Performance of Inductors Attached to a Galvanizing Bath

XINPING ZHOU, SHUO YUAN, CHI LIU, PENG YANG, CHAOQUN QIAN,and BAO SONG

By taking a galvanizing bath with inductors from an Iron and Steel Co., Ltd as an example, thedistributions of Lorentz force and generated heat in the inductor are simulated. As a result, thezinc flow and the temperature distribution driven by the Lorentz force and the generated heat inthe inductor of a galvanizing bath are simulated numerically, and their characteristics areanalyzed. The relationship of the surface-weighted average velocity at the outlet and the tem-perature difference between the inlet and the outlet and the effective power for the inductor isstudied. Results show that with an increase in effective power for the inductor, the surface-weighted average velocity at the outlet and the temperature difference between the inlet and theoutlet increase gradually. We envisage this work to lay a foundation for the study of theperformance of the galvanizing bath in future.

DOI: 10.1007/s11663-013-9938-1� The Minerals, Metals & Materials Society and ASM International 2013

I. INTRODUCTION

THE hot dip galvanizing process is a complexmetallurgical process. In a zinc alloy bath, a steel stripis rapidly immersed and continuously coated by zincwhose temperature is normally between 723 K and753 K (450 �C and 480 �C) (see Figure 1).[1] The oper-ation parameters of various galvanizing lines can varyconsiderably, including product specification, stripspeed, strip temperature, bath configuration, bathchemistry, inductors, wiping systems, as well as chem-istry, placement and dissolution pattern of ingots of zincalloy.[1–3] For example, some operation parameters ofthe galvanizing line studied in this article froman Iron andSteel Co., Ltd are different from those found in literature.For a given hot dip galvanizing line, the flow field, thetemperature field, and the dissolved aluminum (Al) andferrum (Fe) concentration fields (Three Fields) in the bathcan influence the dross formation and the coating qualityof the steel strip surface. The Three Fields are mainlydetermined by the strip speed, strip temperature, stripdissolution, and heat dissipation through walls, induc-tors, and ingots of zinc alloy. Strip speed, strip temper-ature, and heat dissipation through walls can be modeledby setting appropriate boundary conditions.

The simulation of ingots of zinc alloy has beenperformed by assuming an immersion pattern and adissolution pattern. Ajersch et al.[4] assumed the ingots

are added to the bath at ambient temperature; at thesixth minute, the ingots are heated to the melting pointand the ingots start to melt with the effective ingot massflux acted as a parabolic function of time, and at thetwentieth minute, all the zinc and the aluminum in theingots are dissolved in the bath. Ilinca et al.[1] used theexpression of the mean Al mass rate in the ingot meltingvolume as a negative linear function of the Al concen-tration in the bath. Al consumption on the strip surfaceis assumed to take place on the first 0.305 m of the striplength from its entry on the bath (corresponding to 0.2seconds at a trip velocity of 1.524 m/s). The overall Alconcentration of the coating is considered to be 0.4 pctby weight for a coating weight of 0.2 kg/m2 per side.Simulation of the inductors is one of the most

important and complex parts of the bath; the inductorsare referred as to magnetic field, electric field, and ThreeFields. The constant inductor flow rate and temperatureincrease were set in most simulations,[4–8] which weretaken from the data of the previous study.[9] In theirsimulation, the inlet and outlet ports of the channelinductors were modeled and linked with the side walls bythe cowl links. The zinc enters the bath from theinductors at 0.75 m/s and is heated to increase itstemperature by 20 K (20 �C) above the temperature ofthe zinc entering the inductors from bath for 100 pct ofthe inductor power used, and at 0.4 m/s and a temper-ature increase by 8 K (8 �C) for only 20 pct of theinductor power used.[4–8] In Kim et al.,[10] a numericalsimulation of fluid flow and heat transfer in the moltenzinc bath of a continuous hot dip galvanizing line wasconducted; the inlet and outlet ports of the channelinductors were modeled by splitting flow at the side wallsof the bath. This is different from the actual flow field. Auniform velocity was assumed at the inlet and outletports of the channel inductors, and the inlet temperaturewas adjusted considering the heat balance of the bath,i.e., mainly according to the heat inputs of a steel stripwith a certain line speed and a certain width and the heat

