This article is an analysis of the results obtained under the tests aimed at determination of the influence exerted by the current frequency of an induction furnace on the mass transfer coefficient for liquid metallic phase.U radu su analizirani rezultati dobiveni testiranjem provedenim s ciljem utvrditi utjecaj frekvencije industrijske peći na maseni koeficijent prijenosa u tekućoj metalnoj fazi

B. Oleksiak

L. Blacha

R. Przyłucki

S. Golak

U radu su analizirani rezultati dobiveni testiranjem provedenim s ciljem utvrditi utjecaj frekvencije industrijske peći na maseni koeficijent prijenosa u tekućoj metalnoj fazi.This article is an analysis of the results obtained under the tests aimed at determination of the influence exerted by the current frequency of an induction furnace on the mass transfer coefficient for liquid metallic phase

Blacha, L.

Oleksiak, B.

Golak, S.

Przyłucki, R.

Hrčak - Portal of scientific journals of Croatia

67METALURGIJA 51 (2012) 1, 67-70R. PRZYŁUCKI, S. GOLAK, B. OLEKSIAK, L. BLACHAINFLUENCE OF AN INDUCTION FURNACE’SELECTRIC PARAMETERS ON MASSTRANSFER VELOCITY IN THE LIQUID PHASEReceived – Prispjelo: 2011-01-12Accepted – Prihvaćeno: 2011-03-30Preliminary Note – Prethodno priopćenjeISSN 0543-5846METABK 51(1) 67-70 (2012)USD 669:669.3:66.067=111This article is an analysis of the results obtained under the tests aimed at determination of the infl uence exerted by the current frequency of an induction furnace on the mass transfer coeffi  cient for liquid metallic phase.Key words: metallurgy, cooper, refi ning, alloy, induction melting, mass transfer.Utjecaj parametara indukcijske elektro peći na brzinu masenog prijenosa u tekućoj fazi. U radu su analizi-rani rezultati dobiveni testiranjem provedenim s ciljem utvrditi utjecaj frekvencije industrijske peći na maseni koefi -cijent prijenosa u tekućoj metalnoj fazi.Ključne riječi: metalurgija, bakar, rafi nacija, slitina, indukcijsko taljenje, prijenos maseINTRODUCTIONComprehensive analysis of the chemical phenom-ena taking place in the liquid metal/gaseous phase sys-tems typical for metallurgy requires knowledge of the mass transfer phenomena occurring in both the said phases. This results from the fact that in heterogene-ous systems, the components of the individual phases react with one another only after penetrating the inter-face. The parameter characterising the mass transfer velocity in the given phase is the so-called mass trans-fer coeffi cient. Knowing its value with reference to both phases as well as knowing the constant velocity of the reaction taking place in the interfacial area ena-bles calculation of the general mass transfer coeffi cient values characteristic for the given process but also identifi cation of the stages determining the coeffi cient. This paper is an analysis of the results obtained under the tests aimed at determination of the impact exerted by the selected electric and geometric parameters of the furnace on the mass transfer coeffi cient for an in-duction-stirred liquid metallic phase.MODELS OF MASS TRANSFER IN LIQUID PHASEThe fi rst hydrodynamic model applied to describe the phenomenon of mass transfer in a liquid phase was the model of boundary fi lm developed by Nernst [1]. R. Przyłucki, S. Golak - Silesian Technical University Department of Management and Informatics, Katowice, Poland B. Oleksiak, L. Blacha - Silesian Technical University Department of Metalurgy, Katowice, PolandThe model was based on the assumption that concentra-tions of all components in the core of the phase are bal-anced by way of convection, and that only near the in-terfacial area, there is a motionless boundary fi lm through which the transfer in question can take place by diffusion exclusively. According to this model, the mass transfer coeffi cient is described by the following de-pendence: xABlDδβ =  (1)The thickness of the boundary fi lm marked as δx de-pends on the hydrodynamic conditions in the system, and it is a quantity particularly diffi cult to determine. Among the models much more frequently used to de-scribe the mass transfer in the liquid phase than the boundary fi lm model one should mention the penetration models [2, 3]. The fi rst of them (the so-called Higbie mo-del) was based on the assumption that the liquid surface is composed of multiple elements, each being in contact with the gaseous phase for a specifi c period of time, pass-ing deeper into the liquid afterwards. This means that the turbulent swirls occurring deeper in the liquid are not suppressed near the interfacial area and so they approach it causing its continuous restoration. In such a model, the time all elements of the liquid spend near the surface is identical and the dependence describing the mass transfer coeffi cient assumes the following form:  5,0⋅=τπβ ABlD  (2)The second penetration model was proposed by Danckwerts who managed to modify Higbie’s assump-68  METALURGIJA 51 (2012) 1, 67-70R. PRZYŁUCKI et al.: INFLUENCE OF AN INDUCTION FURNACE’S ELECTRIC PARAMETERS ON MASS TRANSFER...tion that time τ is not constant and random. According to Danckwerts, however, the