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11Journal of Siberian Federal University. Engineering & Technologies 5 (2012 5) 568-578
УДК 621.74.011
About Possibility of Mixing of Liquid Metal with External Rotational Magnetic Field
Toksan A. Zhakatayeva* and Klara Sh. Kakimovab
a E. A. Buketov Karaganda State University 28 Universitetskaya Str., Karaganda, 100028 Kazakhstan, b Karaganda State Technical University, 56 Bulvar Mira, Karaganda, 100016 Kazakhstan 1
Received 06.09.2012, received in revised form 13.09.2012, accepted 20.09.2012
Physical justification ofa hypothesis that liquid metal in a melting bath and in a converter can be mixed with external rotational magnetic field is carried out. Numerical solution of the kinetic equations for casual collisions with use of free path length and ofeffective sections ofatoms shows that concentration of impurities can decrease to a desired level in time interval ranging from units of minutes to several seconds. This proves the needforbetter mixing of molten metal with use of magnetic fieldfor increased efficiency ofproduction cycle. It is assumed that free atoms of carbon C and oxygen O in liquid metal don't lose kinetic energy as a result of potential interactions with atoms of Fe. Therefore this model works approximately. Procedure for exact integration of equation of motion for a Brown particle in three-dimensional case is shown.
Keywords: melted, metal, mixing, magnetic, field, circular, impurity, interaction, speed, particle, Brown, travel.
Introduction
Development and implementation of new technologies for effective mixing of liquid metal is still a vital problem. It is known that duringmixingprocess various ingots are formed with different physical and chemical properties. Development of a new theoretical method for calculation of chemical reaction rate of various impurities in liquid metal is an important task in metallurgy. Various atoms and molecules in liquid metal move on trajectories similar to trajectories of Brown particles. Thus when considering motion of atoms and molecules all degrees of freedom need to be considered equally.
Liquid metal (molten metal of crude-iron and steel) can have density around 7000 kg/m3 at the temperature of about 1600 °C [1 - 3]. Processes of carbon oxidation in liquid metal play very important role, C+O = (CO). It is known that carbonization already begins at temperatures of 400 - 600 °C [1-3]. As the process of carbonization goes on the temperature of molten metal goes down. The melting temperature of iron containing 4.3 % carbon can go down to 1147 °C.Carbon oxide plays an important role in reduction reactions of iron [1-3]:
* Corresponding author E-mail address: Toksanzh@yandex.kz
1 © Siberian Federal University. All rights reserved
3Fe2O3 + CO = 2Fe304 -a CO2 - h3740 i'mon
Fe3O4+C0 = 3FoO-rC02+r6680 J/mol, (1)
Fee + CO = Fe +C02 - 16060 Jrmol.
IMietliods! and equipment
Numerical calculations were carried out on PC using Maple f 5 package are used. We expanded an analytical (theoretical) solutionof the differential equation for a Brownian particle into general three-dimeneionpl case.
Results
We have oeceoved the following new differential model for calculation of interaction rate of C and O aloms in liquid mutal
dn3 .r _
dn^-Ti^ •lvne-nrei , (2)
dno 2 _
-=- = -jr-d12-ivn0- vreh (3)
where dj^d;1 i = + rf - effective diiameters op mteractmg atomn, rc =0.7h-l(T10 m-eadtuo o° at carbon atom, n = =.5 8 • 0=~arm-radiu s rf an qaygen etona, t - finee, s.
Avemage spead of oelatier motion ii dafined byequation thi = |v2 - vi| ( In eoa caee i2 = vc -abioluta i^;j:><3«i;<:r oO a cerbon atf!- v 1 = vo - aboolutn speed oi0 ae oxygon atom.
In molten liquid metae ftome of and tnabon san move freely, so their movements can
be described as in moleculaa-einetic's (heory. ia atoms ane ]^ia.:rt^io^l_;-ir ionized, then occillatkng i^ot;ictn one cryntal cus^l wil( ba accompanied by taaesitiou potential ssi^e^r"!^;;-) to kinetic energy end back. Howevea, total col^^noeiic^ raiTe;^t of aeoma wii« be like movement of a Beownian particle. Average kinetec onergy o:^ atoms eemains co notant ant is not lo et to potential energy of interaction with atoms.Excluding moments of time where atoms collide, their motion can be described as in molecular-kinetic's theory.The definition equation for relative speed of motion looks like [4]:
vrel =
8RT( 1 1
(4)
1L ^ H0 |c
where R= 8. 331 J/(mol*K)-a universal gaoconstant; T ^ao^tui-e , K ; | o , | c - molarmas se s
of aL^o^so, kg/mol.
