V I Dubodelov

Yu M Gorislavets

A I Glukhenkii

V M Fikssen

Electromagnetic stirrer of liquid metal with alternate action of traveling and pulsating magnetic fields

Keywords: electromagnetic stirrer of liquid metal, traveling and pulsating magnetic fields, two-limb and three-limb inductors, single-circuit and dual-circuit vortical flows, computer modeling

To improve the quality of the electromagnetic stirring of liquid metal periodically alternate a force to the metal of the traveling and the pulsating magnetic fields has been proposed. Two constructions of the electromagnetic stirrer of the metal in the form of two-limb and three-limb inductors are considered, that allow implementing of such a method of stirring due to the periodic alternation of their multiphase and single-phase power supply. With reference to reverberatory furnace for aluminum based on the 3D computer modeling electromagnetic and hydrodynamic processes in the system "inductora bath of liquid metal" have been investigated. It is shown, that in the case of multiphase power supply both inductors produce in the furnace bath single-circuit vortical flow of metal, but in the case of single-phase power supplydual-circuit vortical flow. Thus stagnant zones of metal that presents in one mode are effectively stirred in another. The durations of transient hydrodynamic processes have been identified. A comparative assessment of the effectiveness of such stirrers for different modes of operation has been executed.

Introduction

Stirring of liquid metal is important technological process, which is performed during the melting of metals and alloys preparation on purpose to intensify technological processes and improve the quality of cast billets. Such operation is carried out mainly by the three-phase or two-phase electromagnetic stirrers that create traveling or rotating magnetic fields [START_REF] Tir | Electromagnetic devices to control the circulation of the melt in electric furnaces[END_REF][START_REF]Non ferrous casting: Equipment MHD Technology[END_REF]. Especially acute need for stirring the melt, as known, occurs in reverberatory furnaces and mixers a heat source of metal in which is located on the top (above the surface). Frequently in such cases electromagnetic stirrer of traveling magnetic field is used, that attaches to the sidewall of the furnace bath, whereby in it a single-circuit vortical flow of liquid metal is created. Such flow involves in motion mostly peripherals (parietal) layers of metal. In the central part of the bath stagnation zone is formed in which metal is not sufficiently stirred. The situation may be improved by using an electromagnetic stirrer with a pulsating magnetic field [START_REF] Dubodelov | [END_REF]. Such stirrer, unlike stirrer with a traveling magnetic field, creates a dual-circuit vortical flow in the bath, which effectively stirs just the central zone of furnace bath (Fig. 1). Fig. 1: Sketch of the bath of the reverberatory furnace with electromagnetic stirrer of pulsating magnetic field In order to further improving of the quality of stirring of the metal melt in the reverberatory furnace bath it is proposed to use alternating electromagnetic effect on the liquid metal of traveling and pulsating magnetic fields [4]. Is obvious that if periodically to alternate single-circuit and dual-circuit vortical flows it result to the more qualitative stirring of liquid metal in the entire volume of the furnace bath. Stagnant zones that occur in one mode of operation will be effectively mixed with the other mode. For realization of such method of stirring, electromagnetic stirrer, which is able to create these flow structures, is required. Such stirrer may be obtained by the inductors basis with two-limb and three-limb magnetic cores (C-and Eshaped cores), on each limb of which are arranged electrical coils (Fig. 2). By connecting each of these inductors to a single-phase voltage in the liquid metal of the furnace is excited pulsating magnetic field, and in the case of multiphase power supplytraveling magnetic field. The aim of this work is to carry out research of electromagnetic and hydrodynamic processes in the system "inductorbath with liquid metal" for the above design solutions of the electromagnetic stirrer in relation to the reverberatory aluminum furnace based on the 3D computer modeling.

Formulation of the problem

Studies were performed on the basis of the numerical solution of the problems of the electromagnetic field and fluid dynamics for each of the electromagnetic system of the stirrer for their single-phase and multiphase power supply. The calculation of the hydrodynamic flow of the liquid metal was carried out for both steady state and unsteady (transient) conditions, which occur by switching from one type of power supply to another. In formulating electromagnetic problem the following simplifying assumptions were taken. The system was considered in the linear approximation. Laminated magnetic core of each inductor was considered electrically non-conductive, magnetic permeability of it is taken constant and hysteresis loss and eddy currents were neglected. Real inhomogeneous structure of coils containing conductive (wires) and non-conductive (frame of coil, insulation) elements was represented a homogeneous non-conductive environment with a uniformly distributed and predetermined current density. Calculations were performed in non-inductive approach excluding the electromotive forces induced in the liquid metal due to its movement. This approximation was based on the evaluation of the magnetic Reynolds number Re m , the value of which in this case turned out to be much less than one. The calculation of the electromagnetic field was carried out by the numerical solution of differential equations for the complex amplitudes of the magnetic vector A  and electric scalar   potentials e r 0 r 0

2 1 r 1 0 ) ( ) ( ) ( J A A                           j j , (1)
where μ 0 and ε 0permeability and permittivity of vacuum, μ r and ε rrelative permeability and relative permittivity, σ -electrical conductivity, ωangular frequency, e J the complex amplitude of the external current density with a given distribution.

The hydrodynamic problem was solved for the area of the liquid metal in the bath of the furnace. The velocity distribution of the melt was determined by means of numerical solution of differential Navies-Stokes equations for viscous turbulent flow

  f u u u u                eff m m ) ( p t ; (2) 0    u , ( 3
)
where u -the velocity of the liquid metal, ρ mmetal density, p -pressure, η effeffectiveness ratio of the dynamic viscosity of the metal, determined using k-ε turbulence model, fspecific electromagnetic forces in the liquid metal, calculated by solving the electromagnetic problem. The boundary conditions on the walls of the furnace bath were set in the form of logarithmic velocity profile for the boundary layer. To simplify the problem the same condition was assumed on the upper surface, that allowed to exclude from consideration the deformation of the free surface of the liquid metal in bath. The equations [START_REF]Non ferrous casting: Equipment MHD Technology[END_REF][START_REF] Dubodelov | [END_REF] The hydrodynamic efficiency of both design variants of the stirrer and of their operation modes was estimated by averaging over the entire volume of the liquid metal V m melt rate

dV u u u V W V z y x     m 2 2 2 m 1 . (4)
As criterion of efficiency, taking into account both the hydrodynamic "performance" and power of the stirrer, was taken relation W S = W/S 1/2 , where Selectromagnetic power, determined from the calculation results of the electromagnetic field in accordance with the expression ) 2

1 ( Abs ) ( Abs c dV j S S V      J A  . ( 5
)
Here S ~complex electromagnetic power, V cvolume of stirrer coils. The real part S ~ in the case of inductionless approximately is equal to the power of heat from eddy currents in the liquid metal. Imaginary part S ~ corresponds to the reactive power of the electromagnetic system. Possibility of taking W S as a generalized specific criterion of stirrer efficiency is caused by turbulent motion of the melt in the furnaces during electromagnetic stirring. Since the friction force herewith, as known, is proportional to velocity squared, and electromagnetic forces in the melt is proportional to ampere-turns of the inductor to the square, then the result the speed of the melt will be proportional to ampere-turns and, respectively, to the square root of inductor power. Thus, for each version of design of the stirrer selected criterion of efficiency W S will remain constant for all values of ampere-turns.

Discussion of the results of simulation

Simulation of electromagnetic and hydrodynamic processes was carried out at the following initial data. The volume of liquid metal (aluminum) located in the furnace bath is (3 × 2 × 0.5) m 3 . The location of inductors at the side wall of the furnace bath was considered symmetrically with respect to the metal both in length and height, at a distance there from of 0.1 m. It was considered that between the inductor and the liquid metal there is a non-conductive insulating wall. The basic geometrical sizes that are shown in Fig. 2, for both inductors were taken the same. They constitute: core length l = 0.6 m, width b = 0.3 m, its height h = 0.3 m, the width and the length of the limb (rod)b r = 0.2 m and l r = 0.2 m. The size of coils are equal, their length l c equal to the length of the rod l r , and the cross-sectional width b c = 0.05 m. All calculations were performed during power supply of stirrers by current of industrial frequency (ω = 2π•50 1/s) at predetermined the total ampere-turns of all stirrer coils, which are uniformly distributed over the coils. It was assumed that the peak value of these ampere-turns for each stirrer kA 60

1 i i i    n w I
(nthe number of stirrer coils). Herewith the value of the current density in the coils computational domain was set as the J e = I i w i / (l c b c ). In the pulsed magnetic field mode currents in adjacent coils of both stirrers were knitted among themselves at an angle ψ I = 180 e. deg., and in the mode of traveling field considered two variants, when this angle is 60 and 120 e. deg. As a result of calculation in such a way was found that both stirrers in case of single-phase power supply (ψ I = 180 e. deg.) creates in the bath furnace dual-circuit vortical flows of the liquid metal, and when the multiphase power supply (as in the case of ψ I = 60 e. deg., and when ψ I = 120 e. deg.)single-circuit. In Fig. 3 the structures of the metal flow (velocity field) in the average height sectional volume of the liquid metal for the three-limb stirrer are shown. Fig. 3а refers to single-phase power stirrer, Fig. 3b to three-phase power of the stirrer with the phase shift angle between the neighboring coils ψ I = 60 e. deg. Qualitatively similar form has also metal flow in the case of two-limb stirrer. The numerical solution of nonstationary hydrodynamic problem make possible to trace in time transformation of vortex flows in furnace bath during changing power supply regiments. In reference [5] you can see the video, which demonstrates the calculated picture of the flow of molten aluminum in the bath furnace as velocity fields generated by the alternate action of the pulsating and traveling magnetic fields for the three-limb inductor. Initially, this video shows how under the influence of a pulsed magnetic field established dual-circuit vortical flow of liquid metal in the bath when to single-phase power inductor is applied. Then inductor switches to the three-phase power supply, resulting in dual-circuit flow transforming to a single-circuit flow. After that by switching back of the inductor to a single-phase power supply the flow is transformed in dual-circuit flow again. And for the completion demonstrates transformation of dual-circuit flow of metal to the single-circuit reverse direction flow when switching to the three-phase power supply with phase reversal (compared with the previous case). The table below shows the obtained estimated values of key parameters of two-limb and three-limb inductors, where Selectromagnetic power of the stirrer, P mpower dissipated in the liquid metal in the form of Joule heat, Вaverage magnetic induction in the magnetic core. Analyzing the table data, it should be noted that both stirrers are more efficient in the mode of traveling magnetic field when two-limb stirrer powered by a two-phase voltage and three-limb stirrerby a three-phase power supply. This follows both from the average speed W and from the criterion of W S . Comparing these inductors together at the mode of pulsed magnetic field turned out to be more effective stirrer with two limbs, and in the mode of traveling field it is three-limb stirrer. As for the multi-phase power mode with different phase angles ψ I , then, based on the criterion of W S , the work of two stirrers at an angle ψ I = 60 e. deg. is more effective according to the obtained data.

Conclusions

Thus, represented designs of inductors, which are able to separately create pulsating and traveling magnetic fields that allow to implement a new method of electromagnetic stirring of liquid metal in the bath of reverberatory furnace by alternately (periodically) change the structure of the vortex flow. The analysis shows that both constructs of the electromagnetic stirrer are approximately equal in efficiency. The final choice in favor of any decision can be made on the basis of the review and including technological aspects of the stirring process, evaluation of the technical capabilities of the power supply system and other features.

