Mathematical models processed results of experimental investigation obtained during ladle gas argon bubbling realized by stir elements situated in the ladle bottom. 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 behavior of steel concentration in pouring ladle during its bubbling. The analysed process was understood in the form of a cybernetic model.Matematičkim su modelima obrađeni rezultati eksperimentalnog instraživanja homogenizacije u loncu s argonskim mješačima. Egzaktan teorijski opis procesa uvođenja mjehurića argona u čeličnu talinu bio bi vrlo kompleksan i vodio bi k sustavu nelinearnih diferencijalnih jednadžbi za opis prijenosa momenta gibanja, topline i elemenata, uz dodatnu funkciju takozvanog determinističkog kaosa (gibanje mjehurića argona). Temeljem dijagramskog modela i koncentracijskih krugova predložen je pojednostavnjen, fizikalno odgovarajući linearni model, koji prikazuje promjenu koncentracije čelične taline u ljevačkom loncu tijekom upuštanja mjehurića. Analizirani process smatran je pritom kibernetičkim modelom

J. Morávka

K. Gryc

K. Michalek

English

Hrčak - Portal of scientific journals of Croatia

K. MICHALEK, J. MORÁVKA, K. GRYCMATHEMATICAL IDENTIFICATIONOF HOMOGENISATION PROCESSES IN ARGON STIRRED LADLEReceived – Prispjelo: 2008-09-09Accepted – Prihva}eno: 2008-12-10Original Scientific Paper – Izvorni znanstveni radINTRODUCTIONPreviously presented paper “Physical Modelling ofBath Homogenisation in Argon Stirred Ladle“ was de-voted to results of bath behaviour in the ladle model in-vestigation during its gas argon bubbling realised by oneor two stir elements situated in the ladle bottom. Studywas performed with use of physical modelling methodon a scale model 1 to 10. Development of homogenisa-tion processes after start of bubbling was evaluated onthe basis of electrical conductivity and temperaturechange, which were measured at three points of the ladlevolume by conductivity and temperature sensors. Theexecuted works were realised for conditions of 180 tonssteel ladles.It was found that the rate of homogenisation in thesteel ladle is significantly dependent on the volume flowof argon. Mathematical method for the identificationprocess was used for determining the optimum flow ofargon.MATHEMATICAL DESCRIPTIONOF PROCESSES AT ARGON BUBBLINGIt was appropriate to create a physical-mathematical(physically adequate) model for the measured timecourses of tracing substance concentrations in physicalmodel of pouring ladle (PL) occurred during argon bub-bling into the steel.Description of situationSchematic representation of situation at argon bub-bling into bath in the model of steel ladle (mLP) isshown in Figure 1.Argon bubbles flow from eccentrically situated stirelement (P) in the bottom of mLP at constant volumetricflow (q). They „disrupt“ the layer of concentration en-riched and coloured water with thickness (hk) and thereoccurs gradual mixing of enriched and clear liquid(steel, water). Two (pressure) forces II and I act basi-cally against each other on molecules of water in prox-imity of the sensors.METALURGIJA 48 (2009) 4, 219-222 219Mathematical models processed results of experimental investigation obtained during ladle gas argon bub-bling realised by stir elements situated in the ladle bottom. Exact theoretical description of processes occurringat argon bubbling into steel would be very complex and it would lead to a system of non-linear partial differen-tial equations describing transfer of momentum, heat, components, and with excitation function in the formof equation of so called deterministic chaos (argon bubbling). On the basis of pouring ladle model diagram andconcentrations courses, the simplified linear physically adequate model was proposed, which described beha-viour of steel concentration in pouring ladle during its bubbling. The analysed process was understood in theform of a cybernetic model.Key words: steel, ladle, gas argon bubbling, stir elements, physical modelling, mathematical