2 and b) CaO-Al2O3-Tl2O3

IV. DISCUSSION The experiments for the thermal incineration of municipal solid waste show the good quality of the combustion products for using in the building tools. The main components are the minerals with the increasing content on aluminum. The chapter II presents the x-ray fluorescence analysis for the total concentrations of elements in slag and the diffraction confirmation for crystallization of pyroxenes, anorthite and melilith [12] in the burning chamber. The relative high inventory of toxic thallium in these materials gives out a danger for the environmental objects and we develop the database for physical properties with aim to investigate the stabilization of thallates and their isomorphic incoming Tl3+Al3+ in the aluminate composites of the bottom ash. According to the structural analogy of Al2O3 and Tl2O3 we notice that an addition of the thallium oxide to the aluminum rich copounds improves their polymerization characteristics (viscosity and porosity) because of the higher reaction ability of Tl2O3 and tightly-filled packing symmetry of its cubic cell as the result of formation for the TlAl3O6 substances. The lower thermochemical properties of the thallium aluminates are compensated with the binding quality of the Tl3+-AlO2- essences and such surface coating is seemed to be more stable against hot products of waste combustion than the water soluble CaO and MgO containing crystals. An inserting of silicon to Al2O3 decreases the exploitation period of the protective film, according to the amorphous structure for andalusite and mullite durig the condensation processes. The limiting factor in these considerations is a saturation of thallium oxide activity to conglomerate the Al2O3 pigments, which are finer than SiO2. Thus, the activity of sand is not enough to coat a defended interface, but after mechanical treatment we become an optimal complex of the physical properties for our anti-corrosion covers. In this department we’d like to pay some attension to the plasma method of treatment for thin layers, which is characterized with high pressures for the processes of formation of complex oxide systems. Such conditions of production for mixture crystals gives us a chance to reach a new polymorphous transformations for common substances, participating the mieral compounds of emeraldite & sapphirite. Let us regard a generator of plasma areas, which are initiated with an arc burning between aluminum electrodes in an environment of thallium containing substances. There is the case of combustion for energetic Al-fuels with Tl technological additives, but in the building of the ionized gas the electrical interactions take place. We calculate a static pressure of these chemical systems in correspondence with the term for temperature at a constant volume. It becomes higher and the reaction products are moving in the direction to a surface, which is prepared for covering. Now we'd like to notice that in the acceleration of the plasma propulsion a dynamic part of pressure plays the main role of executing engine for space velocities. There is the force, which acts to the conducting ions of the aluminum and thallium oxide compounds. This magneto-hydro-dynamic facility of plasma is not only the source of the working body for anti-corrosion coatings, but also a high-speed launcher of macroscopic volumes for the prepared mixtures. In the conditions of high pressures of 40 MPa for our ionized conglomerates during the technological collision with protected materials we expect the invention of new modifications for the routine substances, the crystal cells of which introduce the better

packing symmetries, improving the mechanical properties of the thallium aluminates. In these extreme conditions we combine a Tl+Al-quantity of the agglomerates with opportunity of preparation for homogeneous compositions of the stoichiometric oxide content and the precise mixture for building of the multi-lattice models of complex compositions. A permanent comparison of the experimental quantities for the oxide crystals in combustion slag with calculation results for the equilibrium concentrations of pure substances in the systems Al2O3-SiO2, CaO-Al2O3 and CaO-Al2O3SiO2 involves us in the routine of description for the individual features of tested events so that we input the fugasities in the terms of estimation of chemical potentials for the formation reactions, which describe better the real thermal processes. Thus, we introduce the empiric formulas for the research case of congruent melting of the ash compounds and correct the thermodynamic models according to the kinetic mechanisms of building for computed contents. The localization of intermediate complexes was constructed with help of the diffusion approach for concentration gradients. There is an analytical elaboration, which tasks are the deposition of experimental databases and the numerical prognosis of future investigations. This concequence for the accumulation of theoretic foundations is the first step of the genesis for the new from experience inducted invention. It is the stage of reevalution for coputational algorithms. The second stage of dialectical conclusions is grounded on the calphad technique for calculation of equilibrium concentrations of ionic liquids. We implement it with aim of interpretation for mineral agglomerates, amorphous structures, pressre capacities, electric and magnetic anomalies at the phase boundaries. These tasks require a multipling of sublattice applycation for complex crystals [6], of disordered state of the gluons for galassy matter [14], of phase fields for substrate response [15], of degrees of freedom for electromagnetic fields [16]. Another problem is a coupling of DSC-measure results of the warm capacities for condenced bodies and spectral dates for physical constants of pure substances in gaseous state. In the last case of statistical summation for enthalpies we use the ab initio principles for establishing of the highest levels of electronic, vibration and rotation interactions of atoms in a molecul. There are the limiting factors of registration for atomic symerties and of compilation for correction terms of energetic relation. The mathematical simulation shows a better compatibility of the dates, when the solid-gas equilibriums are regarded in terms of the estimation of the additive activities for all reactions, in which the species were produced. For the example of the aluminum powder with thallium-containing technological ingradients we calculate the Gibbs energies according to the model construction for the contribution of the potential of each agglomerate during the transformation of the metallic propellant in oxygen atmosphere. We propose the methodic of the electro-chemical approximation for numerical investigations of the activities of ionized plasma existences after burning of the Al- fuels and during condensation of the combustion products on a surface. The deference schemes of these differenial equations are constructed with the boundary conditions for computation of equilibrium concentrations in high-T chemical reactors and solved with modified Ritz’s iteration method so that the rounding error is eliminated till the range of E-7. We develop the control strategy for an inconstant width of the step, according which the tolerance of approximation is minimized. The consistence of implemented terms has the higher order of date quality than after spline interpolation. The dependences of mole breaks for the systems Al2O3Tl2O3-SiO2 and CaO-Al2O3-Tl2O3 on temperature are visualized in the form of finite element surfaces. It is necessary to conclude that convergence stability of the assessment (7) is limited with the value of of this argument expands the expression (9) to estimation of the discretization error