Fig. 2 :

 2 Fig. 2: Two-limb and three-limb inductors

Fig. 3 :

 3 Fig. 3: The flow of liquid metal in the furnace bath under the action of the pulsing (a) and traveling (b) magnetic fieldsWhen applying single-phase power supply on every stirrer the establishing time of dual-circuit flow of the metal in the bath with the above sizes was about 100 s, and when they are connected to the multi-phase power supply establishing time of the single-circuit flow was ~150 s. When switching the stirrer from one to the other flow mode transition process time (time of transition from one steady state to another) was approximately equal to 120 s. At unsteady hydrodynamic calculations it was neglected of transient electromagnetic processes, during switching power supply of the inductor, because of their relatively shortness. The table below shows the obtained estimated values of key parameters of two-limb and three-limb inductors, where Selectromagnetic power of the stirrer, P mpower dissipated in the liquid metal in the form of Joule heat, Вaverage magnetic induction in the magnetic core. Analyzing the table data, it should be noted that both stirrers are more efficient in the mode of traveling magnetic field when two-limb stirrer powered by a two-phase voltage and three-limb stirrerby a three-phase power supply. This follows both from the average speed W and from the criterion of W S . Comparing these inductors together at the mode of pulsed magnetic field turned out to be more effective stirrer with two limbs, and in the mode of traveling field it is three-limb stirrer. As for the multi-phase power mode with different phase angles ψ I , then, based on the criterion of W S , the work of two stirrers at an angle ψ I = 60 e. deg. is more effective according to the obtained data.

Table ψ

 ψ

			Two-limb inductor				Three-limb inductor
	I , e. deg.	W, m/sec	W S , m/(sec•kVA 1/2 )	S, kVA	Р m , kW	В, Т	W, m/sec	W S , m/(s•kVA 1/2 )	S, kVA	Р m , kW	В, Т
	180	0.1	0.0054	410	14.9	0.87	0.08	0.0043	358	10.6	0.94
	60	0.12	0.0076	254.5	7.1	0.53	0.24	0.0185	164	4.4	0.56
	120	0.118	0.0062	358.1	10.3	0.77	0.27	0.0155	301	6.8	0.86


A Yu Kizilova
email: kizilova_au@mail.ua

M A Slazhniev

K S Bogdan

Kyung Hyun Kim

Hyun Suk Sim

Dynamic control of operational characteristics in magneto-weighting casting installation (MWI) for aluminium alloys

Keywords: dynamic control, magneto-weighting installation, operational characteristics, electromagnetic forces

In this study by the analysis of the possibilities for direct measurement of the melt weight into magnetodynamic installations was substantiated the ways for realization of direct and indirect dynamic control of pouring processes and managing of it operational characteristics. In the course of studying of the pouring process has been estimated the roles of inertial drain of melt on the the accuracy of dosing by open stream method. The special algorithm for dynamical control by the "tracking" method was developed and tested, which include the 4steps recalculation the difference poured melt mass (m i+1 -m i ) within corresponding time (t mi+i +t mi ). Application the "tracking" method at indirect correction of the melt flow rate at pouring from MDI to casting mold is allowed to reduce the it oscillation to ±2% in comparisons with existing 6% before. Especially for widely using in the industrial conditions has been developed the prototypes of simple devices and methods for dynamic control flow rate of pouring melts into molds, that based on measurement of metal flow rate by determining of "instantaneous value" of force, which is creating by the mass of the melt at it flowing via sloped tray.

Introduction

For providing of controlled electromagnetic pouring and discrete dosage the set portions (dose) of aluminum alloy in PTIMA NAS of Ukraine the magneto-weighting installation (MWI) [1] was developed. The MWI is a magnetodynamic installation (MDN-6A) [2] which are placed on the weight measuring system (WMS) as a scales platform. It is allow producing the continuous static and dynamic (in real time mode) high-precision measurement of the molten melt weight in crucible and channels with accuracy 0.1%. That practically was used for weight control of melt discrete dosing from MDI in a dose range 2~400 kg and accuracy from 2.0% to 0.5%. In the simple execution the weight measuring device there is a scales platform with four strain gauges sensors (BSA Shear beam load cell 2000 kg, with accuracy 0.02%), on which the MDI was placed as shown on Fig. 1. Each load cells to a digital microprocessor unit CAS CI-8000V was connected. The outputted from this device analog/digital signal further using for control of the electromagnetic pump of MDI at pouring of set portion of melt to casting mold by measure decreasing total mass MDI by detaching methods [2,3]. The dosage process by achieving of set unloading value (in range from -2 up to -400 kg) on the weighting systems was produced and by generating the "stop of pumping" command to the control circuit of MDI. By its means the row dosage methods has been tested at application in MDI. There are possible to make the traditional mode of electromagnetic melt pouring from metalduct to casting mold at constant mass flow rate. For precision melt dosing, for example doses 1400 kg with an accuracy of 10.5%, the two-stages pouring mode has been developed. Second stage of pouring included the refilling mode at reduce flow rate (Q) in the output dosing phase. Especially for the pouring melt to die-casting machines can be used the special mode with braking stream of melt in the finishing phase of dosing. For braking stream of melt at the finishing of pouring have used the reversing electromagnetic force action. That to allowing achieved the dosage accuracy greater than ±1.5% for doses of 0.5 kg by the decreasing of inertial process negative overfilling melt from metallduct at stopping of electromagnetic pump. However the existing dosing weighting system not allow producing the wide dynamic controlling of flow rate melt making during pouring from MDI to casting mold. Because it is can be predetermine set up only before starting of pumping and can be changed only manually in little range ±18%.

Presentation of the problem

At exploitation MDI as a mixer-batcher, for example at the dosing aluminium alloy to casting mold on conveyer, the of melt temperature (TC), mass flow rate at pouring (Q (t)) and electromagnetic pressure (p em ) must be maintaining on set position and not changing. There is a total requirement for dosing device in the industries application.

However, the electromagnetic pressure (p em ) in the electromagnetic pump of MDI is possible maintaining as constant value, that is depend from parameters (voltages) of electrical powering on inducers and electromagnet (Pem ≈ f(U ind, Uem). But hydraulic pressures (H) is so very depend from the difference between melt level in crucible (h level ) and output nozzle of metallduct (Fig. 1), exactly at the melt pouring by open stream methods. By this means, the process of pouring alloy from metallduct is equal to overpressure (Δp em ) above the nozzle of metalduct (Fig. 1.). In result at the decreasing of melt level in crucible MDI the useful overpressure also decreasing up to appearing of full stopping outfiling melt from metallduct.

For solving this problem and for automatic maintaining of flow characteristics (Q met ) in MHD-installation was developed and implemented the original methods of dynamic control, by which the maintaining of the setpoint of flow rate characteristics independence from the level of metal in the crucible was provided [1]. Essence of dynamic mass flow control is consisted in the continuously determine the actual difference between the surface of the metal in the crucible and the level in spout of metallduct (L metalduct ) for calculation existed overpressure value p em ≈ H = H emh met (1). Defined by this way p em is compared with initial conditions at start of pouring, and at the it reducing or increasing was produced the correction of voltage supplying to the electromagnetic pump of MDI.

    2 / 3 met metduct v h L Н g 2 b H Q              , kg/sec (1)
wherein Q(H) -flow rate as a function of the pressure, kg/sec; ε v -coefficient of vertical stream compression; b -width of the nozzle of metallduct, m; ε -side compression ratio of stream (ε = 0.85~0.9); φ -flow rate ratio (0.97), Hhydraulic pressure, m; h metlevel of melt in crucible, m; L metductlenght of metallduct, m.

Take into account that measurement of actual height of aluminium alloy level in the crucible by contacting or optical methods is difficult and required of expensive equipment, in study was implemented methods of the in real time recalculation the mass of alloy into MDI to it volume and next to height it level into crucible. For this purpose the experimental pragmatic model as a dependence from metal mass (m met ) was developed -h met.lev (m met ) = 0.569 m met + 5.385. As an incoming the analog signal U WMS from weight measuring devise has been used (m met = f(U WMS ) = [(863× U WMS )/550]-1). For practical application of this methods the special algorithm was developed and programmed into the microcontroller Mitsubishi ALFA AL2-14 MRD (used now as automatic control system ACS), which was installed into the control circuit of magnetodynamic installations with weighting system. Experimental approbation of the developed correction methods to allowed maintained the initial flow rate in the range of from 6.0 to 3.81% (for pouring velocities 2~6 kg/sec), and achieved it independence from melt level in the crucible (h met.lev = 50~350 mm). Accuracy of the discrete dosage melt portions 10.5 kg is makes 0.7%, and without using -2.75%.

Development the methods for indirect dynamic control of flow rate pouring aluminium alloys.

For analysis of the hydrodynamics processes at discrete dosing the experimental dependence of the 2 kg aluminum pouring within time 2.5sec was registered by the means of weight measuring device (Fig. 2, curve 1). Its studying is showed of the oscillations of the mass flow rate of the melt during it pouring from MDI, that at values of flow rate Q=0.8 kg/sec makes 5~15%. These oscillations is a result of the appearing the vortex structures in T-shape working area (WA) of MDI and its interaction [1] at the creation of electromagnetic force. In additional the fluctuation of hydrodynamic characteristics (pem) have a 5 to 15% from nominal electromagnetic pressure and appearing at achieving of linear velocity of the melt moving via working area MDI more than 1~3 m/sec. Frequency analyses of pressure fluctuations shown of 3 main harmonics in range 0.4~0.8 Hz, 2.5~3 Hz, and 4.0~4.4 Hz. Such fluctuations of pressure and flow rate characteristics is seriously are reducing the accuracy of discrete dosing, and there are impacting to the inertial drain of melt from metallduct after stopping of electromagnetic pump. There is shown on Fig. 2 (line 1) that the uncontrollable drain process have a time 0.5sec and 20% of total pouring time. Impact the inertial drain melt dependent from the influence of the force "momentum" J m (2), and characterize the any quantity of aluminum melt, drained under the action of inertia forces in the end of pouring:

J m = (mmet×Q i )/(S met.duct ×ρ Al ), kg (2)
where m met -weight of melt in channel/metalduct, kg; Q i -instant value of flow rate (Q = x met ×S met.duct ×ρ al ), kg/s; x metlinear velocity of melt in metalduct, m/s; S met.duct -cross section of metallduct, m 2 ; ρ al -density of Al alloy, kg/m 3 .

In order to this for reducing negative influencing of inertial drain processes and achieving of high accuracy of dosing it is a pouring time should be increase at increasing value of flow rate Fig. 3. However, for improving possibilities of pouring control in this study was developed the original methods online calculation of presence flow rate flowing melt, as the first derivative of the value of the mass changes by time m'=f(t).

The special algorithm for dynamical control by the "tracking" method was developed and tested, that include the 4-steps recalculation the difference of poured melt mass (m i+1 -m i ) within corresponding time (t mi+i +t mi ), where is flow rate can be presented as -Q i = m' i = (m i+1 -m i )/(t mi+i +t mi ). On Fig. 2 by lines 25 is shown the results of 4-stages recalculation of (m i+1 -m i ) during time (t mi+i +t mi ), for points of i = 1, 2, 3, 4 by expressions Q i,1..4 =∑Q i,i+n /n (n=1..4). The each values of m and t in this case was defined as i-point, which inputted to microcontroller of automatic control system with frequency 50Hz, at renewing of each 0.02sec. The most useful for practical application there are the 4-th constituent of the "tracking" algorithm (Line 5 on Fig. 2), that have been a by time delayed no more than 0.08s, and error in range 5%. Additionally, for dynamic monitoring and control on the pouring process it can be realized by the back-calculation of the expression (1) Q(H), by calculating the pressure caused by electromagnetic pressure (p em ). For the next deeply studying of possibilities dynamic control of operating characteristics in magnetodynamic installation rationally is use the complementary system (3), that also can be programmed as algorithms in the microprocessor unit:

                                           Al met.duct met met 2 / 3 met duct - met v m m i 1 i i S v * v Q h L Н g 2 b H Q ) m m ( m m Q i 1 i , kg/sec (3)
where -Q(v met )* a component of the equation, which can be measured with the use of assistive devices that can be included with the MWI with integrated IWMS.

Experimental tests of "tracking" algorithm at discrete dosing of 1, 3 and 5 kg/s aluminum with automatic correction voltage on electromagnet by deviation real value of flow rate from set has been showed reducing of it oscillation to ±2% in comparisons with existing 6% before.