The equations (2!)) and (3) form a nonlinear system of equations, whose analytical solution is:
n c = nc , 0 • eqp- n • djf • no • vrel • it], (5)
no = n0° • exp [- n • • nc ' Ond • 1; ] 6 )
Where o-°5 — initial value of concentration of impiirisy carbon atoms, n0 0 — initiol u^eiiLlu-i-; of concentration of impurity oxygen atoms.
The system o° oquations (2) and (f) -vfa^sw; solved on Maple 15 package. According to oolutfon dfop of anncenteatsoo of carbon and oxygen atoms from fevel nc0, nt0 (100 %) to a leveC of 2 % occurs in
- 069 e-
t. >
Fig;. 1. Change of concentration ofcariton - 1 and oxygen - 2 atoms in liquid metal as a function of time
very small time interval t~10-11 -10"10 s, Fig. 1. Fig. shows change of relative concentration of carbon and oxygen anoms as a function of time. Element, which. has higher initial concengration than the other, decreases more slowly.
If initial concentration of oxygen atoms is token very tmall, so that only ono collision (one interaction) of C+O is oonsidered at a time, thon aa estimated time of reduction of carbon concentraaion can increase up to 1012 s. By performing calculations at various initial concentrations n,, an estimated time (for reduction of a level of carbon nCO to 0.5 %) equal to 3 minutes is obtained. Thus it is possible to reduce time of technological processing of liquid metal to level of 1-3 minutes.
Careful attention needs to be paid to the fact that our calculations and results correspond to an ideal case: arrangement of impurity atoms is uniform throughout the volume; there are no retarding factors, such as the inverse reaction of CO decomposition;there are no convective currents (loss of atoms);absence of stagnation zones in molten metal;impurity atoms have equal initial concentrations (nc=no) and so on. In reality not every collision of a carbon atom with an oxygen atom results in formation of a CO molecule.
We have dona calculaOions for 1 nf of liquid metal with the following initial sonditions. Liquid metal has three components: nos = npe +nC -one0, pf = '/'000 kg/m3, T=1873 K - density and temperature of liquid metal, oC = 0.04** nae (at 0 %) - mess concenraation nf caobon, NFa = 7.467 • 1028, NC =bO =1.3f4-1028 - (nitial total quaoeity of iron, earbon and oxygen atoms in 1 Mr of Oicjaid metal. For afcrbon erom we obtain: iree paihanagth--. =O .=84-liru m;aue=agtspeed vc =lfl0.41m/s;diffusion caefficient DC = 4.051><ta-8 Oimilarly, for atom an oxygen oiitom we aec/ive: Wf = 7.9t6-10-1i; D0 = 4.a0740"8 m204; v, = =Sn7.7e ro/s, vr/l = 2395(82 m/s.
TTlors tSe ptoblem of more inteosine ond uniform mixing of al1 eolume of8 Hquid metaS, so that all aeome of C and O could enteo inno reac8ion eimukaneoualy aad in ihe shottesO ponsible it^me;, is press iag. We see the solution of a problem in application of rotating variable magnetic field.
AE - winding 1 i C, - winding 2, E® - winding 3, 1,2 - external and internal walls of a melting bath, 3 - molten metal
Fig. 3. Scheme of an arrangement of three-phase winding on ex)ernal side of a case of a melting bath or con-
Fig. 2 shows mag netic field pe netrating a brick with finite wall w idth. These bricks are used in c onve rter walls or in a melting bath in the metallurgical furnac e. Appare ntly at ve rtical position of a brick the magnetic diak does not fall at all under action oa gravity, it is kept with the force of magnetic; attraction. This means that magnetic field possessef significant penetrating capacity through hi brick.
In Fig. 3 the scheme oS an arrangement of exttrnal windings around a case of a melting unit is shown. This arrangement of winning genenctes atoemative sotating magnetic field. Thus we suggest a new applicatton of rotating mageetic field, which is used widely in thretfphase windings of electromagnetic motsert - [5-7]. U he vector ot magnetic endurtfen B = fit S rotates around an axis O witn the fixed frequency cc.
Resultant magnetic fifid rs rotating Use cause magnetic field is generated in three w indings 1, 2, 3 with phase shiet of012!! 0° degrees - [55-7]
Fig. 2. Penetration of magnetic field through a piece of a brick
verter
B J = 13 m sin cot , B2 = Bm sin(cot- 1 20 o) , B3 = Bm sin(ot - 240o),
whe re IBm - anoippjlitiod^ o;f a magnetic field vector, T
The megnitude of a vecSoo of megnetic induction IB doe si not depend on time and is equal [5-7]
B = fBm- (8)
m n 3
The effect of rotation of B(t) fn epace results from — JзlгaLs; ^ it hift toft proj s cjteons Bx = —Bm rlncot
2 2
3 o
and By = — Om loicot, sto tga = —B = itgycot.