XINPING ZHOU, Associate Professor, is with the Department ofMechanics, Huazhong University of Science and Technology (HUST),Wuhan 430074, P.R. China. Contact e-mail: xpzhou08@hust.edu.cnSHUO YUAN, CHI LIU, and CHAOQUN QIAN, PostgraduateStudents, are with the Department of Mechanics, Huazhong Uni-versity of Science and Technology (HUST). PENG YANG, Engineer,is with the Wuhan Iron and Steel Co., Ltd, Wuhan 430074, P.R.China. BAO SONG, Associate Professor, is with the School ofMechanical Science and Engineering, Huazhong University of Scienceand Technology (HUST). Contact e-mail: songbao@hust.edu.cn

Manuscript submitted April 12, 2013.Article published online September 27, 2013.

1580—VOLUME 44B, DECEMBER 2013 METALLURGICAL AND MATERIALS TRANSACTIONS B

consumption of ingot.[10] The zinc flow in a continuousgalvanizing bath was numerically simulated by Dashet al.[12] with a model including mass and momentumalong with the k� e turbulence model by assuming thezinc in the bath to be isothermal. This simulation usesflow barriers to eliminate vortex formation in the flowfield, which does not take the inductors into consider-ation. Park et al.[11] numerically simulated the zinc flowinside the induction channel as well as the flow inside thezinc bath with a commercial computational fluid dynam-ics (CFD) software called FLUENT. Lorentz forceadded to the momentum equations and heat added to theenergy equation have been obtained by modeling theinductor channel using magneto-hydrodynamics. Flowvectors have been demonstrated to direct from the coreto the opposite as a result of Lorentz force. However, theoutlet velocity and temperature difference between theinlet and the outlet related to a given power have notbeen analyzed, which are very important to the simula-tion of the Three Fields in a bath. Except for the work ofPark et al.,[11] few other studies on the simulation ofinductors in the bath have been reported.

In this article, the zinc flow and temperature distribu-tion in the inductors of a galvanizing bath are simulatednumerically using commercial software, which are reli-able tools to perform simulations.[10–12] The Lorentzforce and generated heat are analyzed, and then thevelocity vector of the flow driven by the Lorentz forceand the buoyancy force, and the temperature rise arediscussed. Finally the surface-weighted average velocitiesat the outlet and temperature differences between theinlet and the outlet related to given powers are analyzed.

II. MODELING

The electromagnetic field in the inductor is calculatedwith the Maxwell’s equations[13]:

r�H ¼ Jþ @D@t¼ JS þ Je þ JV þ

@D

@t½1�

r � E ¼ � @B@t

½2�

r � B ¼ 0 ½3�

r �D ¼ q ½4�

where r� is the rotation operator, r� is the divergenceoperator, H is the magnetic intensity vector, J is thetotal current density vector, Js is the current densityvector for external excitation source, Je is the inducededdy current density vector, JV is the velocity currentdensity vector, D is the electric displacement vector, t istime, E is the electric field intensity vector, B is themagnetic induction density vector, and q is the volumecharge density.In the low-frequency field, by neglecting the displace-

ment current, the magnetic vector potential method isused to solve the equations with a calculating frequency

Fig. 2—Flowchart for simulation with ANSYS, FLUENT, andUDFs.

Fig. 3—Computational domain of (a) zinc bath and (b) channelinductor.

Fig. 1—Schema of a galvanizing bath operation.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 44B, DECEMBER 2013—1581

of 60 Hz, and magnetic force is calculated with theMaxwell method.

In time harmonic analyses, the time-average Maxwellforce is calculated by

Fmxav ¼

1

2l0

Z

s

Re n^ �B�� �

BD E

� 1

2B � B�ð Þ n^

� �ds ½5�

where B� is the complex conjugate of B. In harmonic anal-yses, Joule heat is a time average in a period, which is cal-culated with electric field intensity and current density by

Q ¼ 1

2Re JTE�� �

½6�

where Re is the real part of a complex number and E� isthe complex conjugate of E.