Development the prototypes of the methods and devices for direct dynamic control pouring melts

Especially for widely using in the industrial conditions the prototypes of simple devices and methods for dynamic control flow rate of pouring melts into molds has been developed (Fig. 4). According to the proposed methods the measurement of metal flow rate was realized by determining of the "instantaneous value" of force, which is creating by the mass of the melt at it flowing via sloped tray (Fig. 4), during given period of time. Such methods can be realized by two ways, at direct measurement of melt mass flowing on sloped tray melt mass by means of strain gauges or by continuously measurements of the ponderomotive force (Fp) of the interaction of the AC magnetic field (B) from electromagnet that located under of sloped tray (Fig. 4 b) [START_REF] Slazhniev | Ukraine patent for utility model[END_REF].

Metallduct

Q(F p ) = k ph •F p 3/2 + с g •F p , (4)
where Q(Fp) -melt flow rate at the trough, kg / s; F p -ponderomotive force N, (kg • m / s 2 ); k ph -factor for the physical properties of melt and geometric parameters of the trough (k ph ≥ 1), s 2 ×(m / kg) 1/2 ; с g -constant that takes into account the geometrical parameters of the trough, its position relative to the drain spout of metalduct and physical properties of melt (с g ≥ 0), s/m.

Conclusion

The original methods of the maintaining of the set point of flow rate characteristics independence from the level of metal in the crucible MDI was developed and described, that allowed obtaining the stabile dosing high accuracy not less than 1%. By deeply studying of the pouring process has been estimated the roles of inertial drain of melt on the the accuracy of dosing by open stream method. The special algorithm for dynamical control by the "tracking" method was developed and tested, which include the 4steps recalculation the difference poured melt mass (m i+1 -m i ) within corresponding time (t mi+i +t mi ). Application the "tracking" method at indirect correction of the melt flow rate at pouring from MDI to casting mold is allowed to reduce the it oscillation to ±2% in comparisons with existing 6% before. Especially for widely using in the industrial conditions has been developed the prototypes of simple devices and methods for dynamic control flow rate of pouring melts into molds, that based on measurement of metal flow rate by determining of "instantaneous value" of force, which is creating by the mass of the melt at it flowing via sloped tray. By the deeply studying of the pouring process was estimated the roles of inertial drain of melt to the accuracy of dosing by open stream method.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Structural diagram of MWI for aluminium alloy and SAC for discrete/continues dosing and photo of prototype

4 :

 4 The prototypes of measurement devices for dynamic control of the flow rate on sloped tray (a) and scheme for ponderomotive contactless melt flow rate measuring (b).


V Dubodelov

O Smirnov

S Louhenkilpi

A Kolesnichenko

Innovative combinations of MHD technologies and original electromagnetic devices for highly efficient casting on CCM 1

Keywords: continuous casting of steel, CCM mold, solidification, crust, magnetodynamic tundish, electromagnetic stirrer, temperature, casting speed, non-metallic inclusions

Innovative combinations of MHD technologies and original electromagnetic devices for highly efficient casting on CCM

V Dubodelov, O Smirnov, S Louhenkilpi, A Kolesnichenko

Introduction

The modern trends of continuous casting are characterized by tendency to increase in productivity of CCM's strand at synchronous approaching of ingot section to geometry of final wares and improvement of quality of castings. Available data indicates that conventional methods and equipments for continuous casting of steel have reached their technical limits [1,[START_REF] Flick | 7 th European Continuous Casting Conference[END_REF]. Further progress can be provided at using innovative solutions based on application in CCM the combined electromagnetic fields for influence on liquid steell in tundish, mold, and secondary cooling zone. However such complex approach causes refinement of requirements both to operating parameters and to functional modes of MHDdevices and their interactions at pouring and forming of ingot. The accumulated experience and "know-how" in this field confirms that electromagnetic processing of materials and magnetohydrodynamics allows expand continuous casting technology, and increase the effectiveness and competitiveness of products.

Results

The main concept offered to solve the above-mentioned problem based on principle of low metal head pouring of steel from tundish into the mold. For implementation of such concept, it also provides using complex electromagnetic systems. They should provide solutions for other important tasks which are necessary to ensure the effective operation of CCM. In particular, it is concerning to control the temperature and hydrodynamics of steel in tundish, in mold and in liquid metal bath of casting ingot. There is proposed to combine in one CCM the functionality of magnetodynamic tundish (MD-T) [START_REF] Dubodelov | 5 th International Symposium of Electromagnetic Processing of Materials[END_REF] and special complex electromagnetic stirrer [START_REF] Buryak | 3 rd International Symposium of Electromagnetic Processing of Materials[END_REF]. Moreover, in some cases, it is necessary to use original MHD-systems for dispersed pouring of steel into mold (especially at casting of thin slabs). There were compared the conditions of ingot formation at conventional methods of continuous casting and low metal head pouring from tundish into mold. In particular, it was used the mathematical modeling in ANSYS package (main performer is PhD V. Buryak, MHD Department of PTIMA NASU) to study the effect of casting speed on formation conditions of continuous cast ingot. The calculation was performed for the process of casting of round steel billet by diameter 300 mm. The first step is to evaluate process changing of liquid steel hydrodynamics in CCM mold according to linear speed of melt stream (Fig. 1). As can be seen on Fig. 1a, high linear velocity of flow (3 m/sec) causes deep penetration of stream into melt bath in mold. Also there was sizeable length of high energy action of submerged stream and high intensity of secondary flows. Structure of stream at low linear speeds (0.5 m/sec, Fig. 1b) had changes fundamentally. In particular, both depth of stream penetration and length of stream core reduce significantly. It is very important that secondary flows are more branched and they have less intensity. These flows spread on volume of mold and keep their maximum intensity in center of mold. As it is shown, the most unfavorable conditions of heat-removing of poured steel in the mold are at maximum liquid metal head (Fig. 2c). Using low head values (Fig. 2a) does not promote to conditions of effective heat-removing too. At the same time, option on Fig. 2b for this case is an optimum, as it provides heat-removing on height of mold and there is no danger to moving stream of superheated steel outside of mold.

It is well known conditions of formation of solid crust have the special importance for stability process of continuous casting of steel. These conditions are largely dependent from temperature molten steel and hydrodynamics in the mold. Also it is necessary to take into account the impact of mold oscillations on solid crust. For evaluation of these factors, the solid crust formation has been studied on physical model (scale 1:1). As modeling agent, there was used camphene (2.2-dimethyl-3-methylene-bicyclogetan). This material as well as steel is solidifying with formation of dendrites. It has allowed simulating the processes and phenomena occurring in the "mushy" (twophased) zone. Fig. 3 shows formation of oscillation traces at modeling. Fig. 3a shows camphene crust with two oscillation traces formed on intensively cooled wall. It is determined that solidification of solid crust occurs on arc-shaped surface of liquid meniscus. After overflow, this crust creates the edge that entrains into the body of forming ingot. During further solidification, the edge can be either submelted or broken by liquid forced flows as it is shown on Fig. 3b. Broken part of the edge is melting or moving to lower melt layers where it can be caught by growing branches of dendrites (Fig. 3c) and can provoke occurrence of interdendritic cracks. Generally, the solid crust growth in mold on length of ingots occurs fast (0.5-1.0 mm/sec). It depends on several factors, including features of circulating flows in liquid pool of mold. They can be rather asymmetric. In combination with fluctuations of ingot's withdrawal rate (e. g., due to response of break-prevention system) this can lead to fluctuations of solidification rate in different parts of ingot's surface in mold. As result, this promotes the possibility of so-called "waves" at inner surface of ingot along solidification front (Fig. 4). Fig. 4: General view of the inner surface of ingot forming in mold (samples selected from ingot after break) Such evolution at solid crust growth in mold can be considered as one of main causes for origin of hot horizontal cracks and break outs under the mold. At our opinion, the reduction of rate fluctuations of solid crust growth can be reached at low head pouring of molten steel into mold in combination with electromagnetic stirring. This stirring process should realize such melt circulation mode which provides temperature averaging inside mold and prevents either submelting or erosion damage of crust. Moreover, it should be considered the negative influence of mold oscillation. The implementation of principle of low head pouring of steel from tundish into mold does not preclude the application of electromagnetic stirrer. Such device is called as complex electromagnetic stirrer, and it has developed by one of report's authors [START_REF] Buryak | 3 rd International Symposium of Electromagnetic Processing of Materials[END_REF]. However, its functional application changes considerably. In this case, there is not necessary to use main part of electric power of stirrer for braking the high-speed melt stream and meniscus stabilization (waves' suppression). The task will be confined mainly to creation of moderate azimuthal electromagnetic stirring of molten steel in mold for temperature averaging and amplification of acoustic waves' energy directed into the depth of liquid pool of solidified ingot. In this case, along with pulsations moving metal stream keeps rotational motion [START_REF] Kolesnichenko | 5 th International Symposium of Electromagnetic Processing of Materials[END_REF]. The mechanism of this phenomenon has been previously studied by the authors [START_REF] Kolesnichenko | 5 th International Symposium of Electromagnetic Processing of Materials[END_REF] and it is the resonance effect (coincidence of frequencies generated by complex electromagnetic stirrer and natural oscillations of liquid metal bath). The amplitude of pressure near the bottom of bath is in 4-5 times higher than pressure at meniscus. It provides the high intensity of liquid phase pulsations at solidification front along the whole length of solidifying ingot. This motion changes fundamentally the phase transformation mode and causes morphological transformation. At that zone of ultrafine crystals fills the whole cross-section of continuously cast ingot and eliminates cavities, porosity, and carbon saturation of ingot's center. Such processing deletes the conditions for growth of large columnar crystals, which are the cause of intergranular defects (segregation, cracks, porosity). In practice, such result was observed for medium and high carbon steels especially. The effectiveness of low head pouring of steel into mold was confirmed under natural conditions. At that metal pouring via magnetodynamic tundish was simulated by switching of complex electromagnetic stirrer at "braking mode". It provides the reduction of liquid steel speed poured into mold from 4-5 m/sec to 0.15-0.20 m/sec. As result hydrodynamics in mold changes significant and disturbances of molten steel on the meniscus was reducing. Due to stabilization of crust formed in mold the heat flow through the walls of mold increases on 10-18%. Also, there are decreasing the backgrounds for capture from meniscus of nonmetallic inclusions, especially large oxide, and their entrainment into ingot's depth by intense vortex flows [START_REF] Wang | [END_REF]. Researches conducted by the authors have shown that presence of inclusions in cast ingot leads to appearance of internal macrocracks during hot rolling. At the same time, this is difficult to detect such inclusions in cast ingot, as their quantity is sufficiently small in steel [START_REF] Sahai | Technology for Clean Steel Production[END_REF][START_REF] Zhang | [END_REF]. Example of location and size of large inclusions (70-100 m) cracks at surface is shown on Fig. 5. Generally, the combined use of low head pouring of steel into the mold and combine electromagnetic stirrer switched in special modes will provide: flexibility at control of hydrodynamics and temperature during casting; -guaranteed presence of liquid steel inside mold for heat-removing; forming the stable solid crust; due to stabilization of meniscus, prevention of capture from this zone of non-metallic inclusions and particles of slag-making mix; preferential growth of equiaxed crystals and eliminating of irregularities at distribution of carbon in different zones of steel ingot (carbon content in ingot's center without electromagnetic stirring is in 1.32-1.36 times more than its content in peripheral areas. For ingots which had casting with electromagnetic stirring it consists 1.07-1.12 times). Also significant positive effect at application of combine electromagnetic stirrer marked consists in suppression of microporosity (Fig. 6), which took place in ingots without electromagnetic stirring. Area of such pores is depth of 15-50 mm from surface of billet. Application of electromagnetic stirring had reduced quantity of micropores in 120-180 times.