identificationMatemati~ka analiza procesa homogenizacije u loncu s argonskim mije{anjem. Matemati~kim mo-delima obra|eni su rezultati eksperimentalnog instra`ivanja homogenizacije u loncu s argonskim mje{a~ima.Egzaktan teorijski opis procesa uvo|enja mjehuri}a argona u ~eli~nu talinu bio bi vrlo kompleksan i vodio bi kasustavu nelinearnih diferencijalnih jednad`bi za opis prijenosa momenta gibanja, topline i elemenata, uz do-datnu funkciju takozvanog deterministi~kog kaosa (gibanje mjehuri}a argona). Temeljem dijagramskog mode-la i koncentracijskih krugova predlo`en je pojednostavnjen, fizikalno odgovaraju}i linearni model, kojiprikazuje promjenu koncentracije ~eli~ne taline u ljeva~kom loncu tijekom upu{tanja mjehuri}a. Analiziraniprocess smatran je pritom kiberneti~kim modelom.Klju~ne rije~i: ~elik, lonac, upu{tanje argonskih mjehuri}a, mje{a~i, fizikalno modeliranje, matemati~ka identifi-kacijaISSN 0543-5846METABK 48(4) 219-222 (2009)UDC-UDK 669.18 : 621.746.32 : 542.63 : 621.22.015.5 : 517.442=111K. Michalek, K. Gryc, Faculty of Metallurgy and Materials Engineering,V[B-TU Ostrava, Czech RepublicJ. Morávka, Material & Metallurgical Research Ltd., Ostrava, Czech Re-publicMeasured dataAnalysis and synthesis of mathematical models wasrealised with use of the data measured on physicalmodel, where courses of the measured concentrations,(with period of sampling   0,5 s), had on the sensorsK1, K2 and K3 are shown in Figure 2.Several facts are obvious from the development ofconcentration in the sensors:– start and progress of gas argon bubbling into steelcan be approximately considered in the form ofthe Heaviside unit step function and it is thereforepossible to consider the development of concen-tration as a unit step response,– the sensors reacted only after elapsing of certain“dead time”, which is proportional to the distance ofindividual sensors from the liquid level in the mLP,– the overshoot of courses (apparently proportionalto the magnitude of the force I or rather to the dif-ference of the forces I and II) also descends withthe distance from the liquid level in the mLP,– steady-state (final) value of concentration is pro-portional to the proportion of volumes of pure wa-ter and water with enriched concentration.Physical-mathematical modelExact theoretical description of processes occurringat argon bubbling into steel would be very complex andit would lead to a system of non-linear partial differen-tial equations describing transfer of momentum, heat,components, and with excitation function in the form ofequation of so called deterministic chaos (argon bub-bling). On the basis of the mLP diagram and concentra-tions courses, the simplified linear physically adequatemodel was proposed, which described behaviour of steelelements concentration in PL during its bubbling. Ana-lysed process was understood in the form of a cyberneticmodel, which can be transparently visualised by socalled block diagram shown in Figure 3.It is series-parallel connection of three components,namely component of time delay and two parallel pro-portional (inertial) components acting against eachother.Two parallel and antagonistically connected sim-plest proportional systems with inertia of the 1st order(with transfers GI a GII) are assumed for the part of themodel without time delay 1:G sksG sksI II( ) , ( )11221 1 (1)where:k1, k2 - coefficients of transfer (amplification)of systems / %×s/m3,1, 2 - time constants of systems / s,s - complex variable in the Laplacetransform / s-1.For this part of the model it is then possible to com-pose on the algebra basis of transfer the following con-tinuous L-transfers (for zero initial conditions):G sksksk k k k ss( )( ) ( )( )(  11221 2 1 2 2 111 11   2 1s  )(2)Wherefrom for the L-image and original of transferfunction H(s) and h(t) we get the final expression forstandard concentration transfer function, which can beused at the same time as non-linear regression model F1with three parameters k1, 1 and 2:h t c t k t k tn( ) ( ) exp( / ) ( ) exp( / )   	    	 