The zinc flow was modeled as an incompressible fluidand is described by Navier–Stokes equations. One halfof the bath was modeled because the zinc bath issymmetrical with respect to the mid plane. The unsteadyincompressible flow and heat transfer can be describedusing the three-dimensional (3D) mathematical modelgoverning equations including continuity, momentum,energy, and k–e turbulence equations[14]:

Continuity equation:

r � u ¼ 0 ½7�

Momentum equations:

q@u

@tþ u � ru

� ¼ �rpþr � ½2ðlþ lTÞ _cðuÞ�

� qgbðT� T0Þ þ Smom

½8�

Energy equation:

qcp@T

@tþ u � rT

� ¼ r � ½ðkþ kTÞrT� þ Senergy ½9�

Equation for the turbulent kinetic energy K:

q@j@tþ u � rj

� ¼ r � lþ lT

rj

� rj

� �þ Gk þ Gb � qe

½10�

Equation for the energy dissipation e:

q@e@tþ u �re

� ¼r� lþlT

re

� re

� �þCe1

ejðGkþCe3GbÞ

�Ce2qe2

j½11�

In the above equations, t, u, q, p, T, l, cp, and k denotetime, velocity, density, pressure, temperature, viscosity,constant-pressure specific heat, and conductivity, respec-tively, and g, b, and T0 denote the gravity vector, thethermal expansion coefficient, and the reference temper-ature, respectively. Gk and Gb represent the generation ofturbulence kinetic energy due to the mean velocitygradients and the generation of turbulence kinetic energydue to buoyancy, respectively. The turbulent (or eddy)viscosity, lT, is calculated by lT ¼ qCl

j2

e . The constantsin the above equations are given by: Ce1 = 1.44,Ce2 = 1.92, Cl = 0.09, rj = 1.0, and re = 1.3. Ce3 iscalculated by Ce3 ¼ tanh v

u

� �where v and u are the

components of the flow velocity respectively parallel andperpendicular to the gravity vector.The buoyancy force comes into effect by the term

qgbðT� T0Þ. The source term in the momentum equa-tions is the Maxwell electromagnetic force, which isobtained from ANSYS (ANSYS Inc., Canonsburg, PA).The source term in the energy equation is the Joule heat,which is obtained from ANSYS. The source terms areimplemented in FLUENT (Fluent Inc., Lebanon, NH)through user-defined functions (UDFs).[14] Figure 2shows the flowchart to illustrate how the ANSYS,FLUENT, and UDFs are integrated together to per-form the simulation.In our model, the numerical scheme for calculation

used is the finite volume method. The solutions ofequations carried out in this work are based on thealgorithm Semi-Implicit Method for Pressure LinkedEquations (SIMPLE) proposed by Patankar.[15] Pres-sure is discretized with a second-order scheme, whereasthe density, momentum, and energy equations arediscretized with a second-order upwind scheme.The iterative process is stopped when the convergence

criterion defined as

fqþ1 � fq

fqþ1

� e ½12�

is satisfied. In Eq. [12], f stands for continuity, velocitycomponents, and energy; q is the iteration number; and eis the convergence criteria, which is 10�7 for the energyequation and 1 9 10�3 for the momentum equations andthe continuity equation.

Table I. Thermophysical Properties Used in this Calculation

Thermal conductivity (W/mÆK) zinc 60Thermal conductivity (W/mÆK) steel 40Resistivity (XÆm) zinc 5.45 9 10�8

Heat capacity (J/kgÆK) zinc 512Density (kg/m3) zinc 6600

Fig. 4—Grid independence test for velocities at three representativepoints 1, 2, and 3.

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To reduce the effect of the flow and temperature fieldsin the zinc bath on the performance of the zinc in thechannel inductors, proper boundary conditions aregiven for the model. All walls for the zinc bath wereset as constant temperature and no-moving boundarywhere no slip condition is given, and the zinc ingot’seffect is not considered.

III. RESULTS AND DISCUSSION

Taking a galvanizing bath with inductors from anIron and Steel Co., Ltd as an example, we study theperformance of inductors attached to a galvanizingbath. The inductors in a galvanizing bath are modeled asshown in Figure 3. The bath is 4.876 m long, 3.976 mwide, and 2.192 m deep. The channel inductors areinstalled at the two sides of the bath, with an inclinedangle of 60� to the vertical plane. The distance frompoints A or D to the right bath wall is 1.526 m, thatfrom points B or C to the right bath wall is 0.954 m, andthe distances from points A, B, C, and D to the bottom

bath wall are 1.156 m, 1.082 m, 0.712 m, and 0.650 m,respectively. The cross section of the bath containing thelongitudinal symmetry plane of the inductor center ishighlighted in red. The selected inlet and two same-sizeoutlets, respectively, denoted by I and O are shown on ahorizontal line. The area of the O outlet section reaches57.08 cm2, and the area of the I inlet section is88.80 cm2. The inlet and two outlets are 47.85 cm awayfrom the bottom centerline of the inductors; that is,l = 47.85 cm.