Liquid metal head

Practically it provides required characteristics of billets for subsequent rolling [START_REF] Anast | 6 th European Conference on Continuous Casting[END_REF]. Suppression of microporosity at electromagnetic stirring should be considered as confirmation of fact melt circulation flows formed in CCM mold extend deep into liquid pool.

1 :Fig. 2 :

 12 Flow field of liquid steel at pouring from conventional tundish at high liquid metal head (a) and from magnetodynamic tundish at low liquid metal head (b) (both longitudinal and cross sections of ingot)As result at decreasing of linear speed of steel, there are creating the more favourable conditions for retention of formed crust integrity and for melt retention in the area of mold for remove the overheat of liquid steel. At the same time, there was made the calculation of temperature field in steel ingot at pouring from tundish with different values of low metal head and constant mass flow rate (Fig.2) (head of works is Head of Mathematical Modeling Department Dr. Sc. M. Tarasevich, PTIMA NASU). a b c Temperature fields in the solidifying ingot at continuous casting of steel with different liquid metal head in tundish at constant mass flow rate

3 :

 3 Forming of oscillation traces and solid crust

Fig. 5 :

 5 Fig. 5: Large non-metallic inclusions on the border of crack (steel Grade 1004)

Fig. 6 :

 6 Fig.6: Some examples of microporosity in the CC ingot from high carbon steel Conclusion So, the proposed concept of low head pouring and original MHD-devices for its realization in modern continuous casting technologies allows improving the quality and productivity of process.


V Dubodelov

O Smirnov

M Goryuk

V Pogorsky

V Seredenko

V Buryak

Features of functioning of magnetodynamic tundish at continuous casting of steel

Keywords: magnetodynamic tundish, mold, continuous casting, modeling, inclusions, slag, electromagnetic stirrer

Magnetodynamic tundish (MD-T) for continuous casting of steel had been developed in PTIMA NASU. Such aggregate has some features in contrast to conventional tundish and provides different-in-principle modes of work of continuous casting machines (CCM). There were carried out both physical and mathematical modeling continuous casting of steel from MD-T. In comparison with traditional technology, it was shown that reducing in 3.0-3.5 times of metal static pressure at pouring leads to decreasing in 1.7-1.8 times the depth of penetration of high-speed and high-temperature melt stream into solidifying ingot. In this case, for keeping (or rising) of mass pouring rate and ingot's withdrawal rate, it is necessary to provide increasing of melt volume which incomes into CCM mold. Results of researches had indicated that MD-T can decide this problem. Electromagnetic systems of such aggregate allow to forming the flat stream of melt in width of 0.5-0.8 m. Such features of MD-T allow changing cardinally the terms of ingot forming. So, there is increased the heat load onto mold, but at the same time the heat load is decreased in the secondary cooling zone. Also, it is possible to eliminate application of high-expensive electromagnetic systems for melt braking in the CCM mold. This fact creates preconditions for reduction of CCM length.

Introduction

In last 15-20 years main requirements for quality continuous casting ingot becoming more stringent. This is due to the development requirements of the global market and new technological constructions in which the combined process of casting and subsequent rolling. Many researchers are making extra efforts to enhance the technological capabilities of the tundish in protecting steel from contamination by nonmetallic inclusions, control the temperature of the metal during the casting and refining of molten steel directly into the tundish [START_REF] Mclean | Proc. Steelmaking Conf., Iron and Steel Society[END_REF][START_REF] Sahai | Technology for Clean Steel Production[END_REF][START_REF] Marukawa | [END_REF][4]. The functioning process of the tundish during the casting cycle can be divided into the following characteristic cycle: filling the tundish at the beginning casting when you open steel ladle be slide gate; steady state casting while maintaining the nominal level of the metal in the tundish; a period of intense decreasing of loading level steel in the tundish, which caused during the replacement of the ladle; set filling level of the metal in the tundish due to the start of the new steel ladle up to the stabilization of the entire process; fall in the level of metal in the tundish at the end of the casting process. In a liquid bath of the tundish we can be divided into three or four zones, which characterized by considerable difference velocity gradient (Fig. 1). Fig. 1: Key transfer zones in a liquid steel bath of the tundish: Ithe zone of falling liquid steel jet and intensive suppression of its speed; IIarea of active moving of melt flows in the tundish volume; III, IVstagnant ("dead") area, which boundaries are due to the design of the tundish and the presence of the dams and tundish-box (1ladle shroud; 2 tundish; 3stopper; 4nozzle; 5tundish-box)

In liquid metal bath of tundish, there are formed a few areas where melt moves very slowly. Such areas are called stagnant or "dead". In these areas the metal remains in 5-10 times longer than the average residence time of the steel in the a b tundish. This phenomenon leads to the fact that non-metallic inclusions coagulate. The size of non-metallic inclusions can be increased by 10-15 times or more. By the way the "dead" area in the tundish is not constant. It moves slowly from the middle part of tundish to the zone of nozzle and then returning. It is making very difficult to quantify its borders. The existence of such "dead" volumes significantly reduces the active volume of the liquid bath, and thus reduces residence time main part of steel. For this reason, is is necessary to minimize "dead" volume of the liquid bath in the tundish. It will provide the best conditions for flotation non-metallic inclusions and give possibility to reduce the volume of the tundish and its geometrical dimensions.

In present time the most actual issues that require innovative solutions in the development of the tundish, are as follows: effective methods forced flotation of non-metallic inclusions in a liquid bath of the tundish through the use of various dams and tundish box; -application MHD methods to optimize flows movement in the tundish; -refining molten steel in the tundish through the argon injection; -regulation temperature of steel in the tundish during the casting process;optimization configuration of the inner cavity of the tundish to ensure rational motion of convective flows; -development of new methods of metal dosing from the tundish into the mold, which including the provision of the braking effect of the jet and its dispersed supply into the mold. Thus, the most important task for the system of continuous casting technologies for the near future is the problem of optimizing the tundish as multi functional unit, which providing manufacture of high condition continuous casting ingots from pure and high purity grade steel.

Development of tundish design

Modern tundish must perform a variety of technological features that have been listed above. It is a buffer capacity, which agrees a discrete flow of steel from the ladle into a tundish and a continuous flow of steel from the tundish into the mold. Meanwhile, the residence time of the metal in the tundish is 5-15 minutes. This ensures the averaging of incoming metal portion and to prevent ingress of slag in the mold. It is assumed that the tundish design should ensure minimal heat loss during the metal casting cycle.

From the practice of continuous casting well known that quite large areas of the liquid bath of tundish very poorly mixed by convection flows and thus reduce the efficiency of refining operations and averaging process (zone III and IV, Fig. 1). Many investigations found that this area should be stirred forced flows which are formed by injection inert gas to the molten bath [5][6][7]. Meanwhile, possibilities of injection process argon into steel in the tundish sufficiently strongly limited, because with increasing gas flow rate above the critical value may be a metal contamination by extra large inclusions. It should be explained by strong metal and slag wildness at the exit site argon bubbles [7]. Directional movement of forced flows in a liquid bath of the tundish can be achieved through the use of electromagnetic stirring (EMS). Determination of a stirring power, flow directions and places applying a magnetic field should be carried out in accordance with the industrial and quality tasks. Wherein it must be take into account the configuration of the tundish and the presence of auxiliary units (design of tundish box, dams, walls, purging block). In our opinion the most effective technological solution will combine the imposition of electromagnetic stirring using electromagnetic stirring and argon bubbling of metal into a single forced system, which will be located in the bottom of the tundish around the middle part (under the "dead" zone). General view of the design tundish, which was developed with additional electromagnetic stirrer (EMS), is shown on Fig. 2. In the present version a positive effect is achieved primarily by the flow direction of the liquid bath in the tundish (eliminates "dead" zone), and more rapid transport of non-metallic inclusions to the surface of steel in tundish. Also steel refining from non-metallic inclusions occurs due to suppression of bubbling at the boarder slag-metal and reduces the wear of the refractories in zones of high turbulence.

Hot modeling

Evaluate the effectiveness of applying a magnetic field to the molten bath of the tundish were performed in the laboratory equipment. The capacity of the container, which simulates the tundish, has 600 kg of aluminum (Fig. 3). Measuring the speed of the forced flows was performed using Pitot tubes. It was found that the most effective impact on the melt is pulsing magnetic field (EMS-PMF) [START_REF] Dubodelov | 6 th International Conference on Electromagnetic Processing of Materials[END_REF]. Thus, nearly the wall of the tundish, which inductor EMF-PMF with capacity of approximately 20 kW had placed, a result of his actions had formed submerged flow, which ensures mixing of the melt to form a toroidal vortex along both the longitudinal and transverse sides of the container. Moreover, the speed of the melt flow in the core achieves 1 m/sec. This speed is maintained at a distance of not less than 1.2 m from the wall of the tundish. It was research the effect of influence distance from the pole of the inductor to the liquid metal boarder on the velocity of the liquid aluminum in the core flow. It was shown that the critical value of this quantity for the unipolar EMS-PMF is the distance nearly 200 mm, but since this date the speed of the melt had decreased in 3 times. As the comparison, for application running magnetic field inductor in the furnace wall its placed on the distance a 60-100 mm from the liquid metal boarder.

Physical modeling

Due to the fact that the industrial test aimed at the investigation of processes taking place in the tundish, is associated with a number of practical complexities (high test costs, impossibility to visualize specific stages of the process, difficulty to measure physical values), the present paper applies physical modelling method for experimental research [START_REF] Chattopadhyay | [END_REF][10][11].

For the present study, the tundish physical model and its basic functional units were fabricated from transparent material. The model's scale is 1:4. This scale appeared to be sufficient for visual monitoring of all hydrodynamic processes taking place in the tundish, almost in online mode. The behaviour and up floating pattern of non-metallic inclusions was investigated by means of introduction of dedicated mechanical mixture (suspended solids) of hydrophobic particles into the shroud placed between the steel-teeming ladle and the tundish, the mixture density being lower than the water's density. For this purpose, a dedicated multicomponent mixture to include several types of oils and emulsions and very fine solid particles, was developed. Therewith, in the course of tests, the motion trajectory of "slag" particles in the tundish liquid bath was assessed, as well as the approximate portion of particles up floated to the surface and out flown from the tundish together with the water. The motion of liquid convective flows in the tundish originated at the jet location incoming from the steel-teeming ladle, until liquid outflow from nozzles, was recorded by a digital camera. Concurrently, volumes of liquid outflowing from the ladle were painted with coloured dyes.

Results and discussion

The physical model was applied to compare efficiency of various methods of liquid steel refining in the tundish. For this purpose, a certain volume of substance imitating non-metallic inclusions (oil and fine solid particles mixture), was fed into the shroud. Following that, the liquid out flowing from the tundish, was collected in the dedicated vessel and settled down for several hours. Concurrently, the substance imitating non-metallic inclusions floated up to the surface. This part of simulating substance was collected in the dedicated tank and weighted. The highest efficiency of nonmetallic inclusions removal was observed at gas injection and flow forced to the molten bath. This is accounted for a transportation phenomenon, which accompanies flotation of the purged gas bubbles. The cumulative effect of the ring-shaped purging block application can be considered as a reasonably efficient practice to enable flotation of non-metallic inclusions. To enhance performance of the ring-shaped purging lance, it is necessary to control behaviour of bubbles ingress to the liquid surface in the tundish, since this ingress can lead to steel and slag mixing. This process causes entrapment of non-metallic inclusions by steel convective flows. The significant influence on the turbulence of steel and slag mixing is exerted by the injected gas flow-rate. It should be also considered that the minimum mixing of steel and slag was observed under liquid bath purging in the bubbling mode.

