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ISIJ International, Vol. 40 (2000), No. 7, pp. 706–712

Numerical Analysis of Fluid Flow and Heat Transfer in Molten Zinc Pot of Continuous Hot-dip Galvanizing Line

Yong Hee KIM, Young Whan CHO,1) Soon-Hyo CHUNG,1) Jae-Dong SHIM1) and Hyung Yong RA

Division of Materials Science and Engineering, Seoul National University, San 56–1, Shinrim-dong, Kwanak-ku, Seoul 151-742, Korea. E-mail: stldream@netian.com. 1) Metals Processing Research Center, Korea Institute of Science and Technology, P.O. Box 131, Cheongryang, Seoul 130–650, Korea. E-mail: oze@kistmail.kist.re.kr. (Received on October 29, 1999; accepted in ?nal form on February 24, 2000 )

A numerical model adopting a partially staggered grid system for the location of dependent variables has been developed to analyze the ?uid ?ow and temperature distributions in a molten zinc pot of No. 2 CGL of POSCO Kwangyang strip mills. A control volume based ?nite difference procedure was employed to solve the conservation equations transformed by using the boundary-?tted-coordinate (BFC) system. The calculation results have shown that a change in the steel strip velocity has little in?uence on the overall ?ow pattern developed in the pot. The overall temperature distribution was rather uniform as predicted. However, charging cold ingots directly into the pot produced a non-uniform distribution of temperature. The local temperature ?uctuations will promote the formation of intermetallic dross particles. It has been proposed that the non-uniform distribution of temperature could be reduced by selecting an appropriate channel inductor position as well as by optimizing the zinc ingot loading position. KEY WORDS: continuous galvanizing; partially staggered grid; boundary-?tted coordinate; numerical model.

1.

Introduction

For several decades, galvanized steel has been an engineering material for applications demanding corrosion resistance.1–5) Galvanized steel provides low-cost and effective performance by combining the corrosion resistance of zinc with the strength and formability of steel. Recently, it is required to improve the surface quality of galvanized steel more stringently due to an increase in the demand for automotive body panel and household electric appliances of high surface qualities.1) Therefore, the strict control of the operation conditions, particularly the Al concentration and temperature distribution in the zinc pot of continuous galvanizing line, is very important to meet the high surface quality required by the users. In a continuous hot-dip galvanizing line (CGL), the steel strip is coated by passing it through a pot of molten zinc and withdrawing it from the pot through a pair of gas-wiping jets produced by an air knife system. These jets blow off the excess molten zinc that adheres to each side of the steel strip and leave uniform layers of molten zinc with controlled thickness. During this process, the pick up of dross occurs when small particles of Zn–Fe or Al–Fe intermetallics formed in the pot stick to the strip surfaces. This kind of surface defect could be greatly reduced by tight control of the Al concentration and temperature distribution, which suppresses the dross formation in the molten zinc pot. For example, this can be achieved by means of optimizing the ingot loading position and inductor ?ring mode, and by modifying the local ?ow pattern that would ? 2000 ISIJ

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suppress the adhesion of dross particles to the strip surfaces. There have been several reports about fundamental studies of the ?uid ?ow and temperature distributions in CGL pots6–12) including water model experiments. Their results11,12) were apt to be compared with the numerical ones and showed a good agreement with the data from the real molten zinc pot. Numerical simulation studies can provide rather comprehensive information on the ?ow pattern as well as the heat and mass transfer phenomena. However, there are various limits in developing the numerical programs such as the complex geometry of the zinc pot. Very often, they do not include the submerged structures or ignore the turbulent effects.7,9) In the present study, a 3D numerical model has been developed including almost all the details of the submerged structures such as sink rolls, stabilizing rolls, roll frame and the channel inductors. The k–e model has been adopted for the modeling of turbulence and a partially staggered grid scheme for improving the calculation ef?ciency. The in?uences of the strip width and line speed on the ?ow and heat distributions in the pot have been calculated. The effects of the ingot loading position and channel inductor position on the temperature distribution in the pot have also been investigated in order to ?nd optimum conditions for suppressing the formation of intermetallic dross particles. 2. Mathematical Formulation A schematic geometry of the molten zinc pot of No. 2

ISIJ International, Vol. 40 (2000), No. 7

Table 1. Dependent variables, effective diffusion coef?cients and source terms in conservation equations.

