les modelling of turbulent flow, heat exchange and

MAGNETOHYDRODYNAMICS Vol. 47 (2011), No. 4, pp. 399–412

LES MODELLING OF TURBULENT FLOW,HEAT EXCHANGE AND PARTICLE TRANSPORT

IN INDUSTRIAL INDUCTION CHANNEL FURNACES

S. Pavlovs 1, A. Jakovics 1, E.Baake 2, B.Nacke 2, M.Kirpo 1,3

1 Laboratory for Mathematical Modelling of Environmentaland Technological Processes, University of Latvia

8 Zellu str., LV-1002 Riga, Latviawww.modlab.lv; [email protected]

2 Institute of Electrotechnology Leibniz, University of HannoverWilhelm-Busch-Str. 4, D-30167 Hannover, Germanywww.etp.uni-hannover.de; [email protected]

3 Bosch Solar Energy AG, Robert-Bosch-Str. 1, 99310 Arnstadt, Germanywww.bosch-solarenergy.com; [email protected]

The paper presents new results of the long-term computations of turbulent flow andlow-frequency oscillations of the temperature field in the industrial induction channelfurnace (ICF) with a widened channel branch and different iron yoke positions. Thecomputations of turbulent heat and mass exchange in the melt are performed using a 3Dtransient Large Eddy Simulation (LES) approach. A 3D electromagnetic (EM) modelwas used for Lorentz force density computations, which act as a source term in theNavier-Stokes equations of the melt flow.

The distributions of alloying additions into the melt and disjointed impurities dueto channel erosion are discussed for a symmetrical ICF and an ICF with one widenedbranch of the channel. Cloud distributions and particle trajectories are obtained using aLagrangian approach along with LES modelled velocity and thermal fields. A long-termanalysis of the particle transport for industrial ICFs has been performed for the firsttime.

Introduction. It is well known from the computer simulations of industrialturbulent flows that steady two equation turbulent numerical models are not ableto predict a proper temperature distribution for flows with pronounced recircu-lation patterns of the mean flow. On the other hand, transient simulation forthe complete industrial process starting from very beginning will not be possiblefor many years. However, a combined approach, where steady simulation acts asinitial conditions for advanced transient simulation of flow development, is oftenused with success in industrial and academic research [1].

The authors hereby present a further development of long-term computations[2] for industrial ICFs, which target large-time scale processes in the flow. Theexpanded set of considered ICF models consists of three different geometry designswith identical electrical induced power in the melt (≈ 215 kW):

(i) A model with two symmetric branches of the channel (Figs. 1, 2a) presentsthe original design of the ICF, which has been previously introduced in [3].

The top part of the ICF is a large cylindrical melt vessel or a bath (Fig. 1),which is taken into account only in hydrodynamic (HD) and thermal modelling.

The bottom part of the ICF consists of a channel, a throat, an inductor and aniron yoke (Fig. 2a), which are taken into account at EM modelling. Originally, theiron yoke is located around the right branch of the channel at the angle α = −45◦.

To retrace, the origin of the Cartesian coordinate system is placed at the

399

S.Pavlovs, A. Jakovics, E. Baake, B.Nacke, M.Kirpo

Fig. 1. Original design of the ICF with a symmetrical channel.

(a)

+α

(b)

+α

(c)

+α

Fig. 2. ICF geometry for EM modelling: (a) ICF with a symmetrical channel; (b),(c)ICF with a left widened branch of the channel; (a),(b) original position (α = −45◦) ofthe iron yoke; (c) moved iron yoke (α = 45◦), whose position differs from the originalone by 90◦ clockwise rotation.

geometrical centre of the ICF channel loop, where the x-axis corresponds to thelong side of the channel, the y-axis to the short one, and the z-axis to the verticaldirection. The central angle α is counted clockwise, as shown in Fig. 2a, startingfrom the intersection of two perpendicular cross-sections y = 0 and x = 0.

(ii) A model with a considerably widened left branch of the channel (Fig. 2b)differs from the original design (Fig. 2a) with gradual expansion of one branchfrom 100% cross-sectional area to 200% of the cross-sectional area at the junctionto the throat [3].