constant quantity is the rela-tion between the area restored in a unit of time and the liquid’s overall area. In this model, the mass transfer co-effi cient is described by the following dependence [3]: ( )5,0sDABl ⋅=β  (3)The penetration models use quantities that are diffi -cult to determine. They include both time τ and propor-tionality factor s. Therefore, using these models to de-scribe the mass transfer in a liquid metallic phase in hydrodynamic systems, usually occurring in metallurgi-cal aggregates, is limited in scope. One of the excep-tions to this principle is the liquid metal barbotage with gases. Therefore, for instance, Higbie’s model can be effi ciently applied in describing mass transfer in liquid metal for refi ning aluminium with gases [4] or evapora-tion of volatile components from a liquid alloy blown with gas [5, 6].Among the solutions more and more commonly ap-plied for the sake of describing mass transport, one must mention empirical correlations developed in accordance with the principles of the similarity theory [7].Based on the penetration models, Machlin [7] suc-ceeded in preparing a mass transfer model for an induc-tion-stirred liquid metallic phase. Under this model, com-monly referred to as a “streamline fl ow model”, it was assu med that the liquid elements move as a rigid solid bo dy along the contact surface between the bath and the gas, or between the bath and the melting pot, and the ve-locity gradient normal towards the surface is close to ze-ro. In accordance with Machlin’s theory, the mass trans-fer coeffi cient is described by the following dependence: 5,08⋅⋅=mmABl rνDπβ  (4)Machlin’s model was frequently used in kinetic analyses of the evaporation process for volatile contam-inants of iron as well as  lead [8-10].Assuming that one applies dependence (4) to calcu-late the value of coeffi cient βl, a particularly important aspect would be the correct determination of the metal’s velocity near the surface. According to Machlin, this value depends to a slight extend on the electric param-eters of the furnace as well as on its dimensions. How-ever, it has occurred that the nature of the metal motion in an induction furnace as well as its circulation and intensity are the functions of numerous variables, such as the furnace geometry, density of the power emitted or frequency of the current [11-13]. In order to determine the value of velocity νm, one can apply a numerical method based on the so-called electromagnetic fl ow model [14, 15]. The model com-prises the following two subsequent elements:  analysis of the electromagnetic fi eld generated in the liquid metal, analysis of the velocity fi eld occurring in the liq-uid metal.Let us then elaborate on the last method, by applying it to determine the velocity of a liquid, inductively stirred Cu-Pb alloy.The basic furnace operating parameters as well as the physical and chemical quantities characterising the liquid metal, that were used in the respective calcula-tions, have been summarised in Table 1 below.The electromagnetic fi eld calculations were con-ducted in two dimensions, by applying a commercially available numerical programme, Flux. The programme was supplemented with certain procedures enabling de-termination of the mean value for the period of the forc-es volumetric density. Figure 1 is the analytical model diagram. The respective computational areas as well as the model boundaries have been marked as well. Table 1  Furnace operating parameters and selected properties of the Cu-Pb alloy Furnace operating parametersLiquid metal properties(Cu alloy + 2  % wt. of Pb)Frequency 3 kHz Temperature 1 473 KPower 20 kW Metal density 7,9 Mg·m-3Melting pot internal diameter 0,1 m Metal viscosity 0,0015 Pa·sInductor current intensity 155 ASpecifi c conductance 1,25·10-6 H·mElectric conductivity 4·106 S·m-1In the individual computational areas, the following differential equations of magnetic potential were solved for sinusoidal input functions:  01:1 =×∇×∇Ω A  (5) JAA =+×∇×∇Ω σωj1:2 ź (6) 0j1:3 =+×∇×∇Ω AA σω  (7)where: j – imaginary unit ( )1−=j .Figure 1  Computational model of the electromagnetic part of calculations: Ω1  – air; Ω2 — inductor winding; Ω3 – charge material (liquid metal)69METALURGIJA 51 (2012) 1, 67-70R. PRZYŁUCKI et al.: INFLUENCE OF AN INDUCTION FURNACE’S ELECTRIC PARAMETERS ON MASS TRANSFER...At the boundaries of the computational area: DA,CD,BC,AB , the Dirichlet boundary condition of A=0 was assumed.The components of the volumetric density of elec-tromagnetic forces (mean values for the whole period) were calculated according to the vector potential based on formulas 8 and 9. ( ) ( ) ( )( ){ }r BJBJf ImReRe21+=  (8) ( ) ( ) ( )( ){ }rrz BJBJf ImReRe21+−= (9)where: J = jωσA,B = ∇×A,Re, Im –  real and imaginary part of the complex vec-tors’ components,rB , B  – components of the magnetic induction vector.Components of the forces were determined in area Ω3 for a uniform, rectangular mesh of 181 x 199 nodes (in line with axes r and z respectively). The areas of fl ows were calculated numerically us-ing the Fluent 6,2 