Fig. 1. Schematic diagram of CGL pot.

CGL of POSCO Kwangyang strip mills is shown in Fig. 1. On the entry side, the steel strip is dipped into the molten zinc pot through the snout and goes around the sink roll and between the lower and upper stabilizing rolls and ?nally comes out of the bath on the exit side. The frame with one large sink roll, two small stabilizing rolls and other structural parts is submerged and two channel inductors which have two outlet and one inlet ports at the center are located on both side walls approximately in the middle of the pot. There is also a scraper just above the sink roll. In practice, the line speed is changed continually between about 60 and 150 mpm (1–2.5 m/s) and one-ton alloys or pure zinc ingots are loaded at intervals as required. In the present study, all calculations were carried out assuming a steady state for the ?ow and the distribution of temperature. 2.1. Conservation Equation The governing equations for a 3D steady and incompressible turbulent ?ow in the Cartesian coordinate take the following forms13); Continuity equation:

equations, Gf includes the molecular (m l) and turbulent (m t) viscosity and the turbulent viscosity was calculated using the standard k–e model. Table 1 describes Gf and Sf for f variables.14) The last three terms of Sf in the momentum transport equations are the contributions from turbulence and the last term Fy in the v-momentum transport equation is the thermal buoyancy force. This buoyancy force term was obtained using the Boussinesq approximation13) which uses the ?rst-order Taylor’s series expansion. In the energy transport equation, Prt is the turbulent Prandtl number and was assumed to be unity in the present study. It is generally held that the ?uid ?ow near the steel strip is turbulent. Thus, the turbulent effects on the ?ow pattern and energy transport have to be taken into account. Since the direct simulation of turbulent ?ow is not feasible at present, the time-averaged transport equations along with the k–e turbulence model are solved. The main purpose of introducing a turbulence model is to estimate the Reynolds stresses appearing in the time-averaged equations. According to the literatures on various turbulent models,15,16) the two-equation k–e eddy-viscosity model is preferable from the viewpoints of accuracy and calculation speed. This model uses the Kolmogorov-Prandtl relation to estimate the eddy viscosity as m l?fm Cm k2/e , in which the turbulent kinetic energy (k) and the rate of dissipation of the turbulent energy (e ) are obtained by solving two differential equations for k and e . In the governing equations for k and e , G (generation of turbulent kinetic energy) is de?ned as:

2 2 2? ? ? ? ?? ? u ? ? ? v ? ? ? w ? ? G?? t ?2 ? ? ?? ? ?? ? ? ?x ? ? ? y ? ? ?z ? ? ? ? ? ? ?

? ?u ? w ? ? ?v ?u ? ? ? w ?v ? ?? ? ? ? ? ? ?? ? ?? ? ?z ?x ? ? ?x ? y ? ? ? y ?z ?

2

? · (r V)?0 ..................................(1)

General transport equation :

? ? ? ? ...........................................(3)

2

2?

? · (r Vf )?? · (Gf ?f )?Sf ......................(2)

where, r is density and f represents variables such as u, v, w, k, e and T. Gf is the effective diffusion coef?cient and Sf denotes the source term of f . In the momentum transport

707

And the coef?cients in the k and e equations (see Table 1) are given as CD?1.0, Cm ?0.09, C1?1.44, C2?1.92, s t? 0.9, s k?1.0 and s e ?1.3. 2.2. Coordinate Transformation As it is dif?cult to describe the complex geometry of the CGL pot in the Cartesian coordinate, a curvilinear coordi? 2000 ISIJ

ISIJ International, Vol. 40 (2000), No. 7

Fig. 2. Three numerical grid schemes.

