Modeling of bubble behaviors and size distribution in a slab continuous casting mold

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Modeling of bubble behaviors and size distribution in a slab continuous casting mold Z.Q. Liu a, F.S. Qi a, B.K. Li a,∗, S.C.P. Cheung b a b

School of Materials and Metallurgy, Northeastern University, Shenyang 110819, China School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Victoria 3083, Australia

a r t i c l e

i n f o

Article history: Received 23 January 2015 Revised 13 July 2015 Accepted 28 July 2015 Available online xxx Keywords: Population balance Two-phase flow Bubble coalescence Bubble breakage Bubble size distribution Continuous-casting mold

a b s t r a c t Population balance equations combined with Eulerian–Eulerian two-phase model are employed to predict the polydispersed bubbly flow inside the slab continuous-casting mold. The class method, realized by the MUltiple-SIze- Group (MUSIG) model, alongside with suitable bubble breakage and coalescence kernels is adopted. A two-way momentum transfer mechanism model combines the bubble induced turbulence model and various interfacial forces including drag, lift, virtual mass, wall lubrication, and turbulent dispersion are incorporated in the model. A 1/4th scaled water model of the slab continuous-casting mold was built to measure and investigate the bubble behavior and size distribution. A high speed video system was used to visualize the bubble behavior, and a digital image processing technique was used to measure the mean bubble diameter along the width of the mold. Predictions by previous mono-size model and MUSIG model are compared and validated against experimental data obtained from the water model. Effects of the water flow rate and gas flow rate on the mean bubble size were also investigated. Close agreements by MUSIG model were achieved for the gas volume fraction, liquid flow pattern, bubble breakage and coalescence, and local bubble Sauter mean diameter against observations and measurements of water model experiments. © 2015 Elsevier Ltd. All rights reserved.

Introduction Continuous casting (CC) has been widely accepted as the most important production process in the steel industry. In the casting process, molten steel from the ladle flows through the tundish into a mold. Within the mold, the molten steel freezes against the watercooled copper mold walls forming a solid shell. Details of the main complex phenomena in the continuous casting process of steel can be found in some previous works (Szekely and Yadoya, 1972; Thomas et al., 1990; Li and Tsukihashi, 2005). As demonstrated in previous studies, fluid dynamic of the molten steel plays an important role in the continuous casting process. In the molten steel at a temperature above 1500 °C, flow structures and its characteristics such as turbulence, mixing, vortexing, fluid flow separation, and generation of recirculation zones, are of decisive importance for the quality of the final steel product. The physical–chemical reactions, phase changes, mixtures, and floatation processes of non-metallic inclusions suspended in the molten steel are affected by the fluid flow. Consequently, the flow field of molten steel in a mold is one of the important factors for the control of slab qualities.

∗

Corresponding author. Tel.: +86 24 83672216; fax: +86 24 23906316. E-mail address: [email protected] (B.K. Li).

In most of the modern system, argon gas is usually employed in the continuous casting process to prevent nozzle clogging, encourage mixing and promote the floatation of non-metallic inclusion particles from the molten steel by changing the flow field (Thomas et al., 1994; Iguchi and Kashi 2000; Liu et al. 2013; Krishnapisharody and Irons, 2013). The argon gas is injected into the molten steel which enters into the continuous casting mold through the submerged entry nozzle (SEN). After the SEN, due to intense shear forces exerted by molten steel, the argon gas disintegrates into swarm of bubbles with different diameters, as shown in Fig. 1. However, previous studies in Iguchi and Kashi (2000) had found that the trajectories of bubbles are sensitive to its size. Large bubbles have the tendency to escape from the liquid steel surface through the mold flux power layer, while smaller bubbles follow the main stream of molten steel flowing deep into the mold cavity. However, these small bubbles and non-metallic inclusions adhering to the surface of these bubbles may be entrapped by solidified shell, forming defects in the final product, such as slivers, “pencil pipe” blisters, and other costly defects. In order to improve the quality of the final steel products, it is crucial to gain an in-depth understanding of the structure characteristics of molten steel–argon gas two-phase flows as well as the characteristics of bubble size distribution and its related defects in the current continuous casting process. Extensive experimental studies on the bubble formation in the SEN (Wang et al., 1999; Bai and Thomas 2001; Lee et al., 2010) and

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Please cite this article as: Z.Q. Liu et al., Modeling of bubble behaviors and size distribution in a slab continuous casting mold, International Journal of Multiphase Flow (2015), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2015.07.009

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Fig. 1. Schematic of argon bubbles transport in the continuous-casting mold.