Practical

There were carried out both physical and mathematical modeling continuous casting of steel for magnetodynamic tundish (MD-T). In comparison with traditional technology, it is shown that reducing in 3.0-3.5 times of metal static pressure at pouring leads to decreasing in 1.7-1.8 times the depth of penetration of high-speed & high-temperature melt stream into solidifying ingot. At that, either maintenance or rising both mass pouring rate and ingot's withdrawal rate needs corresponding increase of melt volume melt incoming into CCM's mold. Results of modeling and experimental measures indicate that MD-T can decide this problem due to forming (under action of electromagnetic forces) the flat melt stream in width of 0.5-0.8 m. The general concept of the multifunctional magnetodynamic tundish (MD-T), which provides forced mixing, preheating and controlled casting of metal, and its refining, had been developed in Physico-Technological Institute of Metals and Alloys of National Academy of Sciences of Ukraine (PTIMA NASU). Such type a tundish is most interesting in the case of continuous casting special steels with high requirements for quality. Typically, these grades of steel have been producing at relatively a small quantity, which making them a very difficult for process ladle treatment. Accordingly, in this case some part of technological operations has been transferred into tundish. Schematic diagram of the MD-T is shown on Fig. 4. Fig. 4: General design of tundish with MHD pouring and preheating steel: 1steel ladle; 2 -MD-T; 3liquid steel; 4covering slag; 5 -EMS; 6inductor for preheating; 7unit for EM dosing; 8refractory pipe for transport steel; 9submerged nozzle; 10mold of CCM There is a possibility for effective control of the liquid steel temperature at its moving from crucible through channel for subsequent pouring into CCM mold; regulation of out flow of liquid steel from MD-T is made without application nozzle-stopper equipment and sliding shutters. Besides there is possibility for control of pouring pressure and pouring mass flow rate of metal.

Conclusion

Trends in the development of the global steel market will demand from steel manufacturers to improve the technology and equipment for the production of cleaner steel for requests of non-metallic inclusions. A key element of improving the purity of steel, apparently, can be multi-functional tundish with MHD units. Thus such tundish will have some improving design of configuration in combination with a number of improvements in the technology of treatment and casting steel. The effectiveness of the MD-T is substantially extended by the MHD effects. In particular MD-T can be use for controlling the temperature of the steel, control and suppression of turbulent flow and dispensing of the metal during its flow to the mold. Application of MD-T may be of special interest for technologies of manufacturing of small lot (to 10-20 thousand tons steel per year) of high-quality ingots for machine-building and casting enterprises. It will allow organizing the metallurgical micromill by the semicontinuous casting process with using of MD-T. There is developed the technical project of semicontinuous casting machine (SCCM).

Fig. 2 :

 2 Fig.2: Design continuous casting tundish with EMS Fig.3: Laboratory tundish equipped by EMS-PMF [8] and argon injection: 1tundish; 2ladle shroud; 3tundish-box; 4purging plug; 5 -EMS; 6stopper


Dmitrij Celov

Remigijus Leipus

Anne Philippe

Time series aggregation, disaggregation and long memory *

Keywords: AMS classification: 62M10; 91B84 random coefficient AR(1), long memory, aggregation, disaggregation

The paper studies the aggregation/disaggregation problem of random parameter AR(1) processes and its relation to the long memory phenomenon. We give a characterization of a subclass of aggregated processes which can be obtained from simpler, "elementary", cases. In particular cases of the mixture densities, the structure (moving average representation) of the aggregated process is investigated.

Introduction

In this paper we consider the aggregation scheme introduced in the paper of Granger (1980), where it was shown that aggregation of random parameter AR(1) processes can produce long memory. Since this work, a large number of papers were devoted to the question how do micro level processes imply the long memory at macro level and applications (mainly in economics) (see [START_REF] Haubrich | The sources and nature of long-term dependence in the business cycle[END_REF], [START_REF] Oppenheim | Aggregation of random parameters Ornstein-Uhlenbeck or AR processes: some convergence results[END_REF], [START_REF] Zaffaroni | Contemporaneous aggregation of linear dynamic models in large economies[END_REF], Chong (2006), etc.).

Recently, Dacunha-Castelle and [START_REF] Dacunha-Castelle | Mixtures, aggregations and long memory[END_REF] (see also [START_REF] Dacunha-Castelle | Disaggregation of long memory processes on C ∞ class[END_REF]) stated the following problem: which class of long memory processes can be obtained by the aggregation of short memory models with random coefficients?

Let us give a precise formulation of the problem when the underlying short memory models are described by AR(1) dynamics. Let ε = {ε t , t ∈ Z} be a sequence of independent identically distributed (i.i.d.) random variables (r.v.) with Eε t = 0, Eε 2 t = σ 2 ε , and let a be a random variable supported by (-1, 1) and satisfying E 1 1 -a 2 < ∞.

(1.1)

Consider a sequence of i.i.d. processes Y (j) = {Y (j) t , t ∈ Z}, j ≥ 1 defined by the random AR(1) dynamics

Y (j) t = a (j) Y (j) t-1 + ε (j) t ,
(1.2)

where ε (j) = {ε (j) t , t ∈ Z}, j ≥ 1, are independent copies of ε = {ε t , t ∈ Z} and a (j) 's are independent copies of a. Here, the sequences a, a (j) , j ≥ 1 and ε, ε (j) , j ≥ 1, are independent. Under these conditions, (1.2) admits a covariancestationary solution Y (j) t and the finite dimensional distributions of the process X

(N ) t = N -1/2 N j=1 Y (N ) t
, t ∈ Z, weakly converge, as N → ∞, to those of a zero mean Gaussian process X t , called the aggregated process (see [START_REF] Oppenheim | Aggregation of random parameters Ornstein-Uhlenbeck or AR processes: some convergence results[END_REF]).

Assume that the distribution of r.v. a admits a mixture density ϕ, which by (1.1) satisfies

1 -1 ϕ(x) 1 -x 2 dx < ∞.
(1.

3)

The covariance function and spectral density of aggregated process X t are given, respectively, by

γ(h) := Cov(X h , X 0 ) = σ 2 ε 1 -1
x |h| 1 -x 2 ϕ(x)dx

(1.4) and

f (λ) = σ 2 ε 2π 1 -1 ϕ(x) |1 -xe iλ | 2 dx.
(1.5)

Note that an aggregated process X t possess the long memory property (i.e.

∞ h=-∞ |γ(h)| = ∞) if and only if 1 -1 ϕ(x) (1 -x 2 ) 2 dx = ∞.
(1.6)

If the mixture density ϕ is a priori given and our aim is to characterize the properties of the induced aggregated process (moving average representation, spectral density, covariance function, etc.), we call this problem an aggregation problem. And vice versa, if we observe the aggregated process X t with spectral density f and we need to find the individual processes (if they exist) of form (1.2) with some mixture density ϕ, which produce the aggregated process, then we call this problem a disaggregation problem. The second problem, which is much harder than the first one, is equivalent to the finding of ϕ such that (1.5) (or (1.4)) and (1.3) hold. In the latter case we say that the mixture density ϕ is associated with the spectral density f .

Sections 2 and 3 are devoted to the disaggregation problem. Equality (1.4) shows that the covariance function γ(h) can be interpreted as a h-moment of the density function ϕ(x)(1 -x 2 ) -1 supported by (-1, 1), and thus finding of the mixture density is related to the moments' problem (see Feller (1976)). In Section 2, we prove that, under some mild conditions, the mixture density associated with the product spectral density can be obtained from the "elementary" mixture densities associated with the multipliers. In Section 3, we apply the obtained result to the spectral density which is a product of to FARIMA-type spectral densities. Using the form of mixture density for the classical FARIMA model we show that one can solve the disaggregation problem for more complex long memory stationary sequences.

For the aggregation problem, it is important to characterize, for a given class of mixture densities, the behavior of the coefficients in its linear representation and the behavior of the spectral density. We address this problem to Section 4. In the Appendix, we provide a proof of the form of the mixture density in FARIMA case.

2 Mixture density for the product of aggregated spectral densities

Let X 1,t and X 2,t are two aggregated processes obtained from the independent copies of AR(1) sequences

Y 1,t = a 1 Y 1,t-1 + ε 1,t and Y 2,t = a 2 Y 2,t-1 + ε 2,t , respec- tively,
where a 1 , a 2 satisfy (1.1), ε i = {ε i,t , t ∈ Z} and a i are independent, i = 1, 2. Denote σ 2 i,ε := Eε 2 i,t , i = 1, 2. Assume that ϕ 1 and ϕ 2 are the mixture densities associated with spectral densities f 1 and f 2 , respectively, i.e.

f i (λ) = σ 2 i,ε 2π 1 -1 ϕ i (x) |1 -xe iλ | 2 dx, i = 1, 2.
(2.1)

The following proposition shows that, if ϕ 1 and ϕ 2 in (2.1) are supported by [0, 1] and [-1, 0] respectively, then the stationary Gaussian process with spectral density f (λ) = f 1 (λ)f 2 (λ) can also be obtained by aggregation of the i.i.d. AR(1) processes with some mixture density ϕ and noise sequence ε.

Proposition 2.1 Let ϕ 1 and ϕ 2 be the mixture densities associated with spectral densities f 1 and f 2 , respectively. Assume that supp(ϕ

1 ) ⊂ [0, 1], supp(ϕ 2 ) ⊂ [-1, 0] and f (λ) = f 1 (λ)f 2 (λ). (2.2)
Then the mixture density ϕ(x), x ∈ [-1, 1] associated with f is given by equality

ϕ(x) = 1 C * ϕ 1 (x) 0 -1 ϕ 2 (y) (1 -xy)(1 -y/x) dy + ϕ 2 (x) 1 0 ϕ 1 (y) (1 -xy)(1 -y/x)
dy ,

(2.3) where C * :=

1 0 0 -1 ϕ 1 (x)ϕ 2 (y)(1 -xy) -1 dy dx. The variance of the noise is σ 2 ε = σ 2 1,ε σ 2 2,ε C * 2π .
(2.4)

Proof. Obviously, the covariance function γ(h) = Cov(X h , X 0 ) has a form

γ(h) = 1 2π ∞ j=-∞ γ 1 (j + |h|)γ 2 (j), (2.5)
where γ 1 and γ 2 are the covariance functions of the aggregated processes obtained from the mixture densities ϕ 1 and ϕ 2 , respectively:

γ 1 (j) = σ 2 1,ε 1 0 ϕ 1 (x) x |j| 1 -x 2 dx, γ 2 (j) = σ 2 2,ε 0 -1 ϕ 2 (x)
x |j| 1 -x 2 dx.