nate has been adopted in the present study. By introducing new independent variables x , h and z , the governing equations are changed according to the general transformation x ?x (x, y, z), h ?h (x, y, z) and z ?z (x, y, z) and the transformed governing equations are as follows14); ? 1 ? ? ? ? ( ρG1φ )? ( ρG 2φ )? ( ρG 3φ )? ? Ja ? ?η ?ζ ? ? ?ξ ? ? 1 Ja ? ? ?φ ?φ ?φ ? ? ? ? ? ?g12 ?g13 ? ?Γφ φJ a ? g11 ? ?ξ ? ?ξ ?η ?ζ ? ? ? ? ? ? ? ? ? ? ?φ ?φ ?φ ? ? ?g 22 ?g 23 ?Γφ φJ a ? g 21 ? ?ξ ?η ?ζ ? ? ?? ? ? ? ? ? ?φ ?φ ?φ ? ?? ? ?g 32 ?g 33 ?Γφ φJ a ? g 31 ? ??Sφ ? ?ξ ?η ?ζ ? ? ? ? ? ?? ? ...........................................(4)

?

? ?η ? ?ζ

?

2.3. Boundary Conditions In the present study, the following boundary conditions have been used. At the free surface, the normal gradients of all the variables as well as the normal velocity components were set equal to zero. n · ?f ?0, n · V ?0 ..........................(5)

The boundary conditions at the plane of symmetry were equivalent to those adopted for the free surface. A uniform velocity was assumed at the inlet and outlet ports of the channel inductors and the inlet temperature was to be adjusted considering the heat balance of the pot. That is, as the heat inputs increased with faster line speed and wider steel strip, the inlet temperature of molten zinc jets decreased accordingly. When the ingot is loading, the heat input through the inductor inlet ?ows has to be maximum at the same condition. Of course, the mass ?ow rates between the inlet and outlet ports were matched. The zinc pot has a moving strip, rotating rolls and some other structures. As the ?ow varies suddenly from laminar to turbulent ?ow at these solid walls, it is necessary to construct a dense grid near the solid walls. In the present study, the-law-of-wall has been adopted to avoid this inef?ciency. 3. Numerical Scheme The governing equations were discretized using the con? 2000 ISIJ

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trol-volume-based ?nite volume method proposed by Patankar.17) The velocity components were calculated as dependent variables of the Cartesian velocity and the convection-diffusion terms were treated by the hybrid scheme. In order to resolve the velocity-pressure coupling in the momentum equations, a SIMPLE algorithm, suggested by Patankar,17) was adopted. A partially staggered grid scheme was adopted for the location of dependent variables in which all variables except pressure were evaluated at the main grid points, while the pressure component was calculated at the corners of the control volume. In Fig. 2, three numerical grid schemes are presented. The staggered grid can eliminate the unphysical pressure oscillations successfully. When the curvilinear velocity is chosen, however, a complicated tensor algebra is required to derive the momentum equations in the transformed domain and consequently the resulting governing equations include grid-sensitive terms that tend to ill-behave when the numerical grids are not suf?ciently smooth. The non-staggered grid is less in?uenced by the grid non-orthogonality effect, but this grid scheme confronts some critical issues that indulge in arti?cial damping terms or a control volume cell face interpolation technique is needed to avoid the checkerboard pressure mode.18–21) In the partially staggered grid scheme adopted in the present study, a skew and non-orthogonal grid was best calculated without additional computing efforts such as interpolations. As all variables except pressure share the same geometric information such as the cell volume, cell face dimensions, ?ux components and the metric tensor components, the convection contribution to the coef?cients in the discretized equation is all the same. So it requires less memory and computational time. A computational grid system (87?60?34) designed for the numerical simulation of the molten zinc pot of No. 2 CGL of POSCO Kwangyang strip mills is illustrated in Fig. 3 and Table 2 shows the thermo-physical properties used in the present study. 4. Results and Discussions

The driving force of the molten zinc ?ow in the pot is the movement of the steel strip and the rotation of the sink roll and two stabilizing rolls. The Reynolds number is given as Vsp?1.025?106 in the present study. As the line speed Vsp is between 1 and 2.5 m/s in practice, the Reynolds number has the order of 106. Therefore, the melt ?ow in the pot is believed to have the characteristics of the turbulent ?ow. Figures 4(a) and 4(b) show the typical ?ow patterns in

ISIJ International, Vol. 40 (2000), No. 7

Fig. 3. Computational grid system of CGL pot. (87?60?34)

Table 2.