(iii) A new model with a moved iron yoke (Fig. 2c) is a modification of thesecond model with a widened channel branch (ii) and it is introduced by 90◦

clockwise rotation of the iron yoke from the original position to a new one.New results are presented for the following aspects of numerical modelling of

the industrial ICF:

400

LES modelling of turbulent flow, heat exchange and particle transport

– the low-frequency large scale oscillations of instantaneous temperature maxi-mum and its position in the ICF channel are discussed on the basis of long-termcomputations for model (ii);

– the time distribution of the temperature maximum and its position obtainedfor the newly developed model (iii) are compared with model (ii);

– the transport of impurities peeled off from the channel ceramic lining dueto erosion is analyzed for model (ii);

– the distribution of injected alloying additions is considered for model (i).Note that the long-term analysis of particle transport for the industrial ICF

has been performed for the first time.The numerical simulations are performed using the commercial software packa-

ges ANSYS for the EM field and FLUENT for the HD and temperature fields.Initial distributions of the melt velocity and temperature are obtained using thesteady state 3D standard k-ε model. This model is computationally stable andallows to approach the initial conditions relatively rapidly. For the further com-putations, a transient 3D LES model of turbulence is used. As the flow structure,which is obtained by two parameter models, is not precise enough, the LES com-puted results for the start period (first ∼ 10 – 30 sec of the flow time) are notphysically analyzed. The mesh for HD computations consists of approximately3 million elements for a symmetrical ICF and of 6 million elements for an asym-metrical ICF. The time step for transient HD computations is chosen as 0.005 sec.The computation time to obtain 1 sec of physical flow at a PC cluster with 16processor cores is 4–5 hours for a symmetrical ICF and 36–54 hours for an asym-metrical ICF. For post-processing of profile files prepared by FLUENT, our owndeveloped code is used [2].

1. Temperature maximum and its position in the ICF channel.1.1. ICF with a widened channel branch and original position of the yoke –

model (ii).Peculiarities of computational data processing. The results for the maximum

value Tmax of instantaneous temperature in the channel are shown in Fig. 3, whichare plotted using 40 000 points, i.e. every 0.005 sec of the flow time in the 0−200 secrange.

Tm

ax

[K]

1822

1818

1814

1810

1806

1802

1798

α[ ◦]

150

120

90

60

30

0

–30

–60

t [s]0 30 60 90 120 150 180

Tmax

α

Fig. 3. Maximum temperature Tmax and angle α of its position in the ICF with awidened channel (y = 0) and a standard yoke (α = −45◦): flow time t = 0 − 200 sec.

401


Magnitu

de

500 000

50 000

5 000

Period of oscillations [s]50 5

t = 40 − 195 s

t = 40 − 145 s

Fig. 4. FFT analysis for the angle α of the maximum temperature Tmax position shownin Fig. 3 for the flow time: (dotted line) t = 40 − 145 sec; (solid line) t = 40 − 195 sec.

Fast Fourier Transform (FFT) analysis for the fluctuations of the angle α ofthe maximum Tmax value position (Fig. 4) is performed for 155 sec series signalswith 1Hz resolution for a flow period of 40–195 sec. The selected time resolutionis sufficient to determine main periods of the low-frequency oscillations for α.

The results of LES simulation are analyzed after t ∼ 40 sec when the systemhas “forgotten” about its initial conditions, which are the results of the compu-tation according to the steady state 3D standard k-ε model. These results areillustrated in Fig. 3 at t = 0.

The value of temperature maximum. LES simulation shows that the value ofTmax is stabilized (Fig. 3) at about 1806K with ±2K deviations for the flow timet = 40 − 160 sec and then for the flow time t = 160 − 200 sec at ∼ 1802K with±3K deviations.

The maximum values of the time-averaged temperature T avermax for different

averaging periods are rather close: ∼ 1803.5K and ∼ 1804.6K for the flow timet = 65 − 195 sec and t = 65 − 145 sec, accordingly.

The obtained Tmax extremes – the maximum ∼ 1808K at t ≈ 153 sec and theminimum ∼ 1798K at t ≈ 167 sec – may be interpreted as extremes of Tmax singleoscillation with a period ∼ 50 sec for t ∼ 135 − 185 sec.