bubble size and distribution in the mold (Sánchez-Perez et al., 2003; Ramos-banderas et al., 2005;) have been performed using cold water models. Wang et al. (1999) studied the air-bubble formation phenomenon by injecting air into the flowing water through a porous refractory at the upstream of the acrylic nozzle. They found that considerably uniform-sized bubbles were formed from the porous refractory and mixed with the main water stream. Instead of porous refractory, Bai and Thomas (2001) used a horizontal hole for injecting air into a shearing downward turbulent liquid flow. Detailed experimental analyses were carried to investigate the contact angles, bubble-elongation length, mode of bubble formation, bubble size, and size distribution of bubble formation. Ramos-Banderas et al. (2005) performed a series of water model experiments to analyze the coalescence-breakage phenomena of bubbles. Based on their experimental measurements, they concluded that the population and bubble sizes of bubbles within the model were proportional to the gas loads and the casting speeds. During the past decades, due to the extremely high costs on experimental facilities and its complexity of measurement, only very few hot experimental studies have been performed to measure the bubbly flow inside the mold. Li et al. (2000) measured the velocity fields of argon gas injection in a reduced scale continuous molten tin casting system using a two-dimensional sensor. Recently, Eckert et al. (2014) analyzed the size and the motion of gas bubbles in the SEN using a small-scale mockup X-LIMMCAST where eutectic alloy GaInSn at room temperature was used as model fluid. In terms of modeling, two main approaches (i.e. the Euler– Lagrange and Euler–Euler approach) are widely adopted to simulate the two-phase flow in the continuous casting process. Detailed descriptions of the Euler–Lagrange and Euler–Euler approaches can be found in previous studies of Deen et al. (2001). The Euler–Lagrange approach involves tracking individual bubbles trajectory in the liquid phase and it was followed by many researchers (Miki and Takeuchi, 2003; Wang and Zhang, 2011). This approach gives a direct physical interpretation of the fluid–bubble interaction but it is computationally intensive. It is therefore impractical for simulating the system with high dispersed phase volume fraction including bubbly flows studied in the present study. On the other hand, the Euler–Euler approach assumes both of liquid and gas phases to be interpenetrating continua. It is more economical and hence more popular. The Euler– Euler two-phase model with a constant bubble size has been widely used to study the two-phase flow inside various vessels such as

ladle (Lou and Zhu, 2013) and mold (Thomas et al., 1994; Liu et al., 2014) in the continuous casting process, bubble column (Dhotre et al., 2008; Tabib et al., 2008) in the chemical process. Nevertheless, the size of bubbles varies significantly in these vessels due to bubble coalescence and breakage mechanism. The bubble sizes are also sensitive to the operating conditions (e.g. water and gas flow rate and pressure) as well as physical properties of both phases (e.g. density, viscosity and surface tension). Furthermore, local flow pattern and turbulence characteristics of the two-phase flow are also the crucial factors affecting the argon bubble diameter. Owing to the importance of bubble size in two-phase flows, the predictions of argon bubble size distribution become very important to understand the underlying physics and hydrodynamic in the continuous casting mold. Recently, the application of the population balance approach towards better describing complex bubbly flow inside the mold has received an increasing attention. The MUltipleSIze- Group (MUSIG) model provides a framework in which the population balance equation can be incorporated into the generic computational fluid dynamics solution procedures. Rather than prescribing a dispersed phase size as in the standard Eulerian treatment, the MUSIG model allows us to predict a mean bubble size based on fundamental considerations of coalescence and breakup mechanism. The evolution of the bubble size distribution is determined by the relative magnitudes of bubble coalescence and breakage rates. Since the bubble size distribution is discretized into multiple bubble class, the MUSIG model is capable to simulate polydispersed multiphase flows in which the dispersed phase features a large variation in its characteristic sizes. Several studies (Yeoh and Tu, 2006; Cheung et al., 2007; Moilanen et al., 2008; Duan et al., 2011) based on the MUSIG model have been performed to study the bubble size distribution in bubble column reactors and stirred reactors, which are extensively used in a various chemical, petroleum, mining and pharmaceutical industries. Nonetheless, only a few numerical works have been carried out to investigate the bubble size distribution in the continuous casting model. Yuan et al. (2001) attempted to simulate the bubble dynamics using MUSIG model. However, coalescence of bubbles was neglected in their model. The objectives of this work are twofold: (i) to present a Mono-Size (MS) Euler–Euler two-phase model to further describe the effect of bubble size on the two-phase flow inside the mold; and (ii) to develop a MUSIG model to study the polydispersed bubbly flow and bubble size distribution inside the mold. A 1/4th scale water model was established to observe the breakage and coalescence of bubbles and quantify the bubble size distribution in the width of the mold. Mathematical model formation In order to investigate the two-phase flow behaviors inside a continuous casting mold, two Euler–Euler two-phase models are employed in the present study, named MS model and MUSIG model, respectively. In the MS model, bubbles inside the mold are considered to be identical such that breakage and coalescence between bubbles are not accounted. The effects of argon gas bubble sizes on flowrelated phenomena were investigated using this model. However, the argon gas disintegrates into small bubbles of varying sizes as it issues out of the SEN. Based on the two-phase Euler–Euler approach and population balance principle, the MUSIG model for poly-dispersed bubbly flow is developed and constructed. Bubbles are divided into 10 groups and these groups are used to analyze the bubble size distribution inside the mold through coupling with appropriate coalescence and breakage models. Fig. 2 shows the discrete bubble groups employed to characterize the multi-size bubbly flow. In comparison to conventional two-phase Euler–Euler approach (MS model), it gives additional engineering parameters such as bubble size, interfacial area concentration by describing coupled mechanisms of interphase turbulence, momentum transfer and bubble coalescence and