Thermo-physical properties used in the calculation.

the x–y plane at z?0.2 m from the centerline of the pot when the line speeds are 1 and 3 m/s, respectively. In the entry side region, the molten zinc is accelerated toward the sink roll by the steel strip movement. Naturally, the ?ow from the rear side of the pot is drawn and gathered into a ?ow near the steel strip. The ?ow induced toward the sink roll then moves to the exit side region. This ?ow confronts with the lower stabilizing roll rotating clockwise and is de?ected toward the front wall of the pot. This de?ected ?ow is divided into the upward ?ow to the pot surface and the return ?ow toward the rear side of the pot along the front wall and the bottom. Therefore, a large re-circulating ?ow is developed in the pot. In the enclosed region by the entering and exiting strip, a rather complex ?ow pattern is developed. Three or four re-circulating ?ows are formed owing to the counterclockwise rotating sink roll and higher stabilizing roll and the steel strip movement in the entry side and toward the exit side region. In Fig. 4(b), the magnitude of the velocity vector is presented in 1/3 scale of Fig. 4(a) to compare the general ?ow patterns between them. They have almost same ?ow patterns and it means that the ?ow pattern itself does not in?uenced by changing the line speed only. Figures 5(a) and 5(b) show the temperature distributions when the line speeds are 1 and 1.5 m/s, respectively without the ingot loading. In the CGL pot, the heat input comes from the channel inductors located on both side walls of the pot and the steel strip which is normally maintained at 10–30°C higher than the control temperature of the pot. The heat output consists of the heat required to heat up and melting cold ingots and the heat loss through the walls and the free surface of the pot. The heat input and output are balanced to maintain the pot temperature at a constant

709

Fig. 4. Effect of line speed on typical ?ow patterns in x–y plane (z?0.2 m from centerline of pot).

Fig. 5. Effect of line speed on temperature distribution in x–y plane (z?0.2m) without ingot loading.

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Fig. 6. Effect of strip width on ?ow patterns (a) and (c) in y–z plane (x?2.644), and (b) and (d) in x–y plane (z?0.65 m).

level. In this study, the pot and strip temperatures were set at 460 and 475°C, respectively. The heat loss from the walls of the pot was treated as a constant heat ?ux which was contained in the source term of the energy equation. The inlet temperature of the jet ?ow from the port of the channel inductors was set to balance the heat input and output. As shown in Fig. 5(a), the overall temperature distribution in the molten zinc pot is quite uniform within approximately ?1°C. As the rear region of the pot is cooled down owing to the walls and free surface, through which more heat is lost by conduction and radiation, and less heat transfer from the hotter regions to this region as the ?ow is rather quiescent, there exists a small region below 459°C near the free surface and the rear wall. As the temperature of the steel strip is always higher than that of the pot, the molten zinc temperature very near the strip will be higher than 460°C. Some portion of the heat transferred from the hotter steel strip is dispersed into the front region of the pot by the moving steel strip, and the remaining heat would be accumulated in the enclosed region because of the re-circulating ?ow. Figure 5(b) shows the temperature distribution when the strip velocity is 1.5 m/s. As the line speed increases, the molten zinc in the pot mix better and the temperature distribution becomes more uniform. Figures 6(a), 6(b) and 6(c), 6(d) show the ?ow patterns ? 2000 ISIJ