2 B y
Kinematic viscosity of steelmakjng aJLags in the aange of v = 3*t10-6-T0"4) m2/s and
kinemafic ^viL^cv^isui.il^ ef liquid seeel changes accordingly in the range of 4iO-7-0S-rO-7 m2/s [1-3]. Forwaier this parameten fs equal to 990"7 m2/s. It appeals afiat liquid me7al has low viocotity.
By penetrating into deptO of lifuid metal petiodrcal)y rotating mngno tic fir Cd §(10 will drag behind Otsele and cause to rotate tome layeo of tiquid mets" The givon ]^i^1il::iLC)(c3h if goed tn that tha magnetic fie7d will hotoaie layers ef liquid metff which adsom or stick to ae internal wall ot a converter ot a melting bath. Tfui layeos of molten meSaO which fire rhe lef st involved rn mixing before, be mixod the mose.
Fig. 4 is the photo of a constant magnet with which we have conducted experiment uaing thin metal grif. Distortion of magnetic field by thin layer of a metal sheet was sfudied. As experimenie have shown, flat thin iron sheets do not weakennoticeably magnetic field B. Similar experiments were done using the ring magnet in Fig. 2. Thus thin layered metal sheets on external sides of melting baths and converters will not reduce substantially penetration of magnetic field into internal volume of liquid metal.
Interaction of an external exciting magnetic field B with the magnetic fields of currents I1 and I2 is shown in Fig. 5. The current I1 is generated when magnetic flux ®=B^S increases, and the current I2 is induced when flux decreases. It follows from Faraday's law of induction for" boundary parts of external magnetic field -vector, which are shown in Fig. 6, part 1 and 3. In these boundary parts of external magnetic field the doughnut-shaped internal induced field. Bp = Bj + B2 turns to 90o and "sticks" to a wall oil a melting; brth 22 (Fig. 5). This phenomenon is similar to Helmholtz vortices obrerved in fluid
Fig. 4. the source of cylindrical magnetic field
Fig. 5. Interaction scheme of extarnal and of induced internal magnetic fields in a melting bath 1 - winding s for generation of an axteanal magnetic field B ,2 - wall ofmelting bath, 3 - dire ction of rotation of a aectoa B , B+, B2 - ifcduitj^d mf.nalic fiatdS1 OO' - the axir of amelting bath
mechanics . We note that currents Ii and s2 are results of superposition of two currents: Ii = I1e + Iu I2 = I2,e + ^2,1- Thf index "e" corresponds to current of free electrons and the index "i" corresponds to current of positive or negative ions.
In the central (aonstant)partofthe external magnetic field B (Fig. 6, part 2)the direction of currents I1 and I2 depends on a direction of rotation of tht vector B, a designation 3 in Fig. 5. The direction of currents cans Is e determined by the rule of the righc hand. ID irection oh the mognetic volumetric force acting on an element of volume can be defined by the left hand rule. Interaction of the internal induced magnetic field vector Bp with the external magnetic field vectot B leads to attraction, repulsion or cross-sectional (relative to B) movement of some finite layer of liquid metal. As a result this volume of liquid is involved in mechanical motion and there will be mixing of various layers of liquid metal. The thickness of layer of liquid involved in motion depends on the cross-sectional size of a winding 1 in Fig. 5. Hence rotating magnetic field and volume of liquid metal brought to motion by this field has finite cross-sectio nal sizes.
Thn proposed method of intensification of intermixing of liquid metal does not only apply for intaracrion of C+Q It is possible to apply the method to improve mixing and to increase the rate of chemical reaction of all other impurities in various molten metals and their species.Calculation of mtxing ratnof impunities plays an important role in the manufacture of semiconductor crystals, which are wfdely u ted i n electronic indusrry.