Please cite this article as: Z.Q. Liu et al., Modeling of bubble behaviors and size distribution in a slab continuous casting mold, International Journal of Multiphase Flow (2015), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2015.07.009

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3

Lift force in terms of the slip velocity and the curl of the liquid phase velocity is described by:

FL = αg ρl CL (ug − ul ) × (∇ × ul )

(7)

The virtual mass force accounts for relative acceleration, the additional work performed by the bubbles in accelerating the liquid surrounding the bubble. The acceleration of the liquid is taken into account through the virtual mass force, which is given by:

FV M = αg ρl CV M

 Du

g

Dt

−

Dul Dt



(8)

Wall lubrication force, which is in the normal direction away from the wall and decays with distance from the wall, is expressed by:

FW L = −

Fig. 2. Graphical presentations of the MUSIG model.

breakage behaviors. However, the details of MS model can be found in many previous works (Thomas et al., 1994; Bai and Thomas, 2001; Lou and Zhu, 2013; Liu et al. 2013, 2014) which therefore will not repeat in here. The following section presents the details of the MUSIG model and the breakup and coalescence model. Euler–Euler two-phase model The Euler–Euler two-phase model treating both the liquid and gas phases as continua solves two sets of conservation equations governing mass and momentum, which are written for each phase as: Continuity and momentum equations of liquid phase:

∂(αl ρl ) + ∇ · (αl ρl ul ) = 0 (1) ∂t ∂(αl ρl ul ) + ∇ · (αl ρ l ul ul ) = −αl ∇ P + αl ρl g + ∇ · (αl τl ) + Flg ∂t (2) Continuity and momentum equations of gas phase:

∂(αg ρg fi ) + ∇ · (αg ρg ug fi ) = Sdi (3) ∂t ∂(αg ρg ug ) + ∇ · (αg ρ g ug ug ) = −αg ∇ P + αg ρg g + ∇ · (αg τg ) + Fgl ∂t (4) where α , ρ , t, u, τ , P and g are the volume fraction, density, time, velocity, shear stress, pressure, and gravity acceleration, respectively. fi is the scalar volume fraction of each bubble size group in gas phase. The subscripts l and g denote the liquid and gas phase, respectively. Sdi is the additional source due to coalescence and breakage of bubbles based on the formulation, which is described in the next sections. Flg = −Fgl represents the interfacial forces between two phases as:

Flg = −Fgl = FD + FL + FV M + FW L + FT D

(5)

The terms on the right hand side of Eq. (5) are drag force, lift force, virtual mass force, wall lubrication force, and turbulent dispersion force. A brief description of each interfacial force component is presented below. Detail descriptions of these forces can be found in previous works (Deen et al., 2001; Cheung et al., 2007; Liu et al., 2014). The momentum transfer between gas and liquid due to drag force is given by:

FD = −

C 3 αg ρl D |ug − ul |(ug − ul ) 4 db

(6)

αg ρl (ug − ul ) DS



max 0, Cw1 + Cw1



DS nw yw

(9)

Turbulent dispersion force results in additional dispersion of phases from high volume fraction regions to low volume fraction regions due to turbulent fluctuations. The turbulent dispersion force is given based on Favre Averaged Drag Model proposed by Burns et al. (2004):

FT D = CT DCD

  υt,g ∇αl ∇αg − σt,g αl αg

(10)

The drag force coefficient CD in Eq. (6) has been correlated for several distinct Reynolds number regions for individual bubbles according to Ishii and Zuber (1979) model. The present work concentrates on bubbly flow regime with small spherical bubbles, where only positive lift force coefficient is sufficient; the constant CL is set to 0.8. The virtual mass force coefficient CVM is taken to be 0.5 for individual spherical bubbles. The wall lubrication constants Cw1 and Cw2 as suggested by Antal et al. (1991) are 0.01 and 0.05, respectively. By default, the turbulent dispersion coefficient CT D = 1 and the turbulent Schmidt number σt,g = 0.9 are adopted. And db is bubble diameter, υt,g is gas turbulence kinetic viscosity, nw is outward vector normal to the wall. The stress term of liquid phase is described as follows:

  τl = μeff,l ∇ ul + (∇ ul )T

(11)

whereμeff,l is the effective viscosity. The effective viscosity of the liquid phase is composed of three contributions: the molecular viscosity (μL, l ), the turbulence viscosity (μT, l ) and an extra term due to bubble induced turbulence (μBI, l ).