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when the steel strip widths are 600 and 1 800 mm, respectively. The y–z plane is sectioned at x?2.644 m from the rear side of the pot and the x–y plane is sectioned at y? 0.65 m from the centerline of the pot. As the width of the steel strip decreases, the blockaded area with the steel strip in the enclosed region decreases and the ?ow develops more freely. When the strip width is 600 mm, the ?ow induced by the rotating sink roll moves faster toward the bottom and the free surface than when the strip width is 1 800 mm. The ?ow directed to the bottom could stir up and ?oat the bottom dross particles. The ?ow toward the free surface could disturb the melt free surface covered with viscous skim (a mixture of oxides and relatively large intermetallic dross particles containing a large amount of aluminum which is lighter than liquid zinc). This upward ?ow is very likely to promote the formation of more oxidation products and will eventually increase the amount of ?oating dross particles in the zinc pot. Figure 7 shows the temperature distribution in x–y plane (z?0.2 m) when the ingot is loading at the center in the rear of the pot. The loading of the cold ingot will produce a sudden decrease in temperature near the ingot. In consequence, a steep temperature gradient from room temperature of the ingot to that of molten zinc is built up around the ingot. As the temperature of the molten zinc around the ingot de-

ISIJ International, Vol. 40 (2000), No. 7

Fig. 7. Temperature distribution in x–y plane (z?0.2 m) when ingot is loading at center in rear region of pot.

Fig. 9. Effect of channel inductor position on temperature distribution in x–y plane (z?0.2 m) with ingot loading.

Fig. 8. Effect of ingot loading position on temperature distribution in x–y plane (y?1.89 m from bottom of pot).

creases signi?cantly, the solubility of Fe will also be decreased and consequently a considerable amount of Fe–Zn intermetallic dross particles would certainly form. To prevent a sudden decrease in temperature around the loading ingot, an ingot pre-melting system could be employed to solve the problem of generating intermetallic dross particles. To investigate the effect of the ingot loading position on the temperature distribution in the zinc pot, two different ingot loading positions are considered. Figure 8 shows the temperature distribution in a half face of x–z plane at y? 1.89 m from the bottom of the pot. As shown in Fig. 8(a), a portion of the molten zinc cooled by the ingot containing more intermetallic dross particles moves toward the entry region of the steel strip when the ingot is loaded in the middle of the rear side of the zinc pot. This could increase the probability of dross adhesion to the strip surface. Therefore, as shown in Fig. 8(b), the position of the ingot loading should be as far away as possible from the entry side of the steel strip such as the corner of the rear side of the pot, which is the case for the zinc pot of No. 2 CGL of POSCO Kwangyang strip mills. In Fig. 8, the higher temperature re711

gion than 460°C exists near the entry side of the steel strip. Figure 9 shows the effect of the channel inductor position on the temperature distribution in the zinc pot. When the inductors are located near the rear side of the pot, the temperature distributions near the rear and front sides are different from those of the case when the inductors are located under the sink roll frame. When the location of the inductors is close to the ingot loading position, the melt ?ow of rather higher temperature is injected to cold region near ingot loading side. This will certainly increase the melt temperature near the ingot loading position and eventually accelerate the melting rate of the ingot. On the contrary, the inductors located away from the rear side will inject the heated melt toward the moving steel strip or rotating rolls. By the steel strip movement, this melt travels toward the front side and then may slightly increase the difference in temperature between the front and rear regions. Therefore, it is desirable to have a more uniform temperature distribution by carefully selecting the position of the inductors in relation to the ingot charging position. 5. Conclusions

A 3D numerical model based on a partially staggered grid system was developed to analyze the ?ow and temperature distributions in the zinc pot of No. 2 CGL of POSCO Kwangyang strip mills. The results of numerical calculations on the ?ow and temperature distributions could be used to predict the distribution and adhesion probability of the intermetallic dross particles to the steel strip surface. The main results obtained from the present simulation ? 2000 ISIJ

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study may be summarized as follows. The variation of the steel strip speed does not produce any discernable change in the overall ?ow pattern of the pot. However, as the steel strip speed increases, the molten zinc will mix better in the pot and the temperature distribution will become more uniform. The strip width signi?cantly changes the ?ow pattern in the pot. When the strip width is narrow, the ?ow induced by the rotating sink roll develops more freely and is likely to stir up and disturb both the bottom and free surface of the pot, which may increase the overall density of the ?oating dross in the pot. The ingot loading position seems have a signi?cant in?uence on the distribution of intermetallic dross particles and the adhesion probability of dross particles to the steel strip surface. The position of the ingot loading should be as far away as possible from the entry side of the steel strip to reduce the dross-related surface defect of the galvanized strip. It is desirable to have a more uniform temperature distribution by carefully selecting the position of the inductors in relation to the ingot charging position.

