RH process is a metallurgical process, which serves for vacuum treatment of liquid steel by
recirculation method, which enhances quality of steel that was made either in basic oxygen converter
(BOF) or in electric arc furnace (EAF). The paper presents a project and verification of appropriate
physically adequate cybernetic anisochronous model of this process. This model was obtained on the
basis of measured and transformed data of dimensionless concentration of model liquid in the
physical model of a pouring ladle with the RH chamber, constructed in the same length (geometric)
scale 1:9 to the original objects. Mathematical model will be used for optimisation of RH process
under operational conditions.RH proces je metalurgický pochod, který slouží k vakuování tekuté oceli recirkulačním
způsobem za účelem zvyšování kvality oceli vyrobené v kyslíkovém konvertoru anebo v elektrické
obloukové peci. V příspěvku je prezentován návrh a ověření vhodného fyzikálně adekvátního
kybernetického anizochronního modelu tohoto procesu. Uvedený model byl získán na základě
naměřených a transformovaných údajů bezrozměrové koncentrace modelové kapaliny ve fyzikálním
modelu licí pánve s RH komorou, sestrojených ve stejném délkovém (geometrickém) měřítku 1:9 k
dílu. Matematický model bude použit k optimalizaci RH procesu v provozních podmínkách

Michalek, Karel

Morávka, Jan

Transactions of the VŠB - Technical University of Ostrava Mechanical Series

DSpace at VSB Technical University of Ostrava

91 Transactions of the VŠB – Technical University of Ostrava, Mechanical Series No. 2, 2008, vol. LIV, article No. 1627 Jan MORÁVKA*, Karel MICHALEK** ANISOCHRONOUS MODEL OF THE METALURGICAL RH PROCESS ANIZOCHRONNÍ MODEL METALURGICKÉHO RH PROCESU Abstract RH process is a metallurgical process, which serves for vacuum treatment of liquid steel by recirculation method, which enhances quality of steel that was made either in basic oxygen converter (BOF) or in electric arc furnace (EAF). The paper presents a project and verification of appropriate physically adequate cybernetic anisochronous model of this process. This model was obtained on the basis of measured and transformed data of dimensionless concentration of model liquid in the physical model of a pouring ladle with the RH chamber, constructed in the same length (geometric) scale 1:9 to the original objects. Mathematical model will be used for optimisation of RH process under operational conditions. Abstrakt RH proces je metalurgický pochod, který slouží k vakuování tekuté oceli recirkulačním způsobem za účelem zvyšování kvality oceli vyrobené v kyslíkovém konvertoru anebo v elektrické obloukové peci. V příspěvku je prezentován návrh a ověření vhodného fyzikálně adekvátního kybernetického anizochronního modelu tohoto procesu. Uvedený model byl získán na základě naměřených a transformovaných údajů bezrozměrové koncentrace modelové kapaliny ve fyzikálním modelu licí pánve s RH komorou, sestrojených ve stejném délkovém (geometrickém) měřítku 1:9 k dílu. Matematický model bude použit k optimalizaci RH procesu v provozních podmínkách.  1 INTRODUCTION During the period 2006/2007, in the Laboratory of physical and numerical modelling of metallurgical processes at the Department of metallurgy (DM) of the Faculty of Metallurgy and Materials Engineering of the VŠB-Technical University of Ostrava, were manufactured physical models of PL and RH chamber in the same length (geometric) scale 1:9 to the original objects. The first series of experiments was realised on these models in May 2007.  