1 11 1 1 2  (3)Regression model for all concentrationsFigure 4 shows courses of the regression functionbased on the model F1, for standard concentration val-ues in all three sensors K1, K2 and K3.220 METALURGIJA 48 (2009) 4, 219-222K. MICHALEK et al.: MATHEMATICAL IDENTIFICATION OF HOMOGENISATION PROCESSES IN ARGON STIRRED LADLEFigure 1. Argon bubbling into ladle modelFigure 2. The time response of concentrations on the sen-sors K1, K2 and K3Figure 3. Block diagram of processes in ladle model (mLP)It is obvious from the results that the model F1 is suit-able and usable for description of concentration coursefor all three sensors. It is also obvious that the transfer co-efficient (gain coefficient) k1 decreases with the distanceof the sensor from the liquid level in the mLP, while thetime constant ô2 increases in this dependence.USE OF MODEL FOREVALUATION OF EXPERIMENTSProposed physically adequate model F1 was used forevaluation of the physical modelling results at argonbubbling in the ladle model. Its parameters were usedfor determination of other aggregated parameters, andall of them where used for definition of an optimumrange of blown argon flow. The paper documents resultsof one stir element for position „A“ in the ladle bottom.Approaches to solutionThree groups of parameters were chosen for charac-terisation of homogeneity „intensity“ measure in respectto blown argon flow q: basic physical parameters ofmodel (2 parameters – see the model in Figure 3 andtime delay, i.e. k1, k2, and d), aggregated parameter de-fining derivative time constant (1 parameter, i.e. D), andaggregated parameters defining maximum of modeltransfer function (2 parameters, i.e. tm, respectively tmax ,and hm). Aggregated parameters were chosen in order toachieve simplicity, clarity and appropriate inter-pretability of homogenisation process.Parameters of physically adequate modelGraphical dependencies of foregoing mathematicalmodel parameters on blown argon flow for the sensorK2 are shown in Figure 5:It is obvious from the Figure 5 that physically ade-quate model, in conformity with the assumed behaviourof real system, has evaluated a trend of dependence ofthe transport delay d on the argon flow rate. At higherflow rate homogenisation starts sooner and runs morerapidly.Derivative time constantOverall L-transfer of the defined model has characterof so called real derivative member with a delay of the2nd order 2, and derivative time constant D is functionof all three parameters. The constantc D an be expressedfrom the equation (2) with use of the relation:   D k  1 2 1 1( ) (4)Lower values of this derivative time constant ôD ex-press faster reaction of the model on change of the inputvalue, which is in this case represented by the volumeflow of argon.Graphical dependence of this time constant on the ar-gon flow (for measurement at the sensor K2) is shown inFigure 6.It is obvious from the Figure 6 that increase of thevolume flow of argon to 1,4 l/min causes a significantstep change of the value D and therefore also more rapidchanges in chemical composition of the model liquid(and development of homogenisation). The value of Ddoes not change significantly with further increase ofthe argon flow rate.Transfer function overshootThis approach uses analytical relations for determi-nation of transfer function position and transfer functionmaximum values. Graphical dependencies of the as-METALURGIJA 48 (2009) 4, 219-222 221K. MICHALEK et al.: MATHEMATICAL IDENTIFICATION OF HOMOGENISATION PROCESSES IN ARGON STIRRED LADLEFigure 4. Courses of regression of F1 model for data Figure 5. Dependencies of parameters d, k1 on argonflow for the sensor K2Figure 6. Dependence of derivative time constant D onargon flow for the sensor K2sumed aggregated maximum parameters of the blownargon flow for the measurement at sensor K2 are shownin Figure 7 and Figure 8.Diagram in Figure 7 shows development of times forachieving of maximum values in the impulse function independence