 nh

hn  1

. The precision

2L  hn max Ф' ' T  , which T I n

does not fit the inequality at initial temperatures and h=0,01. The problem disappears at h=0,001. The mathematical model and the empirical consequences were realized with the Object Pascal program.

REFERENCES [1] Р. Berthod, S. Michon, L. Aranda, S. Mathieu, J.C. Gachon, Thermodynamic calculations for studying high temperature oxidation of superalloys, Calphad 27 (2003) 279-288, DOI: 10.1016/j.calphad.2004.01.001 [2] L. Kjellqvist, M. Selleby, Bo Sundman, Thermodynamic modelling of the Cr–Fe–Ni–O system. Calphad 32 (2003) 577-592, DOI: 10.1016/j.calphad.2008.04.005 [3] R.V. Goleva, V.V. Ivanov et al, Environmental assessment of potential toxity of rural resources, University Press, Moscow, 2002. [4] H. Bockhorn, Principles of Combustion Technique, University Press, Karlsruhe, 1999. [5] R. Rannacher, Numerical Mathematics, University Press, Heidelberg, 2012.

[6] M. Hillert, H. Mao, M. Seleby, Thermodynamic assessment of the CaO-Al2O3-SiO2 system, J. Am. Ceram. Soc. 89/1 (2006) 298-308, DOI: 10.1111/j.1551-2916.2005.00698.x [7] R. G. Berman, T. H. Brown, A Thermodynamic Model for Multicomponent Melts, with Application to the System CaO–Al2O3– SiO2, Geochim. Cosmochim. Acta, 48/4 (1984), 661–678. [8] D. K. Belashchenko, L. V. Skvortsov, Molecular Dynamics Study of CaO–Al2O3 Melts, Inorg. Mater. 37 (2001), 476–481. [9] M. Benoit, S. Ispas, Structural Properties of Molten Silicates from ab initio Molecular-Dynamics Simulations: Comparison Between CaO–Al2O3– SiO2 and SiO2, Phys. Rev. B 64 (2001), 224205. [10] E. Yazhenskikh, K. Hack, M. Müller, Critical thermodynamic evaluation of oxide systems relevant to fuel ashes and slags, Part 5: Silica–alumina system, Calphad 32 (2008) 195–205, DOI: 10.1016/j.calphad.2007.05.004 [11] H. Mao, M. Seleby, B. Sundman, Phase equilibria and thermodynamics in the Al2O3-SiO2 system. Modeling of mullite and liquid, J. Am. Ceram. Soc. 88/9 (2005), 2544-2551, DOI: 10.1111/j.15512916.2005.00440.x [12] Ebener S. Technische und hydrolytische Qualität von Schlacken aus der thermischen Abfallverbrennung durch Hochtemperaturverfahren. Wissenschaftliche Berichte FZKA 6195, 1999, s. 169. [13] Huahai Mao, Malin Selleby, Bo Sundman, A re-evaluation of the liquid phases in the CaO–Al2O3 and MgO–Al2O3 systems, Calphad 28 (2004), 307–312, doi:10.1016/j.calphad.2004.09.001 [14] Steve Feller and Ugur Akgun, A New Glassy State of Matter: The Color Glass Condensate, International Journal of Applied Glass Science 4/1 (2013), 1–4, DOI:10.1111/ijag.12013 [15] Nils Warnken et al. A modelling approach high temperature oxidation and substrate response of Alumina forming Superalloys, Internationale XLII Calphad Conference, San Sebastian, Spain, 2013. [16] Tilmann Hickel, Biswanath Dutta, Fritz Körmann, Jörg Neugebauer, Coupling of magnetic and lattice degrees of freedom: Ab initio derived consequences for phase diagrams, Internationale XLII Calphad Conference, San Sebastian, Spain, 2013.