There are pocticte) composed of several ctoms and molecules within liquid metal, for example, such as CO, SO, FeC, CaO, NO. There are also large enough particles of slag and structures composed of severalatoms of external impurity. AH of ttlrejse; particles and even angle atoms in liquid metal can move likr Brownian particles. Anctysis of Che equations detcribing mntion of Brownian particles is v^al for the meeallurgical branches. In [4, 8, 9] thte following equation foe calculation of distance traveled by a Brownian particle along X axis is received:
x2 = (9)
3n|r
Fig. 6 - Cross-sectional distribution of external magnetic field vector B
1,3 - gradient (boundary) part of the magnetic field, 2 - constant (central) part of the field
where fi-dynamic viscosity oIf 1 i qui d, a - p article rad ius, T - absolute temp erature, t- time. In literature equation for one-dimensional motion of a Brown particle is presented [4, 8]
m^-B^ + FxCt), (10)
dt2 dt
Alsoin literature three-dimensional equation is given [9]
2 r r
m-Bl = -B£ + F(t), (11)
dt2 dt
whe r e r = r (x , y, z ) - ra dius-vect or ff a parttole, li7 (t) = F(t)=( Fx, Fy,Fz) - iec)cce <cs^sit.iil in magnitude and direction. This force chaeacterizes easuhl wandeemf* arajectories of particles [4, 8, 9],
B = 6 n na (12)
coefficient in lhe Stoaks' equation for drag resistanoo force Fr = B • v, v - speed oda moving particle. For sumplifiuaiion purposed equation (12) is muttiplted in scular way on r and after anme transfosmations the [Calar equat ion turyo out ,9]
(13)
2 dt 2 dt
Letls uverage <^nn"t the given equation on tet oa particle/ and then we wril receive
UBditu )+*. A^t, (14)
2 dt 2 1 ' 2 dt ' V '
as nrv2d2 = 3kT/2 - nffsrdinn (to Sho theorem ofuniiitccir-rsr distribution ofthermal energy on deedrees of free dam and = 0 - force avevaged over time ie equal to eerr».
The analysis carried out by us has eUnwn that the equation (14) can't be integrated twice ifintegration it done frem t = 0 up to the nrbitrary tims moment t , nnd then both pafts ofthe uerult art divide dby a time interval t, at it i si done in [C]( In this case after the fieit integration thr equat ion turns out to be [9]
y -mi- (t 2 )+ JL . r 2=3kT. (15)
dH t t ft
Thi s e quatio n c an't b e integ rate d in time any more as a denominator with t is present - [9]. We found that the three-dime nsional equation ( 14) can b e integrated successfully twice in time if a technique for integration of the one-dimensional equation (10)used in [8] ii implemented. Acco rding
d _r
to this method first we wiWl accept a designation z = —r . Then (14) will be rewritten as
m d B T
--z+—z = 3kT .
2 dt 2
This is ordinary differential equation of the first order - [10]
y'+p(x)y = s(x)
(16)
whichhas the solution
y = exp
Jp(x)dx
y o + j Q(v )exP j"p(u)du
dv
Faom ((6) and (i7) we will eeceive precisely
6kT B
(17)
(18)
Integrating (185) and using (12) and using initial conditions ro = 0 at t=0 from (18) we will easily receive
-2 _ _ kTt_
13 jt|oa
(19)
Equation (19) is the solution of a given prob(em and shows an averaged square of radius-vector of three-dimensional motian rf r Brownian paaticle.The same result can be obtainedfrom the equation (9) if folkowing assumptio n is made [4,8]
-2 -2 -2
x = y = z .
Then the equation (19)fo(lows from the factihat r = x + y + z [8,9].
(20)
ieii scuision
As is known, metellurgkaal manufactura is a ret of complicated intebcoanectad technologies and processer demanding considerable material input(. We hope shat ecqu-EiC consideration c^f fundamental and applied sides during ohe solution of she impoetant scientific and technical problem will lead tee incnrasedprobabiaty of vuccess. Thdrefohe we invite for cooperation all interested foreign partners and colleagues.
Theoretical pnoof thst it i f posslnle to mix liquid meial on the basii of influenae of an erxx"te;:rita.l rotation magnetic field was gsiven. vhe magneric fyii^ld. eotaPes on a circle similarly to how the magnetic field in a -tator of hec(iomagnetic motore and generatyrs roSalee. Labomatory experiments have slrown that magnetic fiekl lan pass thsough a tieirk. Numerical calculation which show asymptotic dacreeet
é/x
s-
U J
Fig. 7. Scheme ofa Brownian particle movement. Atoms C, O
in concentration of carbon Nc and oxygen No atoms in time are carried out. The model is constructed on an assumption that frequency of interaction of these atoms approximately ccrresponds to frequency of casual collisionc under the theory of affective sections. Free atoms of carbon C and oxygen O in liquid metal don'r lote kinetic energy as a result of potential interactions. Therefore tire; given model wocks fpproximately. Procedure for exact integration of the equation of motion for the Brown particle in three-dimensional case is shown.