The position of the temperature maximum. The Tmax position after t ∼ 40 secis mainly in the channel branch with the largest cross-section area – the maximumangle is αmax ∼ 143◦. The very short presence of Tmax position for the flow timest ∼ 85, 135, 140, 182, 190sec is obtained in the right branch of the channel with αranging from 0 to −40◦ (Fig. 3).

However, the averaged values of αaver ∼ 45.9◦ for the flow time t = 40−200secand αaver ∼ 44◦ for t = 0− 200 sec are very close to the start position of αinitial ∼∼ 31◦, which is obtained using the k-ε model.

The periods of the low-frequency oscillations of the Tmax position are ratherclosed for different ranges of the flow time (Fig. 4): ∼ 52 sec and ∼ 53 sec fort = 40 − 195 sec and t = 40 − 145 sec, accordingly.

402


Time delay between extreme of Tmax position α and extreme of Tmax. Theanalysis of both curves in Fig. 3 shows the time delay of α extremes (both maximaand minima) relative to corresponding Tmax extremes. The time delay value isτ ∼10−15−20 sec for the considered flow time after 40 sec.

This phenomenon can be explained by the inertia of the melt reaction to thebuoyancy force increase at higher temperature differences or by the buoyancy forcedecrease at lower temperature differences – the displacement of more heated meltcloser to the outlet of the channel’s left branch or the displacement of less heatedmelt closer to the channel loop’s lower region.

Differences in results for model (ii) and model (i). The results for the ICFwith a widened channel branch (model (ii)) substantially differ from those forthe ICF with a symmetrical channel (model (i)) [2], where long-term oscilla-tions are obtained both for the Tmax value and its position α. For the flow timet = 380 − 700 sec, when the system has “forgotten” about the initial conditionsobtained by the k-ε model and when the Tmax position has been moved from itsinitial state in the left branch of the channel to the right one with the maximumof EM forces, the period of oscillation is 161 sec both for the Tmax value and itsposition α.

The maxima of EM force and Joule heat power for the ICF with a widenedchannel branch and for the ICF with a symmetrical channel branches are placedin the same position, i.e. corresponding to the position of the iron yoke (Fig. 5).

For the ICF with symmetrical channel branches, the only non-symmetricalfactor is the position of EM sources’ maxima. Therefore, the time-averaged (forthe flow time ranging 290–700sec) transit velocity vaver

trans ∼ 2.6 cm/s is directed tothe channel branch with the EM sources’ maxima position.

For the ICF with a widened channel branch, the time-averaged (for the flowtime ranging 45–200 sec) transit velocity vaver

trans ∼ −3.3 cm/s is directed to the re-gion with better conditions for the development of thermogravitational convection(TGC). Therefore, this is the stronger factor, which enforces Tmax to remain inthe widened channel branch if compared with the influence of the position of EMsources’ maxima.

|f|×

10−

6[N

/m

3]

1.8

1.2

0.6

0

Q×

10 −

7[W

/m

3]

2.1

1.8

1.5

1.2

0.9

0.6

0.3

0

α [◦]–90 –60 –30 0 30 60 90

|f |Q

Fig. 5. Module EM force density |f | and Joule heat power density Q in the cross-sectiony = 0 at the inner surface of the channel in an ICF with a widened channel: moved yokeα = 45◦ (solid and dotted lines); original yoke α = −45◦ (strip and dotted strip lines).

403


For the case without TGC, the transit flow has been determined for an experi-mental setup with a horizontally located induction channel unit and symmetricalposition of the magnetic yoke with respect to the channel branches [4, 5]. The mea-surements show that the transit flow is directed from the widened channel branchto the narrow one, because the only non-symmetrical factor is the electrovorticalforces’ difference between regions near the channel inlet and outlet with differentcross-sectional areas.

Conclusion. The comparison of the computational results show that the ex-tremes of the temperature maximum and its position as well as the periods of thelow-frequency oscillations, which are obtained from the FFT analysis, are veryclose for two ranges of flow time: t = 40− 145 sec and t = 40 − 200 sec. This maybe interpreted as a quasi-stable state reached for the investigated flow time rangefrom 40 to 200 sec, hence, in contrast to model (i) [2], oscillations with a very longperiod, exceeding 100 sec, are not expected.