Clearly, γ 1 (j) > 0 and γ 2 (j) = (-1)

j |γ 2 (j)|. Let h ≥ 0. Then 2πσ -2 1,ε σ -2 2,ε γ(h) = σ -2 1,ε σ -2 2,ε ∞ j=-∞ γ 1 (j + h)γ 2 (j) = σ -2 1,ε σ -2 2,ε ∞ j=0 γ 1 (j + h)γ 2 (j) + ∞ j=0 γ 1 (j)γ 2 (j + h) + h-1 j=1 γ 1 (-j + h)γ 2 (j) =: s 1 + s 2 + s 3 . (2.6)
We have

s 1 = lim N →∞ s (N )
1 , where

s (N ) 1 = N j=0 1 0 ϕ 1 (x) x j+h 1 -x 2 dx 0 -1 ϕ 2 (y) y j 1 -y 2 dy = 1 0 1 0 ϕ 1 (x)ϕ 2 (-y) 1 1 -x 2 1 1 -y 2 N j=0 (-1) j x j+h y j dx dy = 1 0 1 0 ϕ 1 (x)ϕ 2 (-y) x h 1 -x 2 1 1 -y 2 1 -(-xy) N +1 1 + xy dx dy.
Note that

ϕ 1 (x)ϕ 2 (-y) x h 1 -x 2 1 1 -y 2 1 -(-xy) N +1 1 + xy ≤ 2ϕ 1 (x)ϕ 2 (-y) 1 1 -x 2 1 1 -y 2 and 1 0 1 0 ϕ 1 (x)ϕ 2 (-y) 1 1 -x 2 1 1 -y 2 dx dy = γ 1 (0)γ 2 (0) < ∞.
Therefore, by the dominated convergence theorem, we obtain

s 1 = 1 0 1 0 ϕ 1 (x)ϕ 2 (-y) x h 1 -x 2 1 1 -y 2 lim N →∞ 1 -(-xy) N +1 1 + xy dxdy = 1 0 1 0 ϕ 1 (x)ϕ 2 (-y) x h 1 -x 2 1 1 -y 2 1 1 + xy dxdy = 1 0 x h 1 -x 2 ϕ 1 (x) 0 -1 ϕ 2 (y) 1 1 -y 2 1 1 -xy dy dx. (2.7)
Analogously, we have

s 2 = 0 -1 y h 1 -y 2 ϕ 2 (y) 1 0 ϕ 1 (x) 1 1 -x 2 1 1 -xy dx dy. (2.8)
For the last term in the decomposition (2.6) we have

s 3 = h-1 j=1 1 0 ϕ 1 (x) x -j+h 1 -x 2 dx 0 -1 ϕ 2 (x) x j 1 -x 2 dx = 1 0 1 0 ϕ 1 (x) x h 1 -x 2 ϕ 2 (-y) 1 1 -y 2 (-y/x) 1 -(-y/x) h-1 1 + y/x dx dy = 1 0 x h 1 -x 2 ϕ 1 (x) 0 -1 ϕ 2 (y) 1 1 -y 2 y/x 1 -y/x dy dx - 0 -1 y h 1 -y 2 ϕ 2 (y) 1 0 ϕ 1 (x) 1 1 -x 2 1 1 -y/x dx dy.
(2.9)

Equalities (2.7)-(2.9), together with (2.6), imply

2πσ -2 1,ε σ -2 2,ε γ(h) = 1 0 x h 1 -x 2 ϕ 1 (x) 0 -1 ϕ 2 (y) 1 1 -y 2 y/x 1 -y/x dy + 0 -1 ϕ 2 (y) 1 1 -y 2 1 1 -xy dy dx + 0 -1 y h 1 -y 2 ϕ 2 (y) 1 0 ϕ 1 (x) 1 1 -x 2 1 1 -xy dx - 1 0 ϕ 1 (x) 1 1 -x 2 1 1 -y/x dx dy = 1 0 x h 1 -x 2 ϕ 1 (x) 0 -1 ϕ 2 (y) 1 (1 -y/x)(1 -xy) dy dx + 0 -1 y h 1 -y 2 ϕ 2 (y) 1 0 ϕ 1 (x) 1 (1 -x/y)(1 -xy)
dx dy.

This and (1.4) imply (2.3), taking into account that

1 -1 ϕ(x)dx = 1. 2
3 Seasonal long memory case

In this section, we apply the obtained result to the spectral densities f 1 and f 2 having the forms:

f 1 (λ; d 1 ) = 1 2π |1 -e iλ | -2d1 (3.1) f 2 (λ; d 2 ) = 1 2π |1 + e iλ | -2d2 , 0 < d 1 , d 2 < 1/2. (3.2)
We call these spectral densities (and corresponding processes) fractionally integrated, FI(d 1 ), and seasonal fractionally integrated, SFI(d 2 ), spectral densities. The mixture density associated with the FI(d 1 ) spectral density (3.1) is given by the following expression (for the sketch of proof see Dacunha-Castelle and Oppenheim ( 2001)):

ϕ 1 (x; d 1 ) = C(d 1 )x d1-1 (1 -x) 1-2d1 (1 + x)1 [0,1] (x) (3.3) with C(d) = Γ(3 -d) 2Γ(d)Γ(2 -2d) = 2 2d-2 sin(πd) √ π Γ(3 -d) Γ((3/2) -d) (3.4)
and the variance of the noise

σ 2 1,ε = sin(πd 1 ) C(d 1 )π . (3.5)
For convenience, we provide the rigorous proof of this result in Proposition 5.1 of Appendix.

Similarly, the mixture density associated with the spectral density (3.2) is given by

ϕ 2 (x; d 2 ) = ϕ 1 (-x; d 2 ) = C(d 2 )|x| d2-1 (1 + x) 1-2d2 (1 -x)1 [-1,0] (x) (3.6) and σ 2 2,ε = sin(πd 2 ) C(d 2 )π , (3.7) since f 2 (λ; d 2 ) = f 1 (π -λ; d 2 ) = σ 2 2,ε 2π 1 0 ϕ 1 (x; d 2 ) |1 + xe iλ | 2 dx = σ 2 2,ε 2π 0 -1 ϕ 1 (-x; d 2 ) |1 -xe iλ | 2 dx.
Clearly, mixture densities ϕ 1 (x; d 1 ) and ϕ 2 (x; d 2 ), given in (3.3), (3.6), satisfy (1.3) and, hence, assumptions of Proposition 2.1, whenever 0 < d 1 < 1/2 and 0 < d 2 < 1/2. Moreover, since d 1 > 0 and d 2 > 0, both ϕ 1 and ϕ 2 satisfy (1.6). The mixture density associated with the spectral density

f (λ; d 1 , d 2 ) = f (λ; d 1 )f (λ; d 2 ) = 1 (2π) 2 |1 -e iλ | -2d1 |1 + e iλ | -2d2 , 0 < d 1 , d 2 < 1/2 (3.8) can be derived from representation (2.3). Denote F (a, b, c; x) a hypergeometric function F (a, b, c; x) = Γ(c) Γ(b)Γ(c -b) 1 0 t b-1 (1 -t) c-b-1 (1 -tx) -a dt, where c > b > 0 if x < 1 and, in addition, c -a -b > 0 if x = 1. Proposition 3.1 The mixture density associated with f (•; d 1 , d 2 ) (3.8) is given by equality ϕ(x; d 1 , d 2 ) = C(d 1 , d 2 )x d1-1 (1 -x) 1-2d1 G(-x; d 2 )1 [0,1] (x) +C(d 2 , d 1 )|x| d2-1 (1 + x) 1-2d2 G(x; d 1 )1 [-1,0] (x), (3.9) where G(x; d) := F 1, d, 2 -d; 1 x -xF (1, d, 2 -d; x)
and

C(d 1 , d 2 ) = (C * ) -1 Γ(d 2 )Γ(2 -2d 2 ) Γ(2 -d 2 ) , C * = 1 0 x d1-1 (1 -x) 1-2d1 (1 + x) 1 0 y d2-1 (1 -y) 1-2d2 (1 + y) 1 + xy dy dx.
The variance of the noise is

σ 2 ε = (2π) -1 C * C(d 1 )C(d 2 )σ 2 1,ε σ 2 2,ε = sin(πd 1 ) sin(πd 2 )C * 2π 3 . (3.10) Proof. (2.3) implies that ϕ(x; d 1 , d 2 ) = 1 C * (C(d 2 )ϕ 1 (x)F (-x; d 2 ) + C(d 1 )ϕ 2 (x)F (x; d 1 )), (3.11)
where

F (x; d) := 1 0 y d-1 (1 -y) 1-2d (1 + y) (1 -xy)(1 -y/x) dy, C * = 1 0 ϕ 1 (x) 0 -1 ϕ 2 (y) 1 -xy dy dx = C(d 1 )C(d 2 )C * .
Using equality

1 + y (1 -xy)(1 -y/x) = 1 (1 -x)(1 -y/x) - x (1 -x)(1 -xy) ,
we have

F (x; d) = 1 1 -x 1 0 y d-1 (1 -y) 1-2d 1 -y/x dy - x 1 -x 1 0 y d-1 (1 -y) 1-2d 1 -xy dy = Γ(d)Γ(2 -2d) Γ(2 -d) 1 1 -x (F (1, d, 2 -d; 1/x) -xF (1, d, 2 -d; x)) = Γ(d)Γ(2 -2d) Γ(2 -d) G(x; d) 1 -x .
(3.12)

Now, (3.9) follows from (3.11) and (3.12), whereas (3.10) follows from (2.4), (3.5), (3.7).

To finish the proof note that all the hypergeometric functions appearing in the form of the mixture density are correctly defined. 2

In the next proposition we present the asymptotics of ϕ(x; d 1 , d 2 ) in the neighborhoods of 0 and ±1. Proposition 3.2 Let the mixture density ϕ be given in (3.9). Then

ϕ(x; d 1 , d 2 ) ∼      π C * sin(πd 2 ) x d1+d2-1 , x → 0+, π C * sin(πd 1 ) |x| d1+d2-1 , x → 0-, (3.13) ϕ(x; d 1 , d 2 ) ∼        2 1-2d2 π C * sin(πd 2 ) (1 -x) 1-2d1 , x → 1-, 2 1-2d1 π C * sin(πd 1 ) (1 -x) 1-2d2 , x → -1 + .
(3.14)

Proof. Applying identities (see [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF])

F (a, b; c; 1/x) = x x -1 b F (b, c -a; c; 1 1 -x ), F (a, b; c; 1) = Γ(c)Γ(c -a -b) Γ(c -a)Γ(c -b) , we have that for x → 0+ G(-x, d 2 ) = F 1, d 2 , 2 -d 2 ; - 1 x + xF (1, d 2 , 2 -d 2 ; -x) = x 1 + x d2 F (d 2 , 1 -d 2 , 2 -d 2 ; 1/(1 + x)) + xF (1, d 2 , 2 -d 2 ; -x) ∼ x d2 F (d 2 , 1 -d 2 , 2 -d 2 ; 1) = Γ(2 -d 2 )Γ(1 -d 2 ) Γ(2 -2d 2 ) x d2 = √ π Γ(2 -d 2 ) 2 1-2d2 Γ((3/2) -d 2 )
x d2 , and similarly for x → 0-

G(x, d 1 ) ∼ √ π Γ(2 -d 1 ) 2 1-2d1 Γ((3/2) -d 1 ) |x| d1 .
This and equality (3.9) imply

ϕ(x; d 1 , d 2 ) ∼ C(d 1 , d 2 ) √ π Γ(2 -d 2 ) 2 1-2d2 Γ((3/2) -d 2 ) x d1+d2-1 = C(d 1 )C(d 2 ) C * Γ(d 2 )Γ(1 -d 2 )x d1+d2-1 = (C * ) -1 π sin(πd 2 ) x d1+d2-1 , x → 0+, ϕ(x; d 1 , d 2 ) ∼ (C * ) -1 π sin(πd 1 ) |x| d1+d2-1 , x → 0 -. For x → 1-we obtain ϕ(x; d 1 , d 2 ) ∼ 2C(d 1 , d 2 )F (1, d 2 , 2 -d 2 ; -1)(1 -x) 1-2d1 = C(d 1 , d 2 ) √ πΓ(2 -d 2 ) Γ((3/2) -d 2 ) (1 -x) 1-2d1 = (C * ) -1 2 1-2d2 π sin(πd 2 ) (1 -x) 1-2d1
and similarly for x → -1+

ϕ(x; d 1 , d 2 ) ∼ 2C(d 2 , d 1 )F (1, d 1 , 2 -d 1 ; -1)(1 + x) 1-2d2 = (C * ) -1 2 1-2d1 π sin(πd 1 ) (1 -x) 1-2d2 . Remark 3.1 Clearly, ϕ 1 (x; d 1 ) ∼    C(d 1 )x d1-1 , x → 0+, 2C(d 1 )(1 -x) 1-2d1 , x → 1-, ϕ 2 (x; d 2 ) ∼    C(d 2 )|x| d2-1 , x → 0-, 2C(d 2 )(1 + x) 1-2d2 , x → -1 + .
Hence, by Proposition 3.2, the mixture density ϕ associated with the product spectral density (3.8) behaves as ϕ 1 when x approaches 1, and behaves as ϕ 2 when x approaches -1. However, at zero, ϕ behaves as |x| d1+d2-1 , i.e. both densities ϕ 1 and ϕ 2 count.