on the argon flow rate (tmx on the axis y is ageneral expression of time parameters tm, which is atime of achieving the maximum regardless of transportdelay d, or max). It is possible to obtain values of the pa-rameter tmax by adding the parameter m to the transportdelay d. Gradual convergence of tm and tmax is given bydecrease of the transport delay d with increasing vol-ume flow of argon. Faster circulation of volume of liq-uid results in faster achieving the maximum values ofthe impulse function. This speed, however, steeply fallsat argon flow rates below 1,4 l/min.The Figure 8 is devoted to a description of depend-ence of the maximum value of the impulse function (hm –maximum value of dimensionless concentration) on thevolume flow of argon. At its lowest flow rate (0,3 l/min)the injected substance gets to the sensor K2 in concen-trated condition without being mixed. At increase of theargon flow rate to 0,68 l/min the faster dispersion of themarking substance at the expense of the speed of circula-tion of the volume of model liquid manifests itself themost from the viewpoint of this parameter (hm). Furtherincrease in argon flow rate brings about again a gradualenforcing of speed of circulation. At the highest argonflow rate (3,25 l/min) the rate of circulation is too highand injected substance gets to the sensor K2 without be-ing able to disperse in the volume of the liquid.CONCLUSIONPhysically adequate mathematical model was pro-posed and verified for the measured time response of sub-stance concentrations in the pouring ladle model bath.This model was used for subsequent analysis of theinput simulation parameters influence on its coefficientsand aggregated parameters with consequences on ap-propriate (optimum) working mode setting for steelbubbling in the pouring ladle. Due to pre-set variousflows of argon blown into bath three groups of (aggre-gated) parameters for homogenisation intensity charac-terisation were determined and verified.On the basis of the shape of their course, the argonflow optimum range was deduced – within limits fromapprox. 1,4 – 2,3 l/min (on the ladle model), which canbe transformed with use of the appropriate volumetricflow scale factor to the realistic industrial equipmentconditions, i.e. 243 - 400 l/min.It was established in the previous paper entitled„Physical Modelling of Bath Homogenisation in ArgonStirred Ladle” that after achievement of a certain„breakpoint“ volume flow its further increase bringslow efficiency from the viewpoint of homogenisationtimes. Parameters assumed in this work enable more-over a more detailed investigation of the character of ho-mogenisation process, correct setting of which forms anintegral part of the advanced secondary metallurgy 3,4. For example magnitude of the maximum values ofthe impulse function (hm) at individual argon flow ratesgives evidence about behaviour e.g. of alloying addi-tions injected into the volume of real pouring ladle at theinitial stage of homogenisation.The work was realised within the frame of solution ofthe grant project No.106/07/0407 under financial sup-port of the Czech Science Foundation.REFERENCES1 A. Víte~ek, L. Smutný, J. Kusyn, Teorie øízení I. Základnípojmy a øe{ené pøíklady z Laplaceovy transformace. III.Management theory I. Basic notions and solved examplesfrom Laplace’s transformation. III., V[B-TU Ostrava,Ostrava, 1988.2 S. Kubík, et al., Teorie regulace - I. Lineární regulace The-ory of regulation – I. Linear regulation. SNTL, Praha,1974.3 J. Krajcar, Metalurgija 32 (1993) 3, 87-92.4 V. Ferketi}, J. Krajcar, A. Ivan~an, Metalurgija 35 (1996) 1,43-48.Note: The responsible translator for English language is B. [kandera,Czech Republic.222 METALURGIJA 48 (2009) 4, 219-222K. MICHALEK et al.: MATHEMATICAL IDENTIFICATION OF HOMOGENISATION PROCESSES IN ARGON STIRRED LADLEFigure 7. Dependencies of maximum times (tm, tmax) of theflow of inert gas for the sensor K2Figure 8. Dependencies of maximum values (hm) on theflow of inert gas for the sensor K2