Fig. 5—Distribution of (a) electromagnetic force vector (N) and (b)Joule heat (W/m3).

Fig. 6—Distribution of (a) velocity vector (m/s) and (b) temperaturefield (�C).

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 44B, DECEMBER 2013—1583

The thermophysical properties of materials for thestrip, the rotating rolls, the walls, and the inductors,used in the simulation, are presented in Table I.

The model geometry is meshed with unstructuredgrids. To choose a reasonable number of cells, a gridindependence test is done. By monitoring the zincvelocities at three representative points 1, 2, and 3, theresults of a grid independence test are presented inFigure 4.

It is shown in Figure 4 that the magnitudes of zincvelocities at three points slightly decrease with anincrease in the number of inductor grid elements from80,560 to 156,024. It was therefore reasonable to select avalue of 80,560 for numerical simulations in our work.

For an electric current density of 3,000,000 A/m2 anda frequency of 60 Hz, the electromagnetic force and theJoule heat generated in the inductor are simulated, andtheir distributions on the cross-sectional plane of theinductor are shown in Figure 5.

As shown in Figure 5(a), the electromagnetic forcedirected from the iron core center to outside is large atthe corners. The induced eddy current is produced onlyin the vicinity of the iron core where the zinc is driven bythe electromagnetic force. The electromagnetic forcevectors are represented with the force vectors in eachelement (N), the magnitudes of which are shown to befar smaller than those represented with the force vectorsin a unit volume (N/m3) in Reference 11. As shown inFigure 5(b), the Joule heat produced is large in thevicinity of the iron core and is negligible far away fromthe iron core. For an electric current density of3,000,000 A/m2, by adding the electromagnetic forceand Joule heat calculated with ANSYS into FLUENTas source terms in the momentum equations and energyequation, the velocity vector and temperature fieldobtained on a cross-sectional plane of the inductor areshown in Figure 6.

As shown in Figure 6(a), the zinc enters the inductorinlet in the middle, and exits from the outlets at the twosides. This forms a circulation zone in the zinc bath. Asexpected, the velocity direction is parallel to the direc-tion of the electromagnetic force. The surface-weightedaverage velocity at the outlet reaches 0.18 m/s. Asshown in Figure 6(b), the temperature difference be-tween the inlet and the outlet is 5.58 K (5.58 �C). The

distributions of velocity vector and temperature field asshown in Figure 6 shows similar characteristics to thosereported in Reference 11.For different electric current densities (corresponding

to different effective powers) inducing different electro-magnetic force and Joule heat, the temperature differ-

Table II. Temperature Difference Between the Inlet and the Outlet and Surface-Weighted Average Velocity at the Outlet for the

Inductor

Electric Current Density (A/m2) Effective Power (kW) Outlet Velocity (m/s) Temperature Difference (K)

80,000 0.0179 0.004 0.20160,000 0.0714 0.009 0.24240,000 0.161 0.014 0.42320,000 0.286 0.018 0.70800,000 1.77 0.05 1.181,600,000 7.07 0.1 2.132,500,000 17.3 0.14 5.003,000,000 24.9 0.18 5.584,000,000 44.2 0.22 8.006,000,000 99.5 0.38 10.008,000,000 177 0.5 10.33

Fig. 7—(a) Surface-weighted average velocity at the outlet and (b)temperature difference between the inlet and the outlet vs electriccurrent density for the inductor.

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ence between the inlet and the outlet and the surface-weighted average velocity at the outlet are presented inTable II.

As shown in Figure 7, with an increase in electriccurrent density, the surface-weighted average velocity atthe outlet and the temperature difference between theinlet and the outlet increase gradually. They, respec-tively, reach 0.22 m/s and 8 K (8 �C) for an electriccurrent density of 4,000,000 A/m2 (corresponding to aneffective power of 44.2 kW), and reach 0.38 m/s and10 K (10 �C) for an electric current density of6,000,000 A/m2 (corresponding to an effective powerof 99.5 kW). References 4–8 reported that the liquid zincfrom the inductors entered the bath at a velocity of0.4 m/s and flowed out at a temperature increased by8 K (8 �C) when the inductor power of 80 kW was used.The values, respectively, lie well between the simulatedresults for the two effective powers (Figures 8(a) and(b)). This can demonstrate the validity of the simulationwith the commercial software.

