MODELLING OF TRANSFER PROCESSES IN GAS LADLE BUBBLING AND THEIR MATHEMATICAL IDENTIFICATION BY L-TRANSFORMATION

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1 Acta Metallurgica Slovaca, 4, 8, (4-3) 4 MODELLING OF TRANSFER PROCESSES IN GAS LADLE BUBBLING AND THEIR MATHEMATICAL IDENTIFICATION BY L-TRANSFORMATION Michalek K., Morávka J., Gryc K. Department of Metallurgy, VŠB-Technical University of Ostrava, Czech Republic Třinecký inženýring, a.s., Czech Republic MODELOVÁNÍ PŘENOSOVÝCH DĚJŮ V PLYNEM PROBUBLÁVANÉ LICÍ PÁNVI A JEJICH MATEMATICKÁ IDENTIFIKACE POMOCÍ L-TRANSFORMACE Michalek K., Morávka J., Gryc K. Katedra metalurgie, VŠB-Technická univerzita Ostrava, Česká republika Třinecký inženýring, a.s., Česká republika Abstrakt Článek prezentuje výsledky modelového výzkumu chování lázně v licí pánvi v průběhu jejího prodmýchávání argonem přes dva dmyšné elementy umístěnými v jejím dně. Studie byla provedena využitím metody fyzikálního modelování na modelu v měřítku ku. Příspěvek uvádí také naše vlastní výsledky experimentálního zkoumání dopadu objemového průtoku argonu a vlivu umístění dmyšných elementů na způsob procesu koncentrační homogenizace a změnu charakteru proudění v modelu licí pánve. V další fázi byla pozornost zaměřena na řešení současného dmýchání argonu pomocí dvou dmyšných elementů umístěných ve dně licí pánve. Průběh homogenizačních procesů po začátku dmýchání argonu byl hodnocen na základě elektrické vodivosti a změny teploty. Tyto veličiny byly snímány ve třech pozicích zasahujících do objemu mezipánve pomocí vodivostních a teplotních sond. Práce byly realizovány pro podmínky licích pánví se 8 t tekuté oceli. Výsledky experimentálního zkoumání byly zpracovány pomocí vhodných modelů. Exaktní teoretický popis procesů probíhajících při dmýchání argonu do oceli je ve své podstatě velmi komplexní a měl by vést k systému nelineárních parciálních diferenciálních rovnic popisujících přenos hybnosti, tepla a složek s excitační funkcí ve formě rovnice tzv. deterministického chaosu (probublávání argonu). Na základě diagramu modelu licí pánve a koncentrace byl sestaven zjednodušený lineární fyzikálně adekvátní model, který popsal chování koncentrace oceli v licí pánvi v průběhu jejího prodmýchávání. Analyzovaný proces byl chápán ve smyslu kybernetického modelu, který je možné přehledně vyjádřit pomocí tzv. blokového diagramu. Abstract The article presents results of model investigation of bath behaviour in the ladle during its gas argon bubbling realised by two stir elements situated in the ladle bottom. Study was performed with use of physical modelling method on a scale model to. The paper presents also our own experimental investigation results obtained from argon volumetric flow impact and influence of stir elements situating on the course of the concentration homogenisation process and change of flow characteristics in the pouring ladle model. The second stage was focused on evaluation of simultaneous bubbling through two stir elements in the ladle bottom and to propose the optimum location of the second SE in the ladle bottom. Development of homogenisation processes after bubbling start was evaluated on the basis of