μeff,l = μL,l + μT,l + μBI,l

(12)

The calculation of the effective gas viscosity is based on the effective liquid viscosity as proposed by Jakobsen et al. (1997).

μeff,g =

ρg μ ρl eff,l

(13)

The model proposed by Sato et al. (1975) has been used to take account of the turbulence induced by the movement of the bubbles. The expression is:

μBI,l = ρ l Cμ,BI αg db,i |ug − ul |

(14)

With a model constant Cμ, BI equals to 0.6. And db, i is the bubble diameter of i-th class. In handling two-phase turbulence flow, unlike single-phase fluid flow problem, there is no standard turbulence model. However, the standard k − ε turbulence model has been successfully employed to study the two-phase flow inside the mold (Thomas et al., 1994). When the k − ε turbulence model is used, the turbulent eddy viscosity is formulated as follows:

μT,l = ρ l Cμ,T

k2

ε

(15)

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The turbulent kinetic energy (k) and its energy dissipation rate (ε ) are calculated from their governing equations:

 μ  ∂(ρl αl k) T,l + ∇ · (αl ρl ul k) = −∇ · αl ∇ k + αl (Gk − ρl ε) ∂t σk  μ  ∂(ρl αl ε) T,l + ∇ · (αl ρl ul ε) = −∇ · αl ∇ε ∂t σε ε + αl (Cε1 Gk − Cε2 ρl ε) k

(17)

where Gk is the rate of production of turbulent kinetic energy. Other model constants are: Cμ,T = 0.09, Cε1 = 1.44, Cε2 = 1.92, σk = 1.00 and σε = 1.30. The bubble local Sauter mean diameter DS based on the calculated value of the scalar fraction fi and bubble diameter db, i can be deduced from:

1 DS =  f i i /db,i

(18)

Bubble population balance model

∂ ni + ∇ · (ug ni ) = BC + BB − DC − DB ∂t

(19)

where ni is the number density of group-i bubbles. BB , DB , BC and DC represent the birth rate due to breakup of larger bubbles, the death rate due to breakup into smaller bubbles, the birth rate due to coalescence of smaller bubbles, and the death rate due to coalescence with other bubbles, respectively, as shown in Fig. 1. MUSIG breakup rate The birth and death rates of bubbles due to turbulence induced breakage are formulated as:

(v j : vi )n j

(20)

j=i+1

DB = i ni

(21)

where vi and vj are the volumes corresponding to bubble group-i and j. nj is the number density of group-j bubbles. i is the breakup rate of group-i bubbles. N is the number of bubble groups. The breakup rate of volume vj into vi is modeled according to the Luo and Svendsen (1996) model.



1/3

(1 + ξ )2 2 11/3 db, ξmin ξ j  2/3 2/3 + (1 − fBV ) − 1)σ 12( fBV exp − dξ 5/3 11/3 2ρl ε 2/3 db, ξ j

(v j : vi ) = 0.923FB (1 − αg )n j

i i 1 η jki χi j ni n j 2

ε

1

(22)

where FB is the breakage calibration factor and 0.5 is used in this study. ξ = λe /db, j is the size ratio between an eddy and a bubble in the inertial sub-range, and consequently ξmin = λemin /db, j . fBV is a stochastic breakage volume fraction. σ is the surface tension.

(23)

j=1 k=1

⎧ (v + vk ) − vi−1 ⎪ ⎪ j ⎪ ⎪ ⎨ vi − vi−1 η jki = vi+1 − (v j + vk ) ⎪ ⎪ vi+1 − vi ⎪ ⎪ ⎩

vi−1 < v j + vk < vi

DC =

N

(24)

vi < v j + vk < vi+1

0

else

χi j ni n j

(25)

j=1

where vk is the volume corresponding to bubble group-k. ηjki is the transfer coefficient between bubble groups arising from bubble breakup. χ ij is the coalescence rate of groups-i and j. The coalescence rate considering turbulence collision taken from Prince and Blanch (1990) model can be expressed as:

χi j = FC

To account for the non-uniform bubble size distribution, the MUSIG model employs multiple discrete bubble size groups to represent the population balance of bubbles. For computational efficiency, assuming each bubble group travels at the same velocity field. Based on Kumar and Ramkrishna’s (1996) work, the individual number density of bubble group-i can be expressed as:

N

MUSIG coalescence rate The birth and death rates of bubbles due to turbulence collision induced coalescence are formulated as:

BC = (16)

BB =

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π 4



[di + d j ]

2

(|uti | + |ut j | ) 2

2 0.5

exp −

ti j



τi j

(26)

where the coalescence calibration factor FC is taken to be 1.0. ti j = [(di j /2) ρl /16σ ]0.5 ln (h0 /h f ) is the time required for two bubbles 3

to coalesce with diameter di and dj , τi j = (di j /2) /ε 1/3 is the time constant for two bubbles. The equivalent diameter dij is calculated as suggested by Chesters and Hoffman (1982): di j = 2(di−1 + d−1 )−1 . j The initial film thickness h0 and critical film thickness hf are set at 1 × 10−4 and 1 × 10−8 m, respectively (Cheung et al., 2007). The turbulent velocity ut in the inertial sub-range of isotropic turbulence is √ given by 2ε 1/3 d1/3 . 2/3

Numerical details For gas-liquid flow in the continuous-casting mold, the CFD code ANSYS-CFX (2012) was utilized to handle two sets of equations governing conservation of mass and momentum of each phase. The discrete bubble sizes prescribed in the dispersed phase were further tracked by solving additional continuity equations of gas phase; where these equations were coupled with the flow equations throughout the simulations. Sensitivity study on the number of size groups was performed through the consideration of equally dividing the bubble diameters into 5, 10 and 15 size groups. The analysis revealed that no appreciable difference was found for the predicted maximum bubble Sauter diameter between the 10 and 15 bubble size groups. In the view of computational resources and times, it was therefore concluded that the subdivision of the bubbles sizes into 10 size groups were deemed sufficient and the following computational results are all based on the discretization of 10 bubble size groups. Grid sensitivity was also examined. In the mean parameters considered, further grid refinement did not reveal significant changes to the two-phase flow parameters. So a uniform grid with 499,546 cells was used. Due to the symmetry of the structure and reducing the computational cost, only a quarter of the actual mold was modeled. A mass flow boundary was set at the water inlet of the SEN based on the casting speed. A mass flow boundary was set at the air inlet of the SEN, because the diameters of the injected bubbles were unknown, uniformly distributed bubble size was specified in accordance with the flow conditions, and all bubbles enter into the SEN with an initial bubble diameter of 1 mm. The density of air is used with a constant density considering the conditions of pressure and temperature at the SEN port. A constant pressure outlet condition at the bottom

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Table 1 Geometrical, physical properties and operating conditions in water model and steel caster.

of the calculation domain is applied. The top surface of the mold cavity is modeled as degassing boundary condition, where dispersed bubbles are permitted to escape, but the liquid is not. Along the walls, wall functions are applied for standard k − ε model, while no-slip boundary condition is adopted. The velocity–pressure linkage was handled through the SIMPLE procedure. The hybrid-upwind discretization scheme was used for the convective terms (Zhou et al., 2008). All the numerical simulation was performed for steady conditions. Water model experiment Previous understanding of fluid flow in continuous casting has come about mainly through experiments using physical water models. This technique is a useful way to test and understand the effects of new configurations before implementing them in the process. A 1/4th scaled water model experimental system was established to measure the bubble distribution inside the mold and validate the mathematical model, as shown in Fig. 3. The entire system was established including the simplified tundish, circumferential gas inlet chamber, SEN, and the mold. All parts for water flow and water containment are made of transparent acrylic so that fluid flow behavior is visible. Water was circulated in the circuit through a buffer tank to the tundish. Water level in the tundish and mold was maintained constant with the help of the slide gate nozzle and the value of the buffer tank. Air was injected into the SEN through a circumferential gas inlet chamber which was made of specially-coated samples of mullite porous brick. The air flow rate was measured and controlled using a flow meter and a control valve. Gravitational and inertial forces are mainly responsible for the fluid flow and bubble movement. Hence, the Froude similarity number (Singh et al., 2006; Liu et al., 2014; Liu and Thomas, 2015) is the most important non-dimensional parameter, which is made equal for the model and the plant in the current work. Water flow rate used in the experiment corresponding to actual throughput in the caster was obtained based on the normal Froude similarity number, which is defined as follows:

u2l

(27)

gL

where u is the characteristic velocity, L is the characteristic length. Air (25 °C) was chosen to simulate argon gas (25 °C) in the model, and the modified Froude similarity number was used to obtain the air flow rate, which is defined as follows (Singh et al., 2006):

Frm =

u2l gL

ρg ρl − ρg

1/4th water model

Steel caster

Mold width × thickness Mold/strand height Diameter of SEN Length of SEN Exit angle of nozzle SEN port height × width Submergence depth of SEN Liquid density Liquid viscosity Liquid flow rate Gas density Gas viscosity Gas flow rate Interfacial tension