1.2. ICF with a widened channel branch and moved position of the yoke –model (iii).

Peculiarities of the new model (iii). The main idea of the model is a combina-tion of two factors, which influence the value and the position of the temperaturemaximum in the ICF channel:

– widening of the left channel branch of the ICF ensures better conditionsfor TGC development (model (ii)) in contrast to the ICF design with symmetricalchannel branches (model (i)) and, hence, the possibility for a more intensive heatexchange in this branch of the channel;

– the new position of the moved iron yoke of the inductor (Fig. 2c) ensuresthat maxima of EM force and Joule heat density (Fig. 5) are obtained in thewidened branch of the channel with better conditions for TGC development.

Therefore, both non-symmetrical factors are combined now in one channelbranch.

Choice of initial conditions for the new model (iii). The distributions of HDand thermal fields in the ICF with a widened channel branch and original yokeposition for the flow time t = 63 sec (Fig. 6) have been chosen as the initialconditions.

The reasons for this choice are the following: the flow time point t = 63 seccorresponds to the local maximal value of Tmax and belongs to the range of flowtime from 45 sec to 75 sec (Fig. 3), which is quasi-stable for the position of Tmax

in the channel.The results for models (iii) and (ii) are compared in Fig. 6, which are plotted

using 7400 points, i.e. every 0.005 sec of the flow time in the 63–100 sec range.The estimated time, when the new system (model (iii)) has “forgotten” about

its initial conditions used from the old (or basic) system (model (ii)), is t ∼ 75 sec.Main trends for Tmax and its position α with the new model (iii). The obtained

results for the flow time ranging 75 to 100 sec (that is 25 sec) show the followingtrends (Fig. 6):

– the time-averaged temperature maximum in the channel T avermax ∼ 4− 5K in

the new model (Fig. 6b) is less than one in the old (or basic) model (Fig. 6a);– the time-averaged position αaver of the temperature maximum in the channel

in the new model is shifted to the outlet of the widened channel by ∼ 25◦ ifcompared to the old (basic) model.

To analyze the above trends of the transit velocity, it is necessary to con-sider a longer time-averaging period. Hitherto time-averaged transit velocity isvavertrans ∼ −1.4 cm/s for the flow time ranging 75–100 sec.

404


(a)

Tm

ax

[K]

1808

1806

1804

1802

1800

1798

α[ ◦]

150

120

90

60

30

0

–30

–60

t [s]60 65 70 75 80 85 90 95 100

Tmaxα

(b)

Tm

ax

[K]

1808

1806

1804

1802

1800

1798

α[ ◦]

150

120

90

60

30

0

–30

–60

t [s]60 65 70 75 80 85 90 95 100

Tmaxα

changeof yokeposition

Fig. 6. Maximum temperature Tmax and angle α of its position in the ICF with awidened channel (y = 0) for the flow time t = 64 − 100 sec: (a) original yoke α = −45◦;(b) moved yoke α = 45◦.

Conclusion. If the described trends are confirmed as long-term trends, thenew ICF construction with the moved yoke will provide a smaller overheatingtemperature in the channel in contrast to the old (basic) model with the originalyoke position. The combination of both non-symmetrical factors (channel branchwidening as well as iron yoke position changing) in one channel branch makes itpossible to additionally stabilize the thermal situation and most probably decreasethe amplitude of long-term pulsations.

2. Distributions of particle clouds and separated particle tracks.The LES study of particle transport for the industrial ICF has been performedusing the authors’ experience referred to the cylindrical induction furnaces [6, 7].Distributions of particle clouds and trajectories of separated particles are obtainedalong with LES modelled distributions of turbulent velocity and thermal fieldsusing the Lagrangian approach. The drag, buoyancy and EM forces acting on theparticles are taken into account. The approach has been tested for the ICF [6] andapplied further for a long-term computation of particle transport in the industrialICF [8].

Particle properties. Particle transport simulations have been performed for aICF with symmetrical branches of the channel (model (i)) and an ICF with a wi-

405


dened left branch of the channel and original position of the iron yoke (model (ii)).The complete list of parameters of injected clouds and separated particles is re-presented in Table 1.