Proposition 2.1 allows us construct the mixture density also in the case when the spectral density f of aggregated process has the form

f (λ) = 1 2π 2 sin |λ| 2 -2d g(λ), 0 < d < 1/2,
where g(λ) is analytic spectral density on [-π, π]. In general, the existence of the mixture density associated with any analytic spectral density is not clear. For example, AR(1) is aggregated process only if the mixture density is the Dirac delta function, what is difficult to apply in practice. Similar inference concerns also the ARMA processes, i.e. rational spectral densities. Another class of spectral densities obtained by aggregating "non-degenerated" mixture densities is characterized in the following proposition.

Proposition 3.3 A mixture density ϕ g is associated with some analytic spectral density if and only if there exists 0 < a * < 1 such that supp(ϕ) ⊂ [-a * , a * ].

Proof. For sufficiency, assume that there exists 0 < a * < 1 such that supp(ϕ) ⊂ [-a * , a * ]. The covariance function of the corresponding process satisfies

|γ(h)| ≤ σ 2 ε 1 -1 |x| |h| 1 -x 2 ϕ(x) dx = σ 2 ε a * -a * |x| |h| 1 -x 2 ϕ(x) dx ≤ σ 2 ε a |h| * a * -a * ϕ(x) 1 -x 2 dx = Ca |h| * ,
i.e. the covariance function decays exponentially to zero. This implies that the spectral density f (λ) = (2π) -1 ∞ h=-∞ γ(h)e ihλ is analytic function on [-π, π] (see, e.g., Bary (1964, p. 80-82)).

To prove the necessity, assume that f is an analytic function on [-π, π] or, equivalently, the corresponding covariance function decays exponentially to zero, i.e. there exists θ ∈ (0, 1) and a constant C > 0 such that |γ(h)| ≤ Cθ |h| . Assume to the contrary that supp(ϕ) = [-1, 1].

Let h ≥ 0 be an even integer. Then

|γ(h)| = σ 2 ε 1 -1 |x| h 1 -x 2 ϕ(x) dx ≤ Cθ h implies that 1 -1 |x| θ h ϕ(x) 1 -x 2 dx ≤ C. (3.15)
Rewrite the last integral as

1/θ -1/θ |x| h ϕ(θx) 1 -(θx) 2 dx = 1 -1 |x| h ϕ(θx) 1 -(θx) 2 dx + -1 -1/θ |x| h ϕ(θx) 1 -(θx) 2 dx + 1/θ 1 |x| h ϕ(θx) 1 -(θx) 2 dx =: I 1 (h) + I 2 (h) + I 3 (h).
For every x ∈ [-1, 1] we have

|x| h ϕ(θx) 1 -(θx) 2 ≤ ϕ(θx) 1 -(θx) 2 .
Hence, by the dominated convergence theorem and (1.3), I 1 (h) → 0 as h → ∞. The Fatou lemma, however, implies that both integrals I 2 (h) and I 3 (h) tend to infinity as h → ∞:

lim inf h→∞ I 2 (h) ≥ 1/θ 1 ϕ(θx) 1 -(θx) 2 lim inf h→∞ x h dx = ∞ since 1/θ > 1. This contradicts (3.15).

2

Example 3.1 Assume that the mixture density ϕ g has a uniform distribution on [a, b], where -1 < a < b < 1. By Proposition 3.3, the associated spectral density is analytic function on [-π, π] and can be easily calculated:

f g (λ) = σ 2 ε 2π(b -a) b a dx 1 -2x cos λ + x 2 = σ 2 ε 2π(b -a) sin |λ| arctan b -cos λ sin |λ| -arctan a -cos λ sin |λ| , λ = 0, ±π. f g (0) = σ 2 ε (2π) -1 (1 -a) -1 (1 -b) -1 , f g (±π) = σ 2 ε (2π) -1 (1 + a) -1 (1 + b) -1 .
We obtain the following corollary.

Corollary 3.1 Let ϕ 1 (x; d) (3.3) and ϕ g (x) be the mixture densities associated with spectral densities f 1 (λ; d) (3.1) and analytic spectral density g(λ), respectively. Assume that supp(ϕ g ) ⊂ [-a * , 0], 0 < a * < 1, and

f (λ) = 1 2π 2 sin |λ| 2 -2d
g(λ).

(3.16)

Then the mixture density ϕ(x), x ∈ [-a * , 1] associated with f is given by equality

ϕ(x) = C -1 * ϕ 1 (x; d)) 0 -a * ϕ g (y) (1 -xy)(1 -y/x) dy+ϕ g (x) 1 0 ϕ 1 (y; d) (1 -xy)(1 -y/x) dy ,
where

C * := 1 0 0 -a * ϕ 1 (x; d)ϕ g (y) 1 -xy dy dx.
4 The structure of aggregated process

In order to make further inference about the aggregated process X t , e.g., estimation of the mixture density, limit theorems, forecasting, etc., it is necessary to investigate more precise structure of X t . In particular, it is important to obtain the linear (moving average) representation of the aggregated process.

Behavior of spectral density of the aggregated process

In this subsection we will study the behavior of the spectral densities corresponding to the general class of semiparametric mixture densities of the form (see [START_REF] Oppenheim | Aggregation of random parameters Ornstein-Uhlenbeck or AR processes: some convergence results[END_REF], [START_REF] Leipus | Orthogonal series density estimation in a disaggregation scheme[END_REF])

ϕ(x) = (1 -x) 1-2d1 (1 + x) 1-2d2 ψ(x), 0 < d 1 < 1/2, 0 < d 2 < 1/2, ( 4.1)
where ψ(x) is continuous and nonvanishes at the points x = ±1. As it is seen from the mixture densities appearing in Section 3, this form is natural, in particular (4.1) covers the mixture density in (3.9). The corresponding spectral density behaves as a long memory spectral density.

Lemma 4.1 Let the density ϕ(x) be given in (4.1), ψ(x) is nonnegative function on [-1, 1] and continuous at the points x = ±1 with ψ(±1) = 0. Then the following relations for the corresponding spectral density hold:

f (λ) ∼ σ 2 ε ψ(1) 2 2d2+1 sin(πd 1 ) |λ| -2d1 , |λ| → 0, (4.2) f (λ) ∼ σ 2 ε ψ(-1) 2 2d1+1 sin(πd 2 ) |π ∓ λ| -2d2 , λ → ±π. (4.3)
Proof. Let 0 < λ < π. (4.1), (1.5) and change of variables u = (x -cos λ)/ sin λ lead to

f (λ) = σ 2 ε 2π 1 -1 (1 -x) 1-2d1 (1 + x) 1-2d2 ψ(x) |1 -xe iλ | 2 dx = 2 sin λ 2 -2d1 2 cos λ 2 -2d2 g * (λ),
where

g * (λ) = σ 2 ε π tan λ 2 -cot λ 2 (sin λ 2 -u cos λ 2 ) 1-2d1 (cos λ 2 + u sin λ 2 ) 1-2d2 1 + u 2 ψ(u sin λ+cos λ) du.
By assumption of continuity at the point 1, the function ψ(u sin λ+cos λ) is bounded in some neighbourhood of zero, i.e. ψ(u sin λ + cos λ) ≤ C 1 (λ 0 ) for 0 < λ < λ 0 and λ 0 > 0 sufficiently small. Hence,

(sin λ 2 -u cos λ 2 ) 1-2d1 (cos λ 2 + u sin λ 2 ) 1-2d2 1 + u 2 ψ(u sin λ+cos λ) ≤ C 2 (λ 0 )(1 + |u|) 1-2d1
1 + u 2 for 0 < λ < λ 0 and, by the dominated convergence theorem, as λ → 0,

g * (λ) → σ 2 ε ψ(1) π ∞ 0 u 1-2d1 1 + u 2 du = σ 2 ε ψ(1) 2π Γ(d 1 )Γ(1 -d 1 ) = σ 2 ε ψ(1) 2 sin(πd 1 )
implying (4.2). The same argument leads to relation (4.3). 2

Remark 4.1 Note, differently from Viano and Oppenheim (2004), we do not require the boundedness of function ψ(x) on interval [-1, 1]. In fact, ψ(x) can have singularity points within (-1, 1), see (3.3), (3.6).

Moving average representation of the aggregated process

Any aggregated process admits an absolutely continuous spectral measure. If, in addition, its spectral density, say, f (λ) satisfies

π -π log f (λ)dλ > -∞, (4.4)
then the function

h(z) = exp 1 4π π -π e iλ + z e iλ -z log f (λ)dλ , |z| < 1,
is an outer function from the Hardy space H 2 , does not vanish for |z| < 1 and f (λ) = |h(e iλ )| 2 . Then, by the Wold decomposition theorem, corresponding process X t is purely nondeterministic and has the MA(∞) representation (see Anderson (1971, Ch. 7.6.3))

X t = ∞ j=0 ψ j Z t-j , (4.5)
where the coefficients ψ j are defined from the expansion of normalized outer function h(z)/h(0), ∞ j=0 ψ 2 j ∞, ψ 0 = 1, and Z t = X t -X t , t = 0, 1, . . . ( X t is the optimal linear predictor of X t ) is the innovation process, which is zero mean, uncorrelated, with variance

σ 2 = 2π exp 1 2π π -π log f (λ)dλ . (4.6)
By construction, the aggregated processes are Gaussian, implying that the innovations Z t are i.i.d. N(0, σ 2 ) random variables.

We obtain the following results.

Proposition 4.1 Let the mixture density ϕ satisfies the assumptions of Lemma 4.1. Then the aggregated process admits a moving average representation (4.5), where the Z t are Gaussian i.i.d. random variables with zero mean and variance (4.6).

Proof. We have to verify that (4.4) holds. According to (4.2), log

f (λ) ∼ -C 1 log |1 -e iλ |, |λ| → 0, where C 1 > 0. For any ǫ > 0 choose 0 < λ 0 ≤ π/3, such that - log f (λ) C 1 log |1 -e iλ | -1 ≥ -ǫ, 0 < λ ≤ λ 0 .
Since -log |1 -e iλ | ≥ 0 for 0 ≤ λ ≤ π/3, we obtain (4.8)

When λ ∈ [λ 0 , π -λ 0 ], there exist 0 L 1 < L 2 < ∞ such that L 1 ≤ 1 2π|1 -xe iλ | 2 ≤ L 2
uniformly in x ∈ (-1, 1). Thus, by (1.5), L 1 ≤ f (λ) ≤ L 2 for any λ ∈ [λ 0 , π -λ 0 ], and therefore

π-λ0 λ0 log f (λ)dλ > -∞.
(4.9) (4.7)-(4.9) imply inequality (4.4). 2

Lemma 4.2 If the spectral density g of the aggregated process X t is analytic function, then X t admits representation

X t = ∞ j=0 g j Z t-j ,
where the Z t are Gaussian i.i.d. random variables with zero mean and variance

σ 2 g = 2π exp 1 2π π -π log g(λ)dλ (4.10)
and the g j satisfy | ∞ j=0 g j | < ∞, g 0 = 1.

Proof. From Proposition 3.3, there exists 0 < a * < 1 such that

g(λ) = σ 2 ε 2π a * -a * ϕ g (x) |1 -xe iλ | 2 dx. (4.11) For all x ∈ [-a * , a * ] and λ ∈ [0, π] we have 1 |1 -xe iλ | 2 ≥ C > 0.
This and (4.11) imply

π 0 log g(λ)dλ > -∞. Finally, | ∞ j=0 g j | < ∞ follows from representation g(λ) = σ 2 g 2π ∞ j=0 g j e ijλ 2
and the assumption of analyticity of g. 2 Proposition 4.2 Let X t be an aggregated process with spectral density

f (λ) = 1 2π 2 sin |λ| 2 -2d g(λ), 0 < d < 1/2,
satisfying the assumptions of Corollary 3.1. Then X t admits a representation (4.5), where the Z t are Gaussian i.i.d. random variables with zero mean and variance

σ 2 = 2π exp 1 2π π -π log f (λ)dλ = exp 1 2π π -π log g(λ)dλ = σ 2 g 2π
and the ψ j satisfy

ψ j ∼ ∞ j=0 g j Γ(d) j d-1 , ψ 0 = 1. (4.12)
Here, the g j are given in Lemma 4.2.