2 Acta Metallurgica Slovaca, 4, 8, (4-3) electrical conductivity and temperature change, which were measured at three points of the ladle volume by conductivity and temperature sensors. Executed works were realised for conditions of 8 t steel ladles. The used models processed results of experimental investigation. Exact theoretical description of processes occurring at argon bubbling into steel would be very complex and it would lead to a system of non-linear partial differential equations describing transfer of momentum, heat, components, and with excitation function in the form of equation of so called deterministic chaos (argon bubbling). On the basis of pouring ladle model diagram and concentrations courses the simplified linear physically adequate model was proposed, which described behaviour of steel concentration in pouring ladle during its bubbling. Analysed process was understood in the form of a cybernetic model, which can be transparently visualised by so called block diagram. Keywords: steel, ladle, gas argon bubbling, stir elements, physical modelling, mathematical identification. Introduction Homogenisation processes during argon bubbling into steel in a ladle was investigated by many authors [see -7] and it brought numerous partial pieces of knowledge. Some of them are already generally accepted and are used in daily practice almost in every steelmaking shop. With some simplification it can be summarised that for each geometry of pouring ladle there exists certain optimum axle offset of stir element, which ensures the best results from the viewpoint of homogenisation time and rate. In certain cases it is possible to use also bubbling with two stir elements.. Description of investigated variants and used experimental methodologies of physical modelling Executed works were realised for conditions of 8 t steel ladles. They were aimed at obtaining data about influence of argon volumetric flow and of stir element (SE) location in the ladle bottom on progress of homogenisation in the ladle. The second stage was focused on evaluation of simultaneous bubbling through two stir elements in the ladle bottom and to propose the optimum location of the second SE in the ladle bottom. Simultaneous bubbling is used mostly in large volume ladles with higher ratio D/H (ladle diameter and height). The main effect obtained by bubbling through two SE in these ladles consists in quicker progress of homogenisation, both concentration and temperature, and minimisation or elimination of possible dead areas in steel bath volume, where melt movement stagnates with all detrimental consequences on bath homogeneity. Another advantage of two SE lies in the fact that in case of failure of any one SE it is still possible to continue bubbling through the other functional SE. Fig. shows investigated positions of stir elements location. Determination of physical similarity conditions was based on verified and used procedure published in our works, e.g. in [8-]. Ladle model in length scale M L = : was used for experimental model research. Argon flow measurement was realised by precise mass flow meter with automatic regulation. Scale factor of volumetric flow M Qv =.76 was determined from modified Froude s criterion, which respects also influence of blown argon expansion due to increase of its

3 Acta Metallurgica Slovaca, 4, 8, (4-3) 6 temperature caused by passage through a liquid phase. Flow on the model Q v =.4 l/min then corresponds to the flow of basic operational case Q v = 43 l/min. Calculated time scale factor was M τ =.36. Argon bubbling in the model was made with use of stir element, which was a model equivalent of the element for industrial conditions. Development of homogenisation processes after bubbling start was evaluated on the basis of electrical conductivity and temperature change, which were measured at three points of the ladle volume by conductivity and temperature sensors. Fig. Studied positions of stir plugs in ladle bottom used for modelling study 3. Obtained results 3. Use of one stir element For purposes of industrial interpretation the homogeneity times measured on the model were recalculated with use of already determined volumetric flow scale factors and time (M Qv, M τ ) to industrial conditions. These values were processed graphically and interlaid by regression function of the type τ H = a Q v b see Fig.. A B E 4 4 homogenization time, 3 3 A : B : E : y = 7.7 x -.68 R =.966 y = x -.8 R =.9698 y = 773. x -. R = gas flow rate, l/min Fig. Influence of argon volumetric flow Q v on achieved homogeneity times τ H for three positions of stir elements in the ladle bottom marked A, B and E The graph shows visible decrease of homogeneity times values τ H with increase of argon volumetric flow. In the area above 4 l/min this decrease is not too significant and homogeneity times values vary between to s. Contrary to that in the area below l/min, and particularly l/min the homogeneity times values steeply increase and they achieve up to 3 s in dependence on location of SE. Course of curves indicates also that from the viewpoint of homogeneity times there are no distinct differences between bubbling through SE in positions A, B and E. Position A (and possibly C, D) can be regarded as the most favourable position of SE, as it had somewhat shorter homogeneity times (approx. by to %) in comparison with positions B and E.