2 DESCRIPTION OF MODELS AND PROCESSES Schematic representation of situation at vacuum treatment of steel (in water model) by an inert gas (argon) in the model of the pouring ladle (mPL) with use of the model of the RH chamber (mRH) is shown in Fig. 1. Brief technological description of the process is the following: negative pressure (V) “lifts up” the level of the model liquid (coloured water enriched in concentration) and pushes it into the RH chamber. After stabilisation argon (bubbles) with constant volume flow q1 is brought into the bottom part of the inlet tube (marked A - left) primarily from special nozzles (Tr), located along the perimeter                                                                                                                                                                    * Ing. Ph.D., Třinecký inženýring, a.s., Frýdecká 126, Třinec - Staré Město, tel. (+420) 558 53 2192, e-mail jan.moravka@tzi.trz.cz ** Prof. Ing. CSc., Department of Metallurgy, Faculty of Metallurgy and Materials Engineering, VSB - Technical University Ostrava, 17. listopadu 15, Ostrava-Poruba, tel. (+420) 59 699 5213, e-mail karel.michalek@vsb.cz 92 of the tube. S1S2S3PInmLPmRHA B12VTr Fig. 1 Blowing of argon into the pouring ladle model with the RH chamber It is possible to bring argon (with constant volume flow q2) secondarily also from eccentrically situated blowing element P (blowing brick) at the bottom of the mPL. Tracing element (time 0) is injected at the tube inlet (In) at model experiments. Due to flowing of liquid through the chamber the tracing substance passes with the liquid through the inlet tube (A) and leaves the chamber through the outlet tube (marked B - right) into the ladle (mPL). The electrical conductivity sensors S1, S2, S3 register concentration changes in the bath during the vacuum treatment. After the liquid passes through the chamber it is “pumped” again by the inlet tube (A) and after it has passed the RH chamber the sensors indicate already lower concentrations. This cyclic progressively homogenises the contents of the pouring ladle till stabilisation at the final concentration of the bath. Two pressure forces are in fact exerted on molecules of the liquid near the sensors S2 (situated lower), and mainly S3: 1 (recirculation, cyclic) and 2 (resistant).  3 MEASURED DATA Analysis and synthesis of mathematic model was realised on the basis of the results from the first series of experiments. Time behaviour of the measured concentration c3(t) at the sensor S3 in experiment Test8 was chosen for documentation and representation – see Fig. 2: RH chamber physical model - Test8, sensor 300.020.040.060.080.10.120 20 40 60 80 100 120time at model [s]KCl concentration [% weight.] c3(t) [% weight.]cpcu Fig. 2 Time behaviour of the measured concentration at the lower sensor S3 93 The time behaviour of concentrations at sensors shows several obvious general facts. Start and time behaviour of bubbling of gas through steel can be reflected by approximation in the form of Heaviside unit step and it is therefore possible to understand time behaviour of concentration as unit step response. At the initial phase of the first overshoot of unit step response certain time delay is obvious, which is given by the time from the injection of tracing substance till its registration by the sensor. This fact is not so directly and outwardly obvious at following cycles (overshoots). Although the existing time delay is still taking effect but it is hidden latent “internal”.  4 FIRST IDENTIFICATION OF THE PLANT It is possible to assess on the basis of theoretical knowledge and experience that transient performance of investigated system reminds oscillating (so called inherent) proportional plant with inertia of the 2nd order (Sp2k). This plant could be physically adequate for investigated process since it is possible to define for the liquid both hydraulic capacity and hydraulic inductivity [NOSKIEVIČ 1999]. Verification of this hypothesis is possible by deterministic identification with use of unit step response. The relevant literature – e.g. [NOSKIEVIČ 1999], [VÍTEČKOVÁ 1998] and [FIKAR & MIKLEŠ 2003] states however that evaluation of unit step responses of oscillating proportional regulated plants is very demanding. The above cited literature references give three different simple approaches to identification. But results of the mentioned identification methods indicate unsuitability and inapplicability of this model (due to the fact that it is impossible to achieve with it the necessary damping and at the same time corresponding periods of the unit step response).  