Proof. We have

1 2π 2 sin |λ| 2 -2d = 1 2π ∞ j=0 h j e ijλ 2 with h j = Γ(j + d) Γ(j + 1)Γ(d)
and, recall,

g(λ) = σ 2 g 2π ∞ j=0 g j e ijλ 2 , ∞ j=0 g 2 j < ∞ since, by Lemma 4.2, π -π log g(λ)dλ > -∞. On the other hand, π -π log f (λ)dλ > -∞ implies f (λ) = 1 2π ∞ j=0 ψj e ijλ 2 , ∞ j=0 ψ2 j < ∞
and, by uniqueness of the representation,

ψk = σ g √ 2π k j=0 h k-j g j .
It easy to see that,

k j=0 h k-j g j ∼ h k ∞ j=0 g j ∼ C 2 k d-1 , (4.13)
where

C 2 = Γ -1 (d) ∞ j=0 g j . Indeed, taking into account that h k ∼ Γ -1 (d)k d-1 , we can write k j=0 h k-j g j = Γ -1 (d)k d-1 ∞ j=0 a k,j g j ,
where a k,j = h k-j Γ(d)k 1-d 1 {j≤k} → 1 as k → ∞ for each j. On the other hand, we have |a k,j | ≤ C(1 + j) 1-d uniformly in k and, since the g j decay exponentially fast, the sum ∞ j=0 (1+j) 1-d |g j | converges and the dominated convergence theorem applies to obtain (4.13).

Hence, we can write (5.5) and the variance of the noise is (see formula 6.1.17 in [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF])

f (λ) = σ 2 g (2π) 2 ∞ j=0 ψ j e ijλ 2 , ψ 0 = 1, where ψ j = ψj √ 2π/σ g ∼ C 2 j d-
σ 2 ε = 2 -d ( C(d)) -1 ∞ 0 dz 1 + z 1/d

  |1 -e iλ |dλ = 0. Using (4.3) and the same argument, we get π π-λ0 log f (λ)dλ > -∞.

  1 and (4.5) follows. 2 5 Appendix. Mixture density associated with FI(d) spectral density Proposition 5.1 Mixture density associated with FI(d) spectral densityf (λ; d) = C(d)x d-1 (1 -x) 1-2d (1 + x)1 [0,1] Equality f (λ; d) = 2 -d-1 π -1 (1 -cos λ) -d implies that 1 -cos λ = (π2 d+1 f (λ; d)) -1/d . Hence, rewriting |1 -xe iλ | 2 = (1 -x) 2 1 + 2x (1 -x) 2 (1 -cos λ) ,and assuming supp(ϕ) = [0, 1], we obtain that the spectral density of aggregated process is of the formx) 2 (π2 d+1 f (λ; d)) -1/d dx. (5.2)The change of variables y1/d = 2x/(1 -x) y 1/d (π2 d+1 f (λ; d)) -1/d ) = f (λ; d)σ 2 ε 2 d C(d) ∞ 0 dz 1 + z 1/dafter the change of variables z = y π2 d+1 f (λ;d) . Therefore, FI(d) is an aggregated process and, by (5.3)-(5.4), the mixture density has a formϕ(x) = C(d)d2 d x d-1 (1 -x) 1-2d (1 + x),

  d (d C(d)) -1 (B(d, 1 -d)) -1 = 2 -d (d C(d)) -1 sin(πd) π .Finally, it remains to calculate the constant C(d) to ensure that the mixture density ϕ given in (5.5) integrates to one over the interval [0, 1]. We have1 0 ϕ(x)dx = C(d)d2 d 1 0 x d-1 (1 -x) 1-2d dx + 1 0 x d (1 -x) 1-2d dx = C(d)d2 d B(d, 2 -2d) + B(d + 1, 2 -2d) = C(d)d2 1+d Γ(d)Γ(2 -2d)) (2 -d)Γ(2 -d) 2) -d)and C(d) = C(d)d2 d . 2

scientific program Gilibert, PAI EGIDE 09393 ZF.


D Hernández
email: daniel.hernandez@tu-ilmenau.de

C Karcher

Lorentz force velocimetry using small-size permanent magnet systems and a multi-degree-offreedom force/torque sensor

Keywords: velocity measurement, liquid metal, contactless

Lorentz force velocimetry using small-size permanent magnet systems and a multi-degree-of-freedom force/torque sensor

D Hernández, C Karcher

To cite this version:

Introduction

In local LFV, owing to the rapid decay of magnetic fields, a localized magnetic field distribution in the liquid metal is archived by using magnets comparatively smaller that the cross-section of the flow. The permanent magnet is connected to a sensitive force sensor and a qualitative assessment of the velocity distribution of the liquid metal can be obtained. In this case, the Lorentz force F L is proportional to the velocity V, to the applied magnetic field B 0 to the power of two and to electrical conductivity σ of the liquid metal [1]:

F L ~ V σ B0 2 (1)
This technique has already proven to identify obstacles and the wake behind them in a rectangular duct with an accuracy of 3 cm with a 1 cm cubic magnet [2]. The reference measurement device was a 1D interference optical force sensor reaching a resolution of 0.3 μN at experimental conditions. In this experiment, just one force component acting on the magnet can be measured simultaneously, i.e. the Lorentz force in the streamwise direction. However, velocity profiles of liquid metal in industrial applications are not simple 1D flows but mostly 3D having low and high scale turbulent structures. For that purpose, we are introducing a new concept of local Lorentz velocimetry in which the permanent magnet is connected to a multi-degree-of-freedom force/torque sensor and the common cubic magnet is replaced by an arrangement of permanent magnets (Table 1). In this case, we are experimentally investigating the effect of different magnetic field distributions on the Lorentz force. As a first step, the magnets are compared based on the model experiments at the experimental facility GALINKA (Fig. 1). The reference 1D force measurement system is used for this purpose having GaInSn in eutectic composition as a test fluid [2].

Problem description

As explained before, as a contactless measurement technique, local LFV allow us to obtain local velocity information of electrical conductive liquids. In this approach, the volume subset of the liquid that interacts with the magnetic field has to be considerably smaller that the cross-section of the flow. However, by decreasing the size of the magnet, the force reduces as well making its measurement a big challenge to present techniques. Having as reference the 10 mm cubic magnet used in [2], we propose a cross-shaped magnet and a cross-shaped magnet arrangement while maintaining the volume and the material of the magnet system constant. We are going to investigate the influence of the change of the magnetic field distribution on the Lorentz force at the liquid metal loop GALINKA (Fig. 1). An electromagnetic pump drives GaInSn in eutectic composition through stainless steel pipes. Just before the liquid metal enters the plexiglass test section, it flows across a honeycomb (D h = 3 mm, L = 160 mm) that works as an hydraulic resistance and a typical turbulent flow is achieve. The wall thickness of the duct is 5 mm and has a rectangular cross-section of 50x50 mm 2 . At the beginning of each experiment, the center of the permanent magnet is placed on the bottom of the wall at y = -25 mm (y * = -1) and its outer surface is placed almost touching the wall at z = 0 mm (Fig. 2). Then, the magnet is moved upwards to y = 25 mm (y * = 1) in 2.5 mm steps. In each step we record the average value of the force, obtaining access to velocity information of the liquid metal inside the duct. Finally, the distance between the outer surface of the magnet and the test section is changed from z = 0 mm (almost touching the wall) to z = 6 mm with a 1 mm step size. As a result, we have an insight on how the Lorentz force depends on the position of the magnet system in y and z directions.

Results

The results of the parametrical study are summarized in Fig 3 . According the force signals, a higher symmetry of the velocity profile with the proposed honeycomb was seen in comparison with the ones obtained by Heinicke in [2]. Here, a shorter one (L = 60 mm, D h = 3 mm) was used at the same experimental conditions. Nevertheless, there is still a discrepancy between the force measured at the bottom duct (y * = -1) and the top of the duct (y * = 1). This effect can be explained due to the fact that, there is a minor misalignment of the surface of the duct and the vertical movement of the magnet. As a result, the magnet is slightly closer to the wall at y * = 1 than at y * = -1 which corresponds to an increment of the force. As a next step, we are going to introduce a corrector factor of the force as function of the distance that will minimize this effect. Finally, a visible improvement of the force was not seen by using the proposed cross-shaped permanent magnet systems in comparison with the reference 10 mm cubic magnet. Possible causes of these results may be explained owing to the rapid decay of the magnetic field with the distance. As a consequence, the effect on the magnetic field distribution is negligible for different geometries of magnet systems at distances higher that 5 mm for a 1000m 3 magnet. . There is still some discrepancy of the value of the force at the bottom and top of the duct (y * =-1 and y * = 1) which is expected to be the equal. The fact that on the bottom of the duct we have higher force can be explained by the slightly misalignment of the duct. As a result, the magnet is closer to the liquid metal on the top than on the bottom of the duct. A comparison of all magnets is shown in (d) at z = 0 mm presenting no noticeable difference between them. Galinstan in eutectic composition is flowing through the rectangular plexiglass duct (50mm x 50 mm) at a Re ≈ 7000.

Conclusions

Based on the results of local LFV presented in this paper, a far more symmetrical velocity profile was seen at the rectangular test section of GALINKA using the proposed honeycomb. This fact implies that we may have successfully obtained a typical turbulent velocity profile in the duct. Now we are performing velocity measurements using Ultra Doppler Velocimetry (UDV) in order to validate our results. Even though there was no noticeable difference between the Lorentz force signals using the new permanent magnet systems, we cannot conclude if there is or not any advantage. For this purpose, we are going to test at first the multi-degree-of-freedom sensor and the magnet systems using a rotating electrically conductive disk achieving a higher control and accuracy of the measurements. In this case, we are able to have a better insight of the forces and torques acting on the magnets by a given velocity profile and velocity gradient.

Table 1 :Fig. 1 :

 11 Fig. 1: Experimental facility GALINKA. The liquid metal loop consists of stainless steel pipes and an 80cm long plexiglass rectangular duct (50mm x 50mm). The GaInSn is driven by an electromagnetic pump and flows though the test section, where the force sensor is placed beside and measures the resulting streamwise force acting on the magnet. Fig 2. Set-up for parametric study investigating the influence of different geometries of permanent magnet systems on the Lorentz force. Their volume (1000 mm 3 ) and the flow rate (Re ≈ 7000) of GaInSn are held constant. On the figure the magnet is placed in y = -25 mm and moves vertically (double sided arrow) until reaching the top wall of the duct (y = 25 mm).

Fig 3 .

 3 Fig 3.Results of parametric study investigating the effect of different magnetic field distributions on the Lorentz force using a cubic (a), a cross-shaped magnet arrangement (b) and a cross-shaped magnet (c). All magnet systems share the same volume (1000m 3 ) and material (N52). The center of the magnet moves from the bottom (y * =-1) to the top (y * =-1) of the duct in 2.5 mm steps. The outer surface of the magnet is located almost touching the surface of the duct (z = 0 mm) and then is increased in 1 mm steps. The results show a clearly more symmetrical turbulent velocity profile based on the Lorentz force measurements in comparison with the ones obtained in[2]. There is still some discrepancy of the value of the force at the bottom and top of the duct (y * =-1 and y * = 1) which is expected to be the equal. The fact that on the bottom of the duct we have higher force can be explained by the slightly misalignment of the duct. As a result, the magnet is closer to the liquid metal on the top than on the bottom of the duct. A comparison of all magnets is shown in (d) at z = 0 mm presenting no noticeable difference between them. Galinstan in eutectic composition is flowing through the rectangular plexiglass duct (50mm x 50 mm) at a Re ≈ 7000.

Acknowledgment

Financial support from LIMTECH Alliance of Helmholtz Society and the Deutsche Forschungsgemeinschaft DFG within the RTG Lorentz Force Velocimetry and Eddy Current Testing under Grant GRK 1567/1 is gratefully acknowledged.