IV. CONCLUSIONS

Inductors attached to a galvanizing bath are usuallytreated as a black box. The zinc flow and the temperaturedistribution are driven by the Lorentz force and heatgenerated in the inductors of a galvanizing bath. Thisdiffers completely from the galvanizing bath itself, wherezinc flow and temperature distribution are driven byforced convection on the strip and rotating rolls, the jetsfrom the inductor outlets, and the natural convectionresulting from temperature and concentration differ-ences. Studying the performance of inductors attached toa galvanizing bath allows us to understand the workprinciple of the inductors well. This work is expected tolay a foundation for the estimation of outlet velocity andtemperature difference between the inlet and the outletfor various effective powers for inductors, and then wecan study the performance of the galvanizing bath andthe optimization of the process of the galvanizing line.

ACKNOWLEDGMENTS

This research has been partially supported by theNational Natural Science Foundation of China (No.50908094), the Ph.D. Programs Foundation of Minis-try of Education of China (No. 20100142120071), andthe Fundamental Research Funds for the Central Uni-versities, HUST (Nos. 2012QN022 and 2013NY015).

REFERENCES1. F. Ilinca, F. Ajersch, C. Baril, and F.E. Goodwin: Int. J. Numer.

Meth. Fluids, 2007, vol. 53, pp. 1629–46.2. P. Toussaint, P. Vernin, B. Symoens, L. Segers, M. Tolley, R.

Winand, and M. Dubois: Ironmaking Steelmaking, 1996, vol. 23,pp. 357–60.

3. C. Koutsaris, F. Ajersch, F. Ilinca, and F.E. Goodwim: Proc. of8th International Conference on Zinc and Zinc Alloy Coated SteelSheet, Galvatech 2011, Genova, Italy, 2011.

4. F. Ajersch, F. Ilinca, J.-F. Hetu, and F.E. Goodwin: Proc. of 7thInternational Conference on Zinc and Zinc Alloy Coated SteelSheet, Galvatech ’07, Osaka, Japan, 2007.

5. F. Ilinca, J.-F. Hetu, and F. Ajersch: Numerical Heat Transfer,Part A: Applications, 2003, vol. 44, pp. 463–82.

6. F. Ajersch, F. Ilinca, J.-F. Hetu, and F.E. Goodwin: Can. Metall.Q., 2005, vol. 44, pp. 369–78.

7. F. Ajersch, F. Ilinca, and J.-F. Hetu: Metall. Mater. Trans. B,2004, vol. 35B, pp. 161–70.

8. F. Ajersch, F. Ilinca, and J.-F. Hetu: Metall. Mater. Trans. B,2004, vol. 35B, pp. 171–78.

9. F. Ajersch, C. Binet, F.E. Goodwin, K.S. Turke, and P.S.Kolisnyk: Proc. of Galvatech’98, Iron & Steel Society, 1998, p. 642.

10. Y.H. Kim, Y.W. Cho, S.-H. Chung, J.-D. Shim, and H.Y. Ra:ISIJ Int., 2000, vol. 40, pp. 706–12.

11. H.S. Park, K.-A. Han, J. Lee, and J.-W. Shim: ISIJ Int., 2008,vol. 48, pp. 224–29.

12. S.K. Dash, M. Dutta, and N. Rajesh: ISIJ Int., 2005, vol. 45,pp. 1059–65.

13. Z.W. Yan: Engineering Electromagnetic Analysis Technology andExample Explanation with ANSYS 10.0, China Water Conser-vancy and Hydropower Press, Beijing, China, 2006.

14. Fluent: FLUENT 6.3 User’s Guide, Fluent Inc., Lebanon, NH,2006.

15. S.V. Patankar: Numerical Heat Transfer and Fluid Flow, Taylor &Francis, London, U.K., 1980.

Fig. 8—Comparisons of (a) surface-weighted average velocity at theoutlet and (b) temperature difference between the inlet and the outletwith various effective powers and experimental data (denoted by redpoint) from Refs. [4–8].

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