5 SECOND IDENTIFICATION OF THE PLANT Failure of the first falsely „heuristic“ identification points to the necessity of use of mote complicated mathematical model.  5.1 Approaches to solution Exact physical-mathematical (theoretical) description of processes running at vacuum treatment of steel with use of argon would be very complicated and it would lead to the system of non-linear partial differential equations describing transfer of momentum, heat, components with exciting function in the form of an equation of so called deterministic chaos (argon bubbling). This approach was for the RH process in certain extent already described in the literature. Another suitable solution consists in use of so-called compartments [ZÍTEK 1990]. Compartment is a fictitious element formed by concentration and separation of the object properties in certain zones of the space. In the analysed case of behaviour of processes in the model of PL with RH chamber it is possible to arrive after intuitive spatial discretisation of continuum (bath) to two compartments acting against each other and marked „pressure powers“ 1 and 2 (Fig. 1).  5.2 Anisochronous proportional plant of the 1st order We will use hereinafter a simple basic anisochronous proportional plant (model) with inertia of the 1st order (with time delay at the input and also at the feedback, Sap1) with Laplace (L) transfer:  )()()()()exp()exp()(1111sNsMsUsYsTsTsTkesTeksGdydusTsTdydu ==−+−⋅=+⋅= −−, (1) where: k1 – static sensitivity coefficient (≈ transfer) of the model (plant), T1 – transition time constant ≈ time constant of the plant [s], Tdu – time delay of the input value of the plant [s], Tdy – time delay of the output value in a feedback [s], Y(s) – L-image of the output which is the measured concentration, U(s) – L-image of the input which is the argon flow rate, M(s) – numerator of a transfer function (zeroes of transfer), N(s) – denominator of a transfer function (poles of transfer). Characteristic equation (which is transcendent in the form of the so called quasi-polynomial 94 and allows therefore unlimited number of solutions in the complex range – see [ZÍTEK & VÍTEČEK 1999]) is the following for this plant:  0)exp()( 11 =−+=+= − sTsTesTsN dysTdy . (2) Its solution in the complex range is possible with use of special function of complex variable, so called Lambert W – function, see [WEISSTEIN 2005], from which we get after more transparent and simple notation Tdy = Td, the following:  ddd TTTWTTs⎟⎟⎠⎞⎜⎜⎝⎛−= 11),( . (3) It is possible to choose for more illustrative view in a 2D plane for example T1 = 10 s and to show (see Fig. 3) dependence of real and imaginary part of solution (3) of the above mentioned characteristic equation (2) on the parameter of time delay Td ∈ (0, 20) s: 0 2 4 6 8 10 12 14 16 18 20-0.3-0.25-0.2-0.15-0.1-0.0500.050.10.150.2Solution of the characteristic equation Sap1Td [s], Td/T1 [-] : T1 = 10 sLambertW(-Td/T1)/TdrealimagstrongweakAperiodicity Stable periodicity strong-1/T1  Fig. 3 Real and imaginary part of the solution of characteristic equation Sap1 Fig. 4 shows graphic presentation of marginal values (important points) of changes of character of unit step response: 0 0.59 1π/21.57aperiodicalstrongly weaklyoscillating, periodicaldamped stable unstableTdyT1Transient process of Sap11/e0.37 Fig. 4 Marginal values of transient process of the model Sap1  5.3 Differential compartment model On the basis of the scheme mPL and mRH, as well as time behaviours of concentration, a simplified so-called minimum model was proposed. This model is based on counterwork of compartments („pressure powers“) 1 and 2 (see Fig. 1) and in past this principle was successfully used for the model of the pouring model itself mPL [MORÁVKA, MICHALEK & KOHOUT 2006]. 