The main particle features are:– particles are inert, i.e. they are chemically non-reactive with the melt;– particles are rigid spheres, their diameters are dp = 0.1mm;– for all considered cases, the particle density is ρp ≈ (3/4)ρmelt except for

the cloud in model (i) – ρp ≈ (1/4)ρmelt.The EM force acting on each particle is defined according to [9] as

FEMp =

32

σmelt − σp

σmelt + σpfEMVp, (1)

where Vp is the volume of a particle, fEM is the EM force density at a point withparticle coordinates, which is defined through application of the ICF model forEM computations.

Table 1. Physical properties of particle clouds and separated particles.

ICF channel geometry

symmetrical – widened left branch –Parameter model (i) model (ii)

Particle Separated Particle Separatedclouds particles clouds particles

Obtained flow time t [s] 10–555 95–200Particle electricalconductivity σp [(Ωm)−1 ] 0 0Particle diameter dp [mm] 0.1 0.1

Characteristic length l0 [m]Smaller radius of thechannel cross-section 0.0599 0.0599Radius of the bath 0.88 0.99

Characteristic (maximum) instantaneous velocity for flow time pointFlow time point t [s] 555 200Particle in the channel vp [m/s] 1.08 2.04Particle in the bath vp [m/s] 0.055 0.085Melt in the channel vmelt [m/s] 1.48 2.43Melt in the bath vmelt [m/s] 0.39 0.53

Particle Reynolds number Rep = ρdp|vp − vmelt|/μIn the channel 48.7 47.4In the bath 40.8 54.1

Coordinates of particleinjection center (cloud)or point (x, y, z) [m] (0,0.7,1.6) (0,0.85,1.6) (0,0,–0.285) (0,0,–0.285)Particle density ρp [103kg/m3] 5.25 5.25 1.55 5.25Particle and melt densitiesrelationship ρp/ρmelt [%] 77.1 77.1 22.8 77.1Number of particles N 16000 6 25000 9

Stokes number St = ρpd2pvmelt/(18μl0)

In the channel 8.6·10−4 8.6·10−4 6.2·10−3 2.1·10−2

In the bath 2.3·10−4 2.3·10−4 8.1·10−5 2.4·10−4

406


As all particles are nonconductive and their electrical conductivity is σp = 0,the EM force is oppositely directed and has the maximal value

FEMp = −3

4fEMVp. (2)

Note that the EM skin-layer thickness in the melt corresponds (in order ofmagnitude) to the characteristic radius of the channel transversal cross-section,but the EM force acting on the particle is proportional to the third power of itsdiameter. The estimations made in [10] show that for a small particle the EMforce is of the secondary significance if compared with the drag and buoyancyforces acting on the particle.

The particle Reynolds number, which is estimated separately for the channeland the bath, is Rep ∼ 40 − 50 (Table 1) for the ICF with a different channelgeometry regardless of different levels of maximum values of the melt vmelt andparticles’ vp velocities in the channel and in the bath. Since Rep � 1, this meansthat the drag force is an important factor for the particles’ motion, but particlescannot be interpreted as fluid tracers.

Estimations of the Stokes number, which is defined as the ratio of the particlefluid response time constant to an appropriate turbulence time scale (dissipationtime), are also presented in Table 1: St ∼ 10−2 − 10−4 for different regions of themelt. The Stokes number can be used to estimate the particle movement characterindependently on other factors: if St � 1, the particles’ response to fluid velocitychanges is very fast; in an extreme case, when St → 0, all particles can be treatedas fluid tracers.

2.1. Distribution of disjointed impurities due to erosion of the channel ce-ramic lining. Computations were performed for model (ii) for the flow timet = 95 − 200 sec (Fig. 7) after injecting a cloud of 25 000 particles and separatedparticles (totally 9) at the channel lower point with the coordinates (0; 0;−0.285),as shown in Fig. 8a,c,e,f . The particle injection in t = 95 sec models the impuritydisjoining due to erosion of the channel ceramic lining.