550 × 75 mm 900 mm 20 mm 305 mm 15° down 20 × 17.5 mm 75 mm 1000 kg m−3 0.001 kg m−1 s−1 15–23 L/min 1.184 kg m−3 (25 °C) 1.85 × 10−5 kg m−1 s−1 0.8–2.4 L/min (25 °C) 0.072 N/m (water/air)

2200 × 300 mm Open bottom 80 mm 1220 mm 15° down 80 × 70 mm 300 mm 7020 kg m−3 0.0056 kg m−1 s−1 480–736 L/min 0.56 kg m−3 (1530 °C) 7.42 × 10−5 kg m−1 s−1 16–48 L/min (25 °C) 1.5 N/m (steel/argon)

The increase in volume during heating of the injected argon gas (25 °C) to molten steel temperature (1530 °C) has been considered. Assuming ideal gas law holds good for argon gas:

Fig. 3. Experimental system of the water model.

Fr =

Parameter

(28)

TArgon,1530◦ C QArgon,1530◦ C (1530 + 273)K = 6.05 = = QArgon,25◦ C TArgon,25◦ C (25 + 273)K

(29)

In this model, the Froude number and modified Froude number were kept same as in the actual caster. Thus water flow rate and air flow rate in the model can be calculated corresponding to actual steel flow rate and argon gas flow rate as follows:

Qwater,25◦ C = λ2.5 = 0.03125 Qsteel,1530◦ C

(30)

QAir,25◦ C QAir,25◦ C = 6.05 · = 6.05 QArgon,25◦ C QArgon,1530◦ C



·λ

2.5

·

ρArgon,1530◦ C (ρWater,25◦ C − ρAir,25◦ C ) = 0.05 ρAir,25◦ C (ρSteel,1530◦ C − ρArgon,1530◦ C ) (31)

where Q is the volume flow rate, λ is the geometrical ratio. However, modeling steel–argon two-phase flows in the continuous casting process is very complex. In the future, further research works aims to predict the bubble size distribution inside the actual steel mold would be done using this model. Then the more accurate gas flow through porous refractory should be considered (Liu and Thomas, 2015). Table 1 gives operational details on the water model and the corresponding actual steel caster. However, in order to validate the mathematical model, all the numerical simulations were calculated using the air and water inside the virtual water model. Flow visualization was made in two different ways: (a) watersoluble dye injection and (b) air injection along with the fluid. To start with, adjusting water supply value makes liquid level in the tundish steady and then injects air from circumferential gas inlet chamber into the SEN at a predetermined flow rate. When the system had established a steady stage, the fluid flow patterns were recorded using a video recorder and the distributions of gas and bubble sizes were visually captured by a high-speed camera with 1000 frames per second and a laser light-sheet, which was positioned at the axialsymmetrical plane of the mold, parallel to the wide face. In order to study the bubble size distribution in the width of the mold, the region of interest in the upper recirculation zone was divided into 16 equal zones, the close-up photographs of bubbles have been shown in Fig. 4. The zone-1 is adjacent to the SEN and the zone-16 is close to the narrow wall of the mold. Then, the population and sizes of bubbles of each zone were measured through the image analysis software of ImageJ (Xiang et al., 2013). This image analyzer program uses mathematical filters and combinations of contrast,

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Fig. 4. Schematic and photograph of the bubble size distribution.

brightness and zooms being possible the resolution, definition and identification of individual bubbles or groups of agglomerated bubbles in images. Three different water flow rates (15, 19 and 23 L/min) and gas flow rates (0.8, 1.6 and 2.4 L/min) were employed in the present study. Considering the transient distribution of bubbles inside the mold, five experimental photographs at different times under the same flow conditions were randomly chosen to obtain the mean bubble diameter, and both sides of SEN were analyzed using ImageJ. Effect of bubble size on two-phase flow Water flow pattern inside the mold The influence of injected gas on the water flow pattern inside the mold was investigated by the water model experiment. A small quantity of black ink was injected into the flowing water to trace the flow pattern of water flow inside the mold. Fig. 5(a)–(d) shows the evolution of the flow pattern inside the water model with no gas injection