Let two particles be within the liquif environment and be at initial time t0 at drstance d from each other. It is posrible then to assume that particles are distributed evenly over all liquid volume with step size of d, Fig. 7. Let each particle after some time t to travel to a new position at distance r = d/2. In this case exte rnal bo rders of two volumes will overlap, Fig. 7. D istance r c an b e calculated w ith the equation (19) as distance traveled by Brownianparticle. From there ft is possible to find time t. From time it ir possible to c alculate average frequency (if collisionor impa/t and from thent probabilrties of interaction of Brown particles. Equation (19) can be used for caktulrtion of vra, then model (2-3) will improve.
We stress distinctive and useftil points of our conclusion. 1ft) - the detailed solution of rhe three-dimensional equation (r3, 14t hao udiversal value, unlike when the one-dimensional equation (10) is contidered o nly( In three dimensionalcase it ii not necessary too do lhe assumption (210), and this causes projections 01s a displacement vector to be indepnndenrt and to have any vulue. In pr tnsy to pats from consideration of ii tocontideraiioa of iis components x, y, z. In equation (19) numerical coefficients havr values: S^ , 2d3, 1.0; 2) - In this case if is not nncessary to ieasclr fth ittr to (etect empirically) so me partial solution of the nonuniform equation - [9]
lias; very small size - [8]; 3)) - Based on probability (frequency) of collisions and on chemical interaction of Brown particles it is possible to solve an inverse problem - to find value of viscosity of the liquid.
Conclusions
1. The hypotheris that rotating variable magnetic field can tie; used to intensify mixing of molten metal in a melting bath or convertet is offered and phyficaHly psoved.
2. The numerical solution of kinetic equations for casuai cwllisions with use of free-path length of atoms sliows that concentration of impurities can decsease to desired level in time .nterval ranging
(21)
When usmg aboundary condition ro =0 it is not necessary to apply a condition tCatTa = Bm<10
i )
from about several seconds to units of minutes. This shows necessity for improvement of mixing of molten metal with the use of magnetic field, which can also increase efficiency of the technological process.
3. Procedure for exact integration of the equation of motion for the Brownian particle in three-dimensional case is shown.
References
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[2] Chelishchev E.V., Arsentyev I.P., Jakovlev V.V., Ryzhonkov D.I. General metallurgy. M.: Metallurgy, 1971. 480 p.
[3] Linchevskij B.V., Sobolevskij A.L., Kalmenev A.A. Metallurgy of black metals.-M.: Metallurgy, 1986. 360 p.
[4] Matveev АN Molecular physics. M.: the Higher school, 1981. 400 p.
[5] Molchanov A.P., Zanadvorov P.N. Course of electro-technology and radio-technology. M.: The Science, 1976. 479 p.
[6] Kasatkin A.S., Nemtsov M. V. M. Electrician engineering: The Higher school, 2002. 542 p.
[7] EvsukovA.A. Electrician engineering. M.: Education, 1979. 248 p.
[8] Anselm A.I. Statistical physics and thermodynamics bases. M.: The Science, 1973. 424 p.
[9] SommerfeldA. Thermodynamics and the statistical physics. M.: IL, 1955. 481 p.
[10] Smirnov V.I. Higher mathematics course.- M: The science, 1967. V.2. 655 p.
О возможности перемешивания жидкого металла внешним переменным круговым магнитным полем
Т.А. Жакатаева, К.Ш. Какимова"
а Карагандинский государственный университет
им. Е.А. Букетова
Казахстан 100028, Караганда, ул. Университетская, 28, б Карагандинский государственный технический университет Казахстан 100016, Караганда, Бульвар Мира, 56
Проведено физическое обоснование гипотезы о том, что для интенсификации перемешивания расплава металла в плавильной ванне и конвертере можно применить внешнее, переменное, вращающееся по кругу магнитное поле. Численное решение кинетических уравнений для случайных столкновений с использованием длины свободного пробега и эффективного сечения атома показывает, что концентрация примесей может уменьшиться до требуемого уровня от минут до нескольких секунд. Это доказывает необходимость улучшения перемешивания расплава с использованием магнитного поля для повышения к.п.д. технологического цикла. Принято, что свободные атомы углерода C и кислорода O в жидком металле не теряют кинетическую энергию в результате потенциальных взаимодействий с атомами Fe. Поэтому данная модель работает приблизительно. Показана процедура точного интегрирования для перемещения броуновской частицы в трехмерном случае.
Ключевые слова: расплав, металл, жидкий, перемешивание, магнитное поле, круговое, примеси, взаимодействие, скорость, частица броуновская, смещение.