100%

80%

60%

40%

20%

0

t [s]95 115 135 155 175 195

2.39

2.11

1.83

1.55

1.27

0.99

0.71

0.43

0.15

–0.13

–0.41–1 –0.5 0 0.5 1

Layer

10

9

8

7

6

5

4

3

2

1

cloudinjectionpoint att=95 s

Fig. 7. ICF with a widened channel and original yoke position; the flow time 95–200sec: relative numbers of 25 000 particles’ (ρp ≈ (3/4)ρmelt) cloud distributed into 10vertical layers of equal thickness.

407


(a)

2.39

2.11

1.83

1.55

1.27

0.99

0.71

0.43

0.15

–0.13

–0.41–1 –0.5 0 0.5 1

0.05%

0.00%

0.00%

0.00%

0.00%

0.00%

0.38%

4.09%

19.82%

75.66%

vp [m/s]


(c)

2.39

2.11

1.83

1.55

1.27

0.99

0.71

0.43

0.15

–0.13

–0.41–1 –0.5 0 0.5 1

17.92%

14.88%

8.57%

10.38%

12.96%

14.23%

5.39%

3.18%

4.58%

7.91%


(b)

1

0.6

0.2

–0.2

–0.6

–1–1 –0.6 –0.2 0.2 0.6 1

vp [m/s]

(d)

1

0.6

0.2

–0.2

–0.6

–1–1 –0.6 –0.2 0.2 0.6 1

(e)

2.39

2.11

1.83

1.55

1.27

0.99

0.71

0.43

0.15

–0.13

–0.41–1 –0.5 0 0.5 1

6.92%

12.38%

15.50%

20.32%

18.52%

18.59%

5.93%

0.90%

0.42%

0.51%


(f)

t=200 s

t=95 s

Fig. 8. ICF with a widened channel and original yoke position: cloud particle velocityvectors (a, b); distribution of 25,000 particles’ (ρp ≈ (3/4)ρmelt) cloud for the flow timet = 100 sec (a, b), t = 145 sec (c, d) and t = 200 sec (e) in side (a, c, e) and top (b, d)views; track coloured by the time of single particle (ρp ≈ (3/4)ρmelt) in side view (f) forthe flow time t = 95 − 200 sec.

408


Note that the formulated conditions are only assumptions for estimation ofthe melt contamination time. The real sources of impurities are not discussedhere.

The dynamics of particle migration from the channel to the bath may beobserved from Fig. 7, which illustrates the time distribution of a relative integralnumber [%] of particles for each of the ten horizontal layers with equal thicknesses.

Only about 5 sec (flow time t = 100 sec) are necessary for the first cloud ofparticles disjoined at the lower point of the channel to leave the channel and reachthe throat (Fig. 8a,b).

After ∼ 25 sec (flow time t = 120 sec) about 40% of the particles may be foundin the bath (Fig. 7). Travelling of about 80% of particles to the bath (Fig. 8c,d)took about 50 sec (flow time t = 145 sec), but after 100 sec (flow time t = 200 sec)more than 90% of the particles are in the bath (Fig. 8e).

Velocities of particle clouds are very high only in the channel vp ∼ 3m/s(Fig. 8a,b), which correspond to melt velocities vmelt in the considered cross-sections of the channel [2], that is why despite of the very small melt transitvelocity vtransit

melt < 5 cm/s [2] the most “dexterous” particles may leave the channelfor a very short time, i.e. in a few seconds after peeling off. The example of aseparated particle track is shown in Fig. 8f . The level of particle velocities in thebath is lower by an order of magnitude.

Conclusions. With the chosen model for qualitative estimations, the followingresults are obtained:

– the peeled off impurities travel very rapidly from the channel to the bath;– the contamination time of the melt in the bath (more than 80% of particles

travelled from the channel to the bath) is less than 1 min.2.2. Homogenization of alloying additions after their injection into the melt.

Computations were performed for model (i), the flow time t = 10−555 sec (Fig. 9).After t = 555 sec, the computation process for particle tracing was interrupted dueto the operative memory shortage for particles data storage in FLUENT.