Acknowledgement

Greek symbols Gf : effective diffusion coef?cient e : rate of dissipation of the turbulent energy ml : molecular viscosity mt : turbulent viscosity r : density s k, s e : constants for k–e turbulence model f : variables such as u, v, w, k, e and T

REFERENCES

1) 2) 3) 4) 5) 6) T. Asamura: Proc. of 4th Int. Conf. on Zinc and Zinc Alloy Coated Steel Sheet (GALVATECH'98), ISIJ, Tokyo, (1998), 14. R. F. Lynch: JOM, 39 (1987), 39. G. W. Bush: JOM, 41 (1989), 34. H. Nakamura: Tetsu-to-Hagané, 81 (1995), 397. T. Hada: Proc. of 1st Int. Conf. on Zinc and Zinc Alloy Coated Steel Sheet (GALVATECH’89), ISIJ, Tokyo, (1989), 111. M. Gagné and M. Gaug: Proc. of 4th Int. Conf. on Zinc and Zinc Alloy Coated Steel Sheet (GALVATECH’98), ISIJ, Tokyo, (1998), 90. K. Otsuka, M. Arai and S. Kasai: Proc. of 4th Int. Conf. on Zinc and Zinc Alloy Coated Steel Sheet (GALVATECH’98), ISIJ, Tokyo, (1998), 96. K. Kurita: CAMP-ISIJ, 10 (1997), 1388. A. Paré, C. Binet and F. Ajersch: Proc. of Use and Manufacture of Zinc and Zinc Alloy Coated Steel Sheet Products (GALVATECH ’95), ISS, Warrendale, PA, (1995), 695. P. Toussaint, P. Vernin, B. Symoens, L. Segers, M. Tolley, R. Winand and M. Dubois: Ironmaking Steelmaking, 23 (1996), 357. J. Kurobe, M. Iguchi, S. Matsubara, K. Nakamoto and Z. Morita: Tetsu-to-Hagané, 81 (1995), 733. K. Andachi: CAMP-ISIJ, 8 (1995), 1557. M. R. Aboutalebi, M. Hasan and R. I. L. Guthrie: Metall. Mater. Trans., 26B, (1995), 731. K. Y. Kim: PhD Disseration, Seoul National University, Korea, (1993). V . C. Patel, W. Rodi and G. Scheuerer: AIAA J., 23 (1985), 1308. R. A. W. M. Henkes and C. J. Hoogendoorn: Int. J. Heat Mass Transfer, 32 (1989), 157. S. V . Patankar: Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor & Francis Group, New York, (1980). S. W. Arm?eld: Computers Fluids, 20 (1991), 1. M. Peric, R. Kessler and G. Scheuerer: Computers Fluids, 16 (1988), 389. M. M. Rahman, T. Siikonen and A. Miettinen: Numer. Heat Transfer, 32B (1997), 63. M. C. Melaaen: Numer. Heat Transfer, 21B (1992), 21.

7)

The authors are grateful to Pohang Iron & Steel Co. (POSCO) for ?nancial support of the present work. Nomenclature C1, C2, CD, Cm : constants for k–e turbulence model : contravariant metric tensor gij G : generation of turbulent kinetic energy Gi : ?ux components de?ned on faces of control volume k : turbulent kinetic energy Ja : jacobian of coordinate transformation P : pressure Prt : turbulent Prandtl number Sf : source term of f T : temperature u : velocity in x direction v : velocity in y direction w : velocity in z direction Vsp : line speed of steel strip plate

8) 9)

10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

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