where the water flow rate was running at 19 L/min. At 1.0 s after the tracer was injected, two jets were straightened rushed out of the SEN ports, as shown in Fig. 5(a). After 1.0 s, as shown in Fig. 5(b), the two jets impinged on the narrow wall. Part of the fluid then flows upward to the top surface forming a recirculating flow forming a recirculating “eye” below the top surface as shown in the figure. Another part of fluid flows downward along this wall, as shown in Fig. 5(c). At 20 s, in Fig. 5(d), the jet in the lower recirculation zone moved deep into the mold forming a large recirculating flow below the jet. Furthermore, in agreement with the findings of previous works (Sánchez-Perez et al., 2003; Liu et al., 2013), an asymmetric flow pattern was also observed during the experiment. Fig. 6(a)–(d) shows similar flow visualization as that shown in Fig. 5 where gas was injected into the water at the flow rate of 1.6 L/min. As depicted, the flow patterns with gas injection exhibited very different characteristics in the upper region of the mold in comparison to the flow without gas injection (see also in Fig. 5). Driven by the bubbles which are subjected to buoyancy force, part of liquid moved up toward the top surface after leaving the SEN port forming a recirculating flow near the SEN. On the other hand, as shown in Fig. 6(b), another part of the liquid continues as a jet impinged on the narrow wall at a slightly higher location. After impinging on the narrow wall, in Fig. 6(c), part of fluid flows then upward forming another upper recirculating zone. When this recirculating flow met the previous recirculating flow near the SEN, two recirculating “eye” were formed below the top surface at the upper roll of the mold. Comparing the flow pattern in Fig. 5(d) and Fig. 6(d), except raising the lower recirculating “eye”, one can observe that flow patterns at the lower recirculation zone in both cases are very similar. Therefore, it is expected that gas bubbles would have insignificant effect on the flow pattern in the lower recirculation zone of the mold. This could be attributed to the fact that most bubbles would float upward and escape from the top surface during traveling with the jet from the SEN port to the narrow wall. Effect of bubble size on gas volume fraction distribution Fig. 7(a)–(d) shows the effect of bubble size on the gas volume fraction distribution inside the mold, which is obtained from the

Fig. 5. Evolution of fluid flow pattern obtain from the water model experiment with no gas injection (a) 0.8 s, (b) 1.5 s, (c) 4 s, and (d) 10 s after tracer was injected.

Fig. 6. Evolution of fluid flow pattern obtain from the water model experiment with gas injection (a) 0.8 s, (b) 1.5 s, (c) 4 s, and (d) 10 s after tracer was injected.

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Fig. 7. Gas volume fraction distribution inside the mold with different bubble sizes (a, b, c and d are 0.5, 1.0, 2.0 and 3.0 mm) and MUSIG model (e).

MS model. For all cases (water–air system), the water flow rate is 19 L/min and the gas flow rate is 1.6 L/min. The bubble size is considered as 0.5, 1.0, 2.0 and 3.0 mm. A comparison between Fig. 7(a)– (d) clearly shows that smaller bubbles exhibited a more dispersed pattern and have a tendency to penetrate further across the mold. Subjected to stronger buoyancy forces, large bubbles show a less dispersed pattern and quickly float towards the water surface. Fig. 7(e) shows the predicted gas volume fraction distribution inside the mold obtained from the MUSIG model which is used to validated against the measured bubble distribution obtained from the water model experiment (see also in Fig. 4). The predicted outlines of the volume fraction are in satisfactory agreement with the measurement of water model experiment. After entering the mold with the water jet, most bubbles were concentrated at the vicinity of the SEN and float upward through the upper recirculation zone and escape from the top surface sequentially. One may also notice that the maximum volume fraction exceed 0.2 in some locations. Meanwhile, some of the small bubbles may penetrate further across the upper recirculation zone and migrated to the middle of wide face. To further assess the MUSIG model predictions, the gas volume fraction with injection of different bubble sizes (using MS model) at the centerline of the top surface along the width of the mold is shown in Fig. 8. In the figure, x-coordinate is the ratio of the distance away from the SEN (w) to the thickness of the mold (t = 0.3 m). The result depicted that the bubbles are dispersed more widely in the upper recirculation zone when the bubble sizes are 0.5 mm and 1.0 mm, especially for 0.5 mm, further close to the narrow wall of the mold.

Fig. 8. Gas volume fraction along line 1 obtained using different bubble sizes and MUSIG model.

With the larger bubble size (i.e. 2.0 mm and 3.0 mm), the distance of bubble penetration becomes shorter due to higher buoyancy. Most of the bubbles were escaped from the top surface near the SEN with the maximum volume fraction of 0.076. When the breakage and coalescence of bubbles are considered using the MUSIG model, a spectrum of bubble sizes is formed just in front of the SEN port. Bubbles of different sizes are subject to lateral migration due to forces acting in lateral direction. Consequently, the lateral migration leads to different

Please cite this article as: Z.Q. Liu et al., Modeling of bubble behaviors and size distribution in a slab continuous casting mold, International Journal of Multiphase Flow (2015), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2015.07.009

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Fig. 10. Predicted horizontal velocities of liquid along line 1 with different bubble sizes and MUSIG model.