Matematička analiza procesa homogenizacije u loncu s argonskim miješanjem

Mathematical models processed results of experimental investigation obtained during ladle gas argon bubbling
realised by stir elements situated in the ladle bottom. 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. The analysed process was understood in the
form of a cybernetic model.Matematičkim modelima
obradeni su rezultati eksperimentalnog instraživanja homogenizacije u loncu s argonskim mješačima.
Egzaktan teorijski opis procesa uvodenja mjehuri}a argona u čeličnu talinu bio bi vrlo kompleksan i vodio bi ka
sustavu nelinearnih diferencijalnih jednadžbi za opis prijenosa momenta gibanja, topline i elemenata, uz dodatnu
funkciju takozvanog determinističkog kaosa (gibanje mjehurića argona). Temeljem dijagramskog modela
i koncentracijskih krugova predložen je pojednostavnjen, fizikalno odgovaraju}i linearni model, koji
prikazuje promjenu koncentracije čelične taline u ljevačkom loncu tijekom upuštanja mjehurića. Analizirani
process smatran je pritom kibernetičkim modelom

Michalek, Karel

Morávka, Jan

Gryc, Karel

DSpace at VSB Technical University of Ostrava

K. MICHALEK, J. MORÁVKA, K. GRYCMATHEMATICAL IDENTIFICATIONOF HOMOGENISATION PROCESSES IN ARGON STIRRED LADLEReceived – Prispjelo: 2008-09-09Accepted – Prihva}eno: 2008-12-10Original Scientific Paper – Izvorni znanstveni radINTRODUCTIONPreviously presented paper “Physical Modelling ofBath Homogenisation in Argon Stirred Ladle“ was de-voted to results of bath behaviour in the ladle model in-vestigation during its gas argon bubbling realised by oneor two stir elements situated in the ladle bottom. Studywas performed with use of physical modelling methodon a scale model 1 to 10. Development of homogenisa-tion processes after start of bubbling was evaluated onthe basis of electrical conductivity and temperaturechange, which were measured at three points of the ladlevolume by conductivity and temperature sensors. Theexecuted works were realised for conditions of 180 tonssteel ladles.It was found that the rate of homogenisation in thesteel ladle is significantly dependent on the volume flowof argon. Mathematical method for the identificationprocess was used for determining the optimum flow ofargon.MATHEMATICAL DESCRIPTIONOF PROCESSES AT ARGON BUBBLINGIt was appropriate to create a physical-mathematical(physically adequate) model for the measured timecourses of tracing substance concentrations in physicalmodel of pouring ladle (PL) occurred during argon bub-bling into the steel.Description of situationSchematic representation of situation at argon bub-bling into bath in the model of steel ladle (mLP) isshown in Figure 1.Argon bubbles flow from eccentrically situated stirelement (P) in the bottom of mLP at constant volumetricflow (q). They „disrupt“ the layer of concentration en-riched and coloured water with thickness (hk) and thereoccurs gradual mixing of enriched and clear liquid(steel, water). Two (pressure) forces II and I act basi-cally against each other on molecules of water in prox-imity of the sensors.METALURGIJA 48 (2009) 4, 219-222 219Mathematical models processed results of experimental investigation obtained during ladle gas argon bub-bling realised by stir elements situated in the ladle bottom. Exact theoretical description of processes occurringat argon bubbling into steel would be very complex and it would lead to a system of non-linear partial differen-tial equations describing transfer of momentum, heat, components, and with excitation function in the formof equation of so called deterministic chaos (argon bubbling). On the basis of pouring ladle model diagram andconcentrations courses, the simplified linear physically adequate model was proposed, which described beha-viour of steel concentration in pouring ladle during its bubbling. The analysed process was understood in theform of a cybernetic model.Key words: steel, ladle, gas argon bubbling, stir elements, physical modelling, mathematical identificationMatemati~ka analiza procesa homogenizacije u loncu s argonskim mije{anjem. Matemati~kim mo-delima