95 The described process was interpreted as cybernetic model, which can be illustratively demonstrated by the block diagram – see Fig. 5: G2+_Q(s) C(s)u(t) y(t)G1x1(t)x2(t)Sap1Sp1  Fig. 5 Block diagram of mRH The model is valid for any sensor S1, S2, S3, notation Q(s) means L-image of argon flow, C(s) is L-image of concentration. It is therefore connection of two parallel and counter-acting proportional elements, namely one anisochronous and the second classical isochronous. Generalised functional or anisochronous state formulation of the proposed model of the RH process has the following form:  ⎪⎪⎪⎭⎪⎪⎪⎬⎫⎪⎪⎪⎩⎪⎪⎪⎨⎧−=⋅−⋅−=⋅−⋅=−⋅−⋅=)()()()(1)(1)(1)()()(1)()(21222122222111111txtxtytxTtuTktxTtuTkdttdxTtxTtuTkdttdxd. (4)  5.4 Modelling and simulation identification of the model The differential compartment anisochronous model mPL with mRH was in graphical form modelled in the environment of the program Simulink (see Fig. 5). However, modelling and simulation parametric identification was much simpler in the environment of the simulation program 20-sim 2.3 Pro Shareware [ZÍTEK & PETROVÁ 1996], simply by entering the textual form of the system of equation (4). Fig. 6 shows the graphical output of result of simulation identification in the program 20sim:  Fig. 6 Measured and approximated time behaviour of the unit step response mRH 96 It is obvious from the Fig. 6 that simulation parametric identification offers acceptable and usable results. The following values were obtained for normalized values of concentration from the sensor S3 at the experiment Test8: k1 = 5.2, T1 = 8.2 s, T2 = 12.8 s, Td = 7.5 s.  5 CONCLUSIONS The paper describes compartment physically adequate approach to searching of suitable cybernetic-mathematical model of behaviour of equipment of the pouring ladle with RH chamber with use of signal of measured and normalized concentration of tracing substance in reduced physical model. Dampened oscillating time behaviour of gas (argon) blowing was modelled by combined (differential) compartment model, containing anisochronous proportional plant of the 1st order. Results of analyses will be further developed and used for optimisation of the operating mode of gas blowing (determination of its optimum flow) on the given equipment at the Třinec metallurgical plant (TŽ), a.s. for determination of the so called circulation mass (or volumetric) flow of liquid (steel) through the chamber, which cannot be measured directly (not even in modelling conditions, mainly due to argon supply), also for teaching at the Technical University of Mining and Metallurgy in Ostrava (VŠB-TUO).  This work was realised within the frame of solution of the grant project No. 106/07/0407 under financial support of the Grant agency of Czech Republic.  REFERENCES [1] NOSKIEVIČ, P. Modelování a identifikace systémů. Ostrava : MONTANEX, 1999. 276 p. ISBN 80-72225-030-2. [2] VÍTEČKOVÁ, M. Seřízení regulátorů metodou inverze dynamiky. 1st ed. Ostrava : FS VŠB-TU Ostrava, 1998. 56 p. ISBN 80-7078-628-0. [3] FIKAR, M. & MIKLEŠ, J. Identifikácia systémov. 1st ed. Bratislava : CHTF STU Bratislava, 1998/2003. 114 p. ISBN 80-227-1177-2. [4] ZÍTEK, P. Simulace dynamických systémů. 1st ed. Prague : SNTL, 1990. 420 p. ISBN 80-03-00330-X. [5] ZÍTEK, P. & VÍTEČEK, A. Návrh řízení podsystémů se zpožděními a nelinearitami. 1st ed. Prague : ČVUT Praha, 1999. 165 p. ISBN 80-01-01939-X. [6] WEISSTEIN, E. W. Lambert W-function [online]. MathWorld, 2007 [cit. 2007-05-16]. Available from www: <http://mathworld.wolfram.com/LambertW-Function.html> [7] MORÁVKA, J., MICHALEK, K. & KOHOUT, J. Matematické zpracování přechodových dějů při prodmýchávání oceli v licí pánvi. In Proceedings of 22nd conference with international participants Výpočtová mechanika 2006, Volume II. Plzeň : ZČU Plzeň, 2006, p. 379-386. ISBN 80-7043-477-5. [8] ZÍTEK, P. & PETROVÁ, R. Matematické a simulační modely. 1st ed. Prague : FS ČVUT Prague, 1996. 128 p. ISBN 80-01-01524-6.  Reviewers: prof. Ing. Roman Prokop, CSc., Tomas Bata University in Zlín prof. Ing. Boris Rohaľ-Ilkiv, CSc., Slovak University of Technology in Bratislava 

Anizochronní model metalurgického RH procesu

http://dspace.vsb.cz/bitstream/10084/70851/2/1627_MORAVKA_MICHALEK.pdf

Anizochronní model metalurgického RH procesu

Abstract

Similar works

Full text

Available Versions

DSpace at VSB Technical University of Ostrava