Note that the formulated conditions are only assumptions for estimation of

60%

50%

40%

30%

20%

10%

0%

t [s]10 70 130 190 250 310 370 430 490 550

1.681.47

1.26

1.05

0.84

0.64

0.43

0.22

0.01

–0.20–0.41

–0.88 –0.44 0 0.44 0.88

Layer

10

9

8

7

6

5

4

3

2

1

cloud injected at t=10 s

Fig. 9. ICF with a symmetrical channel and original yoke position; the flow time10–555 sec: relative numbers of 25 000 (ρp ≈ (1/4)ρmelt) particles’ cloud distributed into10 vertical layers of equal thickness.

409


the melt homogenization time. The real sources of impurities are not discussedhere.

A cloud of 16 000 particles was injected at the time point t = 10 sec into theregion with its centre coordinates (0; 0.7; 1.6), as shown in Fig. 10a,b,c. Separatedparticles (totally 6) were injected at the time point t = 10 sec at a point withcoordinates (0; 0.85; 1.6), as shown in Fig. 11.

Most of the particles injected into the bath, which are lighter than the melt,remain in the bath. The estimated time to reach an approximately uniform alloyingaddition distribution in the bath is ∼ 115 sec after particle injection (Fig. 10a,b,c).The intermediate state for the flow time t = 65 sec is shown in Fig. 10d,e. Thedynamics of particle long-term homogenization in the bath is illustrated in Fig. 9,

(a)

1.68

1.47

1.26

1.05

0.84

0.64

0.43

0.22

0.01

–0.20

–0.41–0.88 –0.44 0 0.44 0.88

16.17%

20.59%

19.65%

20.05%

17.55%

2.48%

1.25%

0.39%

0.36%

0.60%


(b)

1.68

1.47

1.26

1.05

0.84

0.64

0.43

0.22

0.01

–0.20

–0.41–0.88 –0.44 0 0.44 0.88

16.17%

20.59%

19.65%

20.05%

17.55%

2.48%

1.25%

0.39%

0.36%

0.60%


(c)

1.68

1.47

1.26

1.05

0.84

0.64

0.43

0.22

0.01

–0.20

–0.410 1/4π 2/4π 3/4π 4/4π 5/4π 6/4π 7/4π 8/4π

16.17%

20.59%

19.65%

20.05%

17.55%

2.48%

1.25%

0.39%

0.36%

0.60%


(d)

1.68

1.47

1.26

1.05

0.84

0.64

0.43

0.22

0.01

–0.20

–0.41–0.88 –0.44 0 0.44 0.88

13.44%

25.99%

22.57%

16.95%

20.45%

0.43%

0.15%

0.01%

0.01%

0.01%

vp [m/s]

(e)

0.88

0.70

0.53

0.35

0.18

0

–0.18

–0.35

–0.53

–0.70

–0.88–0.70 –0.35 0 0.35 0.70

vp [m/s]

Fig. 10. ICF with a symmetrical channel and original yoke position: cloud particlevelocity vectors (d, e); 25 000 particles’ (ρp ≈ (1/4)ρmelt) cloud – initial distribution atthe flow time t = 10 sec is highlighted (a, b, c); distribution at t = 125 sec in the first(a) and and second (b) side views and angular distribution (c); distribution at t = 65 secin side (d) and top (e) views.

410


(a)

t=10 s

t=555 s

(b)

t=10 st=555 s

Fig. 11. ICF with a symmetrical channel and original yoke position: tracks coloured bythe time (a, b) of two selected particles (ρp ≈ (3/4)ρmelt) for the flow time t=10− 555 s.

where the time distribution is plotted for a relative integral number [%] of particlesfor each of the ten horizontal layers with equal thicknesses.

As the velocities of the particles in the bath are by an order of magnitudesmaller (Fig. 10d,e) if compared with those in the channel (Table 1), only themost “dexterous” particles reach the lower point of the channel (Fig. 11a) or evenreturn to the bath after travelling along the channel from the left outlet to theright (Fig. 11b).

Note that the previous model for disjoined impurities in the channel with thewidened branch may be used to estimate the homogenization process after thecloud particles have mostly left the channel (Fig. 7). In spite of the fact that theparticles of this cloud in model (ii) are extremely lighter ρp ≈ (1/4)ρmelt (Table 1)if compared with the particles in model (i) ρp ≈ (3/4)ρmelt, the intermediate state(flow time t = 65 sec) for model (i) (Fig. 10d,e) is similar to the intermediate state(flow time t = 145 sec) for model (ii) (Fig. 8c,d). This fact in the cases underconsideration evidences that the buoyancy force is less significant than the dragforce.