4 Acta Metallurgica Slovaca, 4, 8, (4-3) 7 3. Use of two stir elements Physical modelling was then used for investigation of stir elements combinations AB, AC, AD and AE. Physical modelling results are summarised in Fig. 3. homogenization time, AB AC A D A E gas flow rate, l/min AB : AC : AD : AE : y = x -.97 R =.9666 y = x -.64 R =.989 y = 73. x R =.984 y =.6 x -.6 R =.987 Fig.3 Influence of argon volumetric flow Q v on achieved homogeneity times τ H for four combinations of stir elements position marked AB, AC, AD and AE Combinations AB and AC are practically identical from the viewpoint of obtained homogeneity times. Less favourable is combination AE, which had longer homogeneity times than variants AB and AC. Surprisingly the worst was the variant AD, in which location of SE in position D the y axis was axially symmetrical to the position A. This combination had the longest homogeneity times in full volumetric flows range. It represents increase of time by 3 to 4 % in comparison with combinations AB or AC. Explanation of this apparent anomaly can be found in flow character or in formation of two recirculation zones in the pouring ladle, directions of which are opposite to each other. In this way the speed components of flowing mutually significantly influence or even eliminate each other with negative impact on components and heat transfer. Homogenisation process, both concentration and temperature, slows down and obtaining of homogenous stabilised bath requires longer bubbling and thus bigger blown argon volume, which can cause even greater drop in temperature. 4. Mathematical description of processes at argon bubbling It was appropriate to create physical-mathematical (physically adequate) model for the measured time courses of tracing substance concentrations in physical model of pouring ladle (PL) at argon bubbling into the steel. 4. Description of situation Schematic representation of situation at argon bubbling into bath in model of pouring ladle (mlp) is shown in Fig. 4. Argon bubbles flow from eccentrically situated stir element P in the bottom of mlp at constant volumetric flow q. They disrupt layer of concentration enriched and coloured water with thickness h k and there occurs gradual mixing of enriched and clear liquid (steel, water). Two (pressure) forces II and I act basically against each other on molecules of water in proximity of the sensors.

5 Acta Metallurgica Slovaca, 4, 8, (4-3) 8 K K K3 I II mlp P q h k h c(t) - concentration [% wt.].9 K K K3.8.7 K c u. K K3. c p time [s] Fig.4 Argon bubbling into ladle model Fig. The time response of concentrations on sensors K, K and K3 4. Measured data Analysis and synthesis of mathematical models was realized with use of the data measured on physical model, where courses of the measured concentrations, (with period of sampling t. s), had on the sensors K, K and K3 are shown in Fig.. Several facts are obvious from the development of concentration in the sensors: start and progress of gas argon bubbling into steel can be approximately considered in the form of the Heaviside unit step function and it is therefore possible to consider the development of concentration as unit step response, the sensors reacted only after elapsing of certain dead time, which is proportional to the distance of individual sensors from the liquid level in the ml, the overshoot of courses (apparently proportional to the magnitude of the force I or rather to the difference of the forces I and II) also descends with the distance from the liquid level in the ml, steady-state (final) value of concentration is proportional to the proportion of volumes of pure water and water with enriched concentration. 4.3 Physical-mathematical model Exact theoretical description of processes occurring at argon bubbling into steel would be very complex and it would lead to a system of non-linear partial differential equations describing transfer of momentum, heat, components, and with excitation function in the form of equation of so called deterministic chaos (argon bubbling). On the basis of mlp diagram and concentrations courses the simplified linear physically adequate model was proposed, which described behaviour of steel elements concentration in PL during it s bubbling. Analysed process was understood in the form of a cybernetic model, which can be transparently visualised by so called block diagram shown in Fig. 6. It is series-parallel connection of three components, namely component of time delay and two parallel proportional (inertial) components acting against each other. Two parallel and antagonistically connected simplest proportional systems with inertia of the st order (with transfers G I a G II ) are assumed for the part of the model F without time delay:

6 Acta Metallurgica Slovaca, 4, 8, (4-3) 9 GI ( s) = T k, s + GII ( s) = T k s + () where k, k - coefficients of transfer (amplification) of systems [% s.m -3 ], T, T - time constants of systems [s]. G I 6 Courses of regression for sensors K, K a K3 K.. Q(s) G d G II + - C(s) cn [-] 4 3 K3.7. cn, 3 [-] K time [s]. Fig.6 Block diagram of mlp Fig.7 Courses of regression of F model for data For this part of the model it is then possible to compose on the algebra basis of transfer the following continuous L-transfers (for zero initial conditions) of the model F: k ( ) ( ) ( ) k k k + k T k T s G s = = () T s + Ts + ( T s + )( T s + ) Wherefrom for L-image and original of transfer function H(s) = G(s) /s and h(t) we get the final expression for standard concentration transfer function, which can be at the same time used as non-linear regression model with three parameters k, T and T : h( t) = cn( t) = k exp( t / T ) + ( k ) exp( t / T ) (3) 4.4 Regression model for all concentrations Fig. 7 shows courses of the regression function based on the model F, for standard concentration values in all three sensors K, K and K3. It is obvious from results that F is suitable and usable for description of concentration course for all three sensors. It is also obvious that the transfer coefficient (gain coefficient) k decreases with the distance of the sensor from the liquid level in the ml, while the time constant T increases in this dependence.. Use of model for evaluation of experiments It is evident that the proposed physically adequate model can be used for evaluation of the physical modelling results at argon bubbling in the ladle model. Its parameters were used for determination of other aggregated parameters, and all of them where used for definition of an optimum range of blown argon flow. The paper documents results for position A of one stir element.

7 Acta Metallurgica Slovaca, 4, 8, (4-3) 3. Approaches to solution Four groups of parameters were chosen for characterisation of homogeneity intensity measure in respect to blown argon flow q: basic physical parameters of model (4 parameters see model () + time delay), aggregated parameter defining derivative time constant ( parameter), aggregated parameters defining maximum of model transfer function ( parameters) and aggregated parameter defining surface of model impulse function ( parameter). Aggregated parameters were chosen in order to achieve simplicity, clarity and appropriate interpretability of homogenisation process.. Parameters of physically adequate model Graphical dependencies of foregoing mathematical model parameters on blown argon flow for the sensor K are shown in Fig. 8: Td [s] 3 SE "A", sensor K : Td, k = f(q) min Td [s] k [-] y =.8x -.78 R = k [-] 3 Tx [s] 3 3 SE "A", sensor K : T, T = f(q) max T [s] T [s] Fig.8 Dependencies of parameters T d, k (on the left) and T, T on flow of argon in the sensor K.3 Derivative time constant Overall L-transfer of the defined model has character of so called real derivative member with delay of the nd order, and derivative time constant T D is function of all three parameters. Graphical dependence of this time constant on argon flow (at the sensor K) is shown in Fig. 9. TD [s] "break" SE "A", sensor K : TD = f(q) TD [s] Fig.9 Dependence of derivative time constant T D on argon flow at the sensor K