obra|eni su rezultati eksperimentalnog instra`ivanja homogenizacije u loncu s argonskim mje{a~ima.Egzaktan teorijski opis procesa uvo|enja mjehuri}a argona u ~eli~nu talinu bio bi vrlo kompleksan i vodio bi kasustavu nelinearnih diferencijalnih jednad`bi za opis prijenosa momenta gibanja, topline i elemenata, uz do-datnu funkciju takozvanog deterministi~kog kaosa (gibanje mjehuri}a argona). Temeljem dijagramskog mode-la i koncentracijskih krugova predlo`en je pojednostavnjen, fizikalno odgovaraju}i linearni model, kojiprikazuje promjenu koncentracije ~eli~ne taline u ljeva~kom loncu tijekom upu{tanja mjehuri}a. Analiziraniprocess smatran je pritom kiberneti~kim modelom.Klju~ne rije~i: ~elik, lonac, upu{tanje argonskih mjehuri}a, mje{a~i, fizikalno modeliranje, matemati~ka identifi-kacijaISSN 0543-5846METABK 48(4) 219-222 (2009)UDC-UDK 669.18 : 621.746.32 : 542.63 : 621.22.015.5 : 517.442=111K. Michalek, K. Gryc, Faculty of Metallurgy and Materials Engineering,V[B-TU Ostrava, Czech RepublicJ. Morávka, Material & Metallurgical Research Ltd., Ostrava, Czech Re-publicDSpace VŠB-TUO http://hdl.handle.net/10084/76057 18/06/2012Measured dataAnalysis and synthesis of mathematical models wasrealised with use of the data measured on physicalmodel, where courses of the measured concentrations,(with period of sampling 0,5 s), had on the sensorsK1, K2 and K3 are shown in Figure 2.Several facts are obvious from the development ofconcentration in the sensors:– start and progress of gas argon bubbling into steelcan be approximately considered in the form ofthe Heaviside unit step function and it is thereforepossible to consider the development of concen-tration as a unit step response,– the sensors reacted only after elapsing of certain“dead time”, which is proportional to the distance ofindividual sensors from the liquid level in the mLP,– the overshoot of courses (apparently proportionalto the magnitude of the force I or rather to the dif-ference of the forces I and II) also descends withthe distance from the liquid level in the mLP,– steady-state (final) value of concentration is pro-portional to the proportion of volumes of pure wa-ter and water with enriched concentration.Physical-mathematical modelExact theoretical description of processes occurringat argon bubbling into steel would be very complex andit would lead to a system of non-linear partial differen-tial equations describing transfer of momentum, heat,components, and with excitation function in the form ofequation of so called deterministic chaos (argon bub-bling). On the basis of the mLP diagram and concentra-tions courses, the simplified linear physically adequatemodel was proposed, which described behaviour of steelelements concentration in PL during its bubbling. Ana-lysed process was understood in the form of a cyberneticmodel, which can be transparently visualised by socalled block diagram shown in Figure 3.It is series-parallel connection of three components,namely component of time delay and two parallel pro-portional (inertial) components acting against eachother.Two parallel and antagonistically connected sim-plest proportional systems with inertia of the 1st order(with transfers GI a GII) are assumed for the part of themodel without time delay 1:G sksG sksI II( ) , ( )11221 1 (1)where:k1, k2 - coefficients of transfer (amplification)of systems / %×s/m3,1, 2 - time constants of systems / s,s - complex variable in the Laplacetransform / s-1.For this part of the model it is then possible to com-pose on the algebra basis of transfer the following con-tinuous L-transfers (for zero initial conditions):G sksksk k k k ss( )( ) ( )( )(  11221 2 1 2 2 111 11   2 1s )(2)Wherefrom for the L-image and original of transferfunction H(s) and h(t) we get the final expression forstandard concentration transfer function, which can beused at the same time as non-linear regression model F1with three parameters k1, 1 and 2:h t c t k t k tn( ) ( ) exp( / ) ( ) exp( / )   	    	 1 11 1 1 2  (3)Regression model for all concentrationsFigure 4 shows courses of the regression functionbased on the model F1, for