Conclusion. With the chosen model for qualitative estimations, the followingresults are obtained:

– the estimated homogenization time of alloying additions in the melt bath isabout 2 ÷ 3 min after their injection;

– the buoyancy force is less significant for particle distribution in the consi-dered cases.

3. Final remarks. The performed long-term computations have shown theeffectiveness of the LES approach for predictions of particle transport, transientcharacteristics of turbulent heat and mass exchange in the industrial ICF.

The validity of the LES approach application for HD and thermal fields mo-delling in the ICF has been successfully verified by comparing the experimentaland computational results. However, there is a difficulty concerning the complexexperimental verification of the computational models of particle transport in high-temperature non-transparent electrically conducting melts. This problem awaitsits solution in the nearest future. As to the conclusions for the special casesconsidered, they are formulated above in the corresponding chapters.

411


4. Acknowledgements. The current research was performed under thefinancial support of the ESF project of the University of Latvia, contract No.2009/0223/1DP/1.1.1.2.0/09/APIA/VIAA/008.

REFERENCES

[1] A.Umbrashko, E. Baake, B.Nacke, A. Jakovics. Experimental inves-tigations and numerical modelling of the melting process in cold crucible.COMPEL: The International Journal for Computation and Mathematics inElectrical and Electronic Engineering, vol. 24 (2005), no. 1, pp. 314–323.

[2] E.Baake, A. Jakovics, S. Pavlovs, M.Kirpo. Long-term computationsof turbulent flow and temperature field in the induction channel furnace withvarious channel design. Magnetohydrodynamics, vol. 46 (2010), no. 4, pp. 317–330.

[3] E.Baake, A. Jakovics, S. Pavlovs, M.Kirpo. Influence of channel designon heat and mass exchange of induction channel furnace. COMPEL: TheInternational Journal for Computation and Mathematics in Electrical andElectronic Engineering , vol. 30 (2011), no. 5, pp. 1637–1650.

[4] I.E.Bucenieks, M.Y.Levina, M.Y. Stolov, E.V.Shcherbinin. Phy-sical grounds of MHD and heat phenomena in induction channel furnaces(Preprint LAFI-021, Salaspils, Latvia: Institute of Physics, Latvian Academyof Sciences, 1980), 47 p. (in Russ.).

[5] A.V.Arefyev, I.E.Bucenieks, M.Y.Levina, M.Y. Stolov, V.I. Sha-ramkin, E.V. Shcherbinin. Intensification of heat and mass exchange ininduction channel furnaces (Preprint LAFI-023, Salaspils, Latvia: Instituteof Physics, Latvian Academy of Sciences, 1981), 49 p. (in Russ.).

[6] M.Kirpo, A. Jakovics, E.Baake, B.Nacke. LES study of particle trans-port in turbulent recirculated liquid metal flows. Magnetohydrodynamics,vol. 42 (2006), no. 2, pp. 199–208.

[7] M.Kirpo, A. Jakovics, E.Baake, B.Nacke. Particle transport in recir-culated liquid metal flows. COMPEL: The International Journal for Com-putation and Mathematics in Electrical and Electronic Engineering , vol. 27(2008), no. 2, pp. 377–386.

[8] A. Jakovics, S. Pavlovs, M.Kirpo, E. Baake. Long-term LES study ofturbulent heat and mass exchange in induction channel furnaces with variouschannel design. Proc. of International PAMIR Conference on Fundamentaland Applied MHD (Borgo, Corsica, France, September 5–9, 2011), vol. 1,pp. 283–288.

[9] D.Leenov, A.Kolin. Theory of electromagnetophoresis. I. Magnetohydro-dynamic forces experienced by spherical and symmetrical oriented cylindricalparticles. The Journal of Chemical Physics, vol. 22 (1954), pp. 683–688.

[10] M.Shchepanskis, A. Jakovics, E.Baake. The statistical analysis of theinfluence of forces on particles in EM driven recirculated turbulent flows.Journal of Physics: Conference Series, vol. 333 (2011), p. 012015.

Received 22.12.2011

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