8 Acta Metallurgica Slovaca, 4, 8, (4-3) 3.4 Transfer function overshoot This approach uses analytical relations for determination of transfer function position and transfer function maximum values. Graphical dependencies of assumed aggregated maximum parameters on the blown argon flow for the sensor K are shown in Fig.. tmx [s] max SE "A", sensor K : tm, tmax = f(q) tm tmax min hm [-] SE "A", sensor K : hm = f(q) min hm max Fig. Dependencies of times (on the left) and maximum values on the flow of inert gas at the sensor K. Integral surface of impulse function Value of area determined by the model impulse function is an attempt of the transfer process intensity characterisation in the pouring ladle model. Dependencies of impulse function integral area, as well as maximum values of transfer function, on the blown argon flow for the sensor K are shown in Fig.. S, hm [-] SE "A", sensor K : S, hm = f(q) min hm S max Fig. Dependencies of area of impulse function and maximum values on argon flow at the sensor K 6. Conclusion Physical modelling method was used for model investigation of influence of argon bubbling through a stir element or stir elements situated in the ladle bottom on the course of homogenisation processes in a bath. The best results were obtained in variants with simultaneous bubbling through SE in positions A and B or A and C (Fig. 3). On the other hand the longest homogeneity times were obtained in the variant with location of SE in positions A and D, which can be explained by creation of two re-circulation zones in pouring ladle with important mutual influencing and elimination of components transfer rate. Physically adequate mathematical model was proposed and verified for measured time response of substance concentrations in the pouring ladle model bath. This model was used for subsequent analysis of input simulation parameters influence on its coefficients and aggregated parameters with consequences on appropriate (optimum) working mode setting for steel bubbling in pouring ladle. Due to set various flows of argon blown into bath there were determined and verified four groups of (aggregated) parameters for homogenisation intensity characterisation. On the basis of the shape of their course the argon

9 Acta Metallurgica Slovaca, 4, 8, (4-3) 3 flown optimum range was deduced within limits from approx l/min (on ladle model), which can be transformed with use of the appropriate volumetric flow scale factor to the realistic industrial equipment conditions, i.e. - 4 l/min. Acknowledgement The work was realised within the frame of solution of the grant project No.6/7/47 under financial support of the Grant Agency of Czech Republic. Literature [] Mazumdar, D., Guthrie, R.I.L.: The Physical and Mathematical Modelling of Gas Stirred Ladle Systems. ISIJ International, Vol. 3, (99), No., pp. -. [] Zhu, M.Y., Inomoto, T., Sawada, I. Hsiao, T.Ch.: Fluid Flow and Mixing Phenomena in the Ladle Stirred by Argon through Multi-Tuyere. ISIJ International. Vol. 3 (99). No., pp [3] Rogler, J.P., Heaslip, L.J., Xu, D., Mehrvar, M.: Modeling of Ladle Bubbling for the Elimination of Thermal Stratification. AIST Transactions, 4, April, pp [4] Madan, M., Satish, D., Mazumdar, D.: Modeling of Mixing in Ladles Fitted with Dual Plugs, ISIJ International, Vol. 4 (), No., pp [] Mandal,Y., Patil, S., Madan, M., Mazumdar, D.: Mixing Time and Correlation for Ladles Stirred with Dual Porous Plugs. Metallurgical and Materials Transactions B, Vol. 36B (), No.8, pp [6] Ramírez-Argáez, M.A., Nava-Ramos, H.: Mathematical Modeling of Fluid Flow, Mixing and Inclusions Removal in Gas Stirred Ladles. In METAL 6. th International Metallurgical and Material Conference Hradec nad Moravicí, Czech Republic, pp. -. [7] Zhu, M., Zheng, S., Huang, Z., Gu W.: Numerical simulation of nonmetallic inclusions behaviour in gas-stirred ladles. Steel Research Int. 76,, č., s [8] Benda, M., Michalek, K., Foukal, J.: Modelové studium homogenizačních pochodů v roztavené oceli v pánvi při dmýchání plynu [Model investigation of homogenisation processes in molten steel in ladle at gas blowing]. Hutnické listy, 98, No. 7, pp [9] Benda, M., Michalek, K.: Přibližné fyzikální modelování homogenizačních pochodů v ocelářských pánvích. [Approximate physical modelling of homogenisation processes in steel ladles]. Hutnické listy, 987, No. 6, pp [] Benda, M., Michalek, K. et al.: Vliv vybraných činitelů na průběh teplotní homogenizace prodmýchávané lázně [Influence of selected factors on development of blown bath temperature homogenisation]. Hutnické listy, 989, No., pp