standard concentration val-ues in all three sensors K1, K2 and K3.220 METALURGIJA 48 (2009) 4, 219-222K. MICHALEK et al.: MATHEMATICAL IDENTIFICATION OF HOMOGENISATION PROCESSES IN ARGON STIRRED LADLEFigure 1. Argon bubbling into ladle modelFigure 2. The time response of concentrations on the sen-sors K1, K2 and K3Figure 3. Block diagram of processes in ladle model (mLP)DSpace VŠB-TUO http://hdl.handle.net/10084/76057 18/06/2012It is obvious from the results that the model F1 is suit-able and usable for description of concentration coursefor all three sensors. It is also obvious that the transfer co-efficient (gain coefficient) k1 decreases with the distanceof the sensor from the liquid level in the mLP, while thetime constant ô2 increases in this dependence.USE OF MODEL FOREVALUATION OF EXPERIMENTSProposed physically adequate model F1 was used forevaluation of the physical modelling results at argonbubbling in the ladle model. Its parameters were usedfor determination of other aggregated parameters, andall of them where used for definition of an optimumrange of blown argon flow. The paper documents resultsof one stir element for position „A“ in the ladle bottom.Approaches to solutionThree groups of parameters were chosen for charac-terisation of homogeneity „intensity“ measure in respectto blown argon flow q: basic physical parameters ofmodel (2 parameters – see the model in Figure 3 andtime delay, i.e. k1, k2, and d), aggregated parameter de-fining derivative time constant (1 parameter, i.e. D), andaggregated parameters defining maximum of modeltransfer function (2 parameters, i.e. tm, respectively tmax ,and hm). Aggregated parameters were chosen in order toachieve simplicity, clarity and appropriate inter-pretability of homogenisation process.Parameters of physically adequate modelGraphical dependencies of foregoing mathematicalmodel parameters on blown argon flow for the sensorK2 are shown in Figure 5:It is obvious from the Figure 5 that physically ade-quate model, in conformity with the assumed behaviourof real system, has evaluated a trend of dependence ofthe transport delay d on the argon flow rate. At higherflow rate homogenisation starts sooner and runs morerapidly.Derivative time constantOverall L-transfer of the defined model has characterof so called real derivative member with a delay of the2nd order 2, and derivative time constant D is functionof all three parameters. The constantc D an be expressedfrom the equation (2) with use of the relation:   D k  1 2 1 1( ) (4)Lower values of this derivative time constant ôD ex-press faster reaction of the model on change of the inputvalue, which is in this case represented by the volumeflow of argon.Graphical dependence of this time constant on the ar-gon flow (for measurement at the sensor K2) is shown inFigure 6.It is obvious from the Figure 6 that increase of thevolume flow of argon to 1,4 l/min causes a significantstep change of the value D and therefore also more rapidchanges in chemical composition of the model liquid(and development of homogenisation). The value of Ddoes not change significantly with further increase ofthe argon flow rate.Transfer function overshootThis approach uses analytical relations for determi-nation of transfer function position and transfer functionmaximum values. Graphical dependencies of the as-METALURGIJA 48 (2009) 4, 219-222 221K. MICHALEK et al.: MATHEMATICAL IDENTIFICATION OF HOMOGENISATION PROCESSES IN ARGON STIRRED LADLEFigure 4. Courses of regression of F1 model for data Figure 5. Dependencies of parameters d, k1 on argonflow for the sensor K2Figure 6. Dependence of derivative time constant D onargon flow for the sensor K2DSpace VŠB-TUO http://hdl.handle.net/10084/76057 18/06/2012sumed aggregated maximum parameters of the blownargon flow for the measurement at sensor K2 are shownin Figure 7 and Figure 8.Diagram in Figure 7 shows development of times forachieving of maximum values in the impulse function independence on the argon flow rate (tmx on the axis y is ageneral expression of time parameters tm, which is atime of achieving the maximum regardless