software. The model entailed axial symmetry of the system. Specifi c boundary conditions were introduced into the model in the form of two walls (the melting pot’s bottom and generator) without any skid (i.e. with a zero velocity by the wall), an axis of symmetry and a free surface of the metal simulated by a wall free of friction with the liquid. The impact of the electromagnetic fi eld forces was in-troduced into the model by setting user defi ned functions. The functions used the fi eld of force previously estab-lished by means of the Flux2D programme. Since the structure of the force fi eld mesh developed using Flux2D differed from the mesh of the fl ow model, it was neces-sary to interpolate the results. However, as the me shes only differed in the radial direction and the input mesh of forces was more dense than the mesh of fl ows, the inter-polation error could be considered negligibly small.In order to calculate the velocity and turbulence, a second order discretisation was applied [16]. The pres-sure was determined based on the standard method pro-posed by Rhie and Chow [17]. The analysed method of calculating the velocity of induction-stirred liquid metal was also applied to liquid Fe-Cu alloys [18].INFLUENCE OF THE CURRENTFREQUENCY ON THE METAL VELOCITYFurther analysis of the infl uence exerted by the fur-nace’s electric parameters on the velocity of mass trans-fer involved determination of how the metal velocity near the surface as well as the mean velocity depend on the value of the current frequency. The exemplary  fi elds of velocity have been depicted in Figures 2-3.The simulation performed implied that for the cur-rent frequency up to 3 kHz, as the frequency increases also the value of the velocity near the surface increases considerably. Further increase of the frequency value causes virtually no changes in the velocity - Figure 4. Figure 2 Velocity fi eld obtained for the frequency of 1 HzFigure 3 Velocity fi eld obtained for the frequency of 5 HzFigure 4  Infl uence of the current frequency on the liquid metal velocity70  METALURGIJA 51 (2012) 1, 67-70R. PRZYŁUCKI et al.: INFLUENCE OF AN INDUCTION FURNACE’S ELECTRIC PARAMETERS ON MASS TRANSFER...The values obtained for the velocity near the surface were then used to determine the value of the mass trans-fer coeffi cient for the liquid phase by application of Machlin’s dependence (6). The values obtained varied from 6.15·10-4 m·s-1 for 1 Hz to 4,57·10-4 m·s-1 for 5 kHz (Figure 5).CONCLUSIONSThe tests conducted implied a considerable impact of the current frequency on the mass transfer coeffi cient for the induction-stirred liquid metallic phase. This is caused by the change of the stirring intensity along with the said parameter and thus the stirring velocity as well. Using a numerical method based on the so-called elec-tromagnetic fl ow fi eld enabled determination of the metal velocity near the surface and the mean velocity deeper in the metal. For the current frequency change from 1 kHz to 5 kHz, a decrease in both velocity values was observed, this, however, being more noticeable for the velocity near the surface. REFERENCES[1] D. R. Poirier, G. H. Geiger: Transport Phenomena in Ma-terials Processing, TMS, Warrendale (1994).[2] R. Higbie: Trans AICHE 31 (1935), 365 - 368.[3] P. Danckwerts: Trans. Faraday Soc. 46 (1950), 101 - 103.[4] J. Botor: Prace IMN 1 (1978), 12-16.[5] B. Oleksiak: Metalurgija 47(2008)2, 87 - 90.[6] J. Benavidies, I. Gokhan: Metall. Trans.B 32 (2001), 285 - 295.[7] J. Botor: Podstawy metalurgicznej inżynierii procesowej, Wyd. Pol. Śl.., Gliwice (1999).[8] R. Ward: Journal Iron and Steel Inst. 201 (1963), 920 - 923.[9] R. Ohno, T. Ishida: Japan Inst. Met. 31 (1967), 1164 - 1169.[10] J. Łabaj, J. Botor, R. Sosnowski: Archives of Metallurgy 47 (2002), 307 - 320.[11] J. Szekely, K. Nakanishi: Metall. Trans. B 6 (1975), 245 - 256.[12] J. Szekely, K. Nakanishi: Metall. Trans. B 6 (1975), 245 - 256.[13] E. D. Tarapore, J. W. Evans J: Metall. Trans. B 7 (1976), 343 - 351.[14] S. Golak, R. Przyłucki: A simulation of the coupled prob-lem of magnetohydrodynamics and a free surface for liq-uid metals, Transactions on Engineering Science, WIT 63 (2009).[15] S. Golak: Application of image analysis for the measure-ment of liquid metal surface, Transactions on Modelling and Simulation, WIT 48 (2009).[16] T. J. Barth, D. Jespersen: The design and application of upwind schemes on unstructured meshes, Technical Re-port AIAA-89-0366, AIAA 27th Aerospace Sciences Meet-ing, Reno (1989).[17] C. M. Rchie, W. L. Chow: Numerical Study of the Turbu-lent Flow past an Airfoil with Trailing-Edge Separation, AIAA Journal 21 (1983), 1525 - 1153.[18] J. Łabaj: Kinetyka parowania miedzi z ciekłego żelaza, Wyd. Oldprint, Katowice (2010).Note:  Nowak P. is responsible for English language, Katowice, Poland Figure 5  Infl uence of the current frequency on the mass transfer coeffi  cient for the liquid metal phase (Cu-Pb alloy, 1 473 K)

Utjecaj parametara indukcijske elektro peći na brzinu masenog prijenosa u tekućoj fazi

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