of transportdelay d, or max). It is possible to obtain values of the pa-rameter tmax by adding the parameter m to the transportdelay d. Gradual convergence of tm and tmax is given bydecrease of the transport delay d with increasing vol-ume flow of argon. Faster circulation of volume of liq-uid results in faster achieving the maximum values ofthe impulse function. This speed, however, steeply fallsat argon flow rates below 1,4 l/min.The Figure 8 is devoted to a description of depend-ence of the maximum value of the impulse function (hm –maximum value of dimensionless concentration) on thevolume flow of argon. At its lowest flow rate (0,3 l/min)the injected substance gets to the sensor K2 in concen-trated condition without being mixed. At increase of theargon flow rate to 0,68 l/min the faster dispersion of themarking substance at the expense of the speed of circula-tion of the volume of model liquid manifests itself themost from the viewpoint of this parameter (hm). Furtherincrease in argon flow rate brings about again a gradualenforcing of speed of circulation. At the highest argonflow rate (3,25 l/min) the rate of circulation is too highand injected substance gets to the sensor K2 without be-ing able to disperse in the volume of the liquid.CONCLUSIONPhysically adequate mathematical model was pro-posed and verified for the measured time response of sub-stance concentrations in the pouring ladle model bath.This model was used for subsequent analysis of theinput simulation parameters influence on its coefficientsand aggregated parameters with consequences on ap-propriate (optimum) working mode setting for steelbubbling in the pouring ladle. Due to pre-set variousflows of argon blown into bath three groups of (aggre-gated) parameters for homogenisation intensity charac-terisation were determined and verified.On the basis of the shape of their course, the argonflow optimum range was deduced – within limits fromapprox. 1,4 – 2,3 l/min (on the ladle model), which canbe transformed with use of the appropriate volumetricflow scale factor to the realistic industrial equipmentconditions, i.e. 243 - 400 l/min.It was established in the previous paper entitled„Physical Modelling of Bath Homogenisation in ArgonStirred Ladle” that after achievement of a certain„breakpoint“ volume flow its further increase bringslow efficiency from the viewpoint of homogenisationtimes. Parameters assumed in this work enable more-over a more detailed investigation of the character of ho-mogenisation process, correct setting of which forms anintegral part of the advanced secondary metallurgy 3,4. For example magnitude of the maximum values ofthe impulse function (hm) at individual argon flow ratesgives evidence about behaviour e.g. of alloying addi-tions injected into the volume of real pouring ladle at theinitial stage of homogenisation.The work was realised within the frame of solution ofthe grant project No.106/07/0407 under financial sup-port of the Czech Science Foundation.REFERENCES1 A. Víte~ek, L. Smutný, J. Kusyn, Teorie øízení I. Základnípojmy a øe{ené pøíklady z Laplaceovy transformace. III.Management theory I. Basic notions and solved examplesfrom Laplace’s transformation. III., V[B-TU Ostrava,Ostrava, 1988.2 S. Kubík, et al., Teorie regulace - I. Lineární regulace The-ory of regulation – I. Linear regulation. SNTL, Praha,1974.3 J. Krajcar, Metalurgija 32 (1993) 3, 87-92.4 V. Ferketi}, J. Krajcar, A. Ivan~an, Metalurgija 35 (1996) 1,43-48.Note: The responsible translator for English language is B. [kandera,Czech Republic.222 METALURGIJA 48 (2009) 4, 219-222K. MICHALEK et al.: MATHEMATICAL IDENTIFICATION OF HOMOGENISATION PROCESSES IN ARGON STIRRED LADLEFigure 7. Dependencies of maximum times (tm, tmax) of theflow of inert gas for the sensor K2Figure 8. Dependencies of maximum values (hm) on theflow of inert gas for the sensor K2DSpace VŠB-TUO http://hdl.handle.net/10084/76057 18/06/2012

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Matematička analiza procesa homogenizacije u loncu s argonskim miješanjem

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