Fig. 9. Streamlines of liquid inside the mold without gas injection (a), with different bubble sizes (b, c, d and e are 0.5, 1.0, 2.0 and 3.0 mm) and MUSIG model (f).

penetration depths of small and large bubbles. Large bubbles would move upward to the top surface close to the SEN, but the smaller bubbles would penetrate further across the mold and escape from the top surface some distance away from the SEN. As shown in Fig. 8, results of MUSIG model, there are two peaks of gas volume fraction; it is different from other results of MS model. Effect of bubble size on the fluid flow pattern Flow in the liquid cavity of a continuous casting mold is greatly affected by the argon gas injection. The significant effects of gas flow rate and bubble size on the flow pattern in the mold have been studied in many previous works (Wang et al., 1999; Iguchi and Kashi 2000; Li et al., 2000; Liu et al., 2014). In the current work, both the MS model and the MUSIG model were used to study the effect of bubble size on the fluid flow pattern inside the mold. Fig. 9(a) shows the predicted fluid flow pattern (streamline) without gas injection. From the figure, one can notice that the fluid left the SEN port as a strong jet impinging on the narrow wall of the mold. The water jet was then

split into two vertical streams creating the upper and lower recirculation zone; forming a typical double-roll flow pattern. The influence of the gas bubbles on the fluid flow pattern can be seen in Fig. 9(b) obtained from the MS model assuming a uniform bubble diameter of 0.5 mm. As depicted in the figure, part of fluid moved up toward the top surface after leaving the SEN port, another part of fluid continue as a jet impinge at a slightly higher location on the narrow wall. In order to analyze the effect of bubble size on the fluid flow pattern, Fig. 9(c)–(e) illustrates the fluid predicted flow pattern based on the MS model with the bubble diameter of 1.0 mm, 2.0 mm and 3 mm respectively. With reference to the previous results shown in Fig. 6, it can be seen that smaller bubbles (i.e. 0.5 mm shown in Fig. 9(b)) can penetrate further across the mold and closer to the narrow face and residence inside the model for a longer time and have a greater influence on the liquid flow pattern. Larger bubbles (i.e. 3 mm shown in Fig. 9(e)) have the tendency to leave the mold more quickly and concentrate closer to the SEN, which eventually have less influence on the liquid flow pattern. Unfortunately, all predicted results failed to capture the flow pattern obtained from water model experiment (see also in Fig. 6). Fig. 9(f) shows the flow pattern obtained from the MUSIG model. The flow pattern in the upper recirculation zone shows a satisfactory agreement with the water model experiment. Part of the jet flows upward to the top surface when exiting the SEN port, while another part of fluid continues as a jet impinge on the narrow wall and then forms a recirculation flow. Therefore, it can be concluded that the consideration bubble size distribution in the mold is essential for accurately capture the two-phase flow behavior in the continuous casting process. Fig. 10 shows the comparison of the predicted horizontal velocity profiles at the centerline of the top surface along the width of the mold by the MS and MUSIG models. The positive value of horizontal velocity represents that the direction of velocity is pointing from the SEN to the narrow wall of the mold. In contrast, negative value represents velocities flowing from the narrow wall to the SEN. Without gas injection, the horizontal velocities are all negative indicating that the motion of liquid steel is mostly flowing from the narrow face to the SEN; also refers as “double roll” flow pattern. Meanwhile, one can also notice that there is a clear difference between 0.5 mm bubbles and other sizes bubbles. For 0.5 mm bubbles, the horizontal velocity near the SEN is negative but becomes positive at the vicinity of the narrow. This indicates that there are two jet flows move from the impingement point of jet on the top surface to the SEN and to the narrow wall respectively. However, when the bubble size is larger than 0.5 mm, the horizontal velocity near the SEN is positive and become negative close to the narrow face; showing that there are two jet flows

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Fig. 11. Process of bubbles breakage in the water model experiment.

move from SEN to center and narrow to center (see also Fig. 6(c)). The transition from positive to negative velocity denotes the stagnation/impingement point of two recirculation rolls. With increasing the bubble sizes, the impingement point moves from the narrow wall to the SEN. In case of MUSIG model, the horizontal velocity distribution is similar with the 1.0 mm bubble of MS model. Nonetheless, the predicted impingement point is much closer to the SEN. Bubble breakage and coalescence behavior To understand the complex processes associated with the twophase flow inside the mold, some visualizations and observations derived from the high-speed photography are presented herein. Fig. 11 shows the typical bubble distribution photography in a half width of the mold with the water and gas flow rate of 19 L/min and 1.6 L/min respectively. A clear slug two-phase flow pattern can be observed inside the SEN. Moreover, a swirling fluid jet with strong turbulence kinetic energy was also found at the bottom of SEN. The slug flow would breakup into various sizes of bubbles and escaped by the strong turbulence and swirling motion of the water jet (as shown from the visualization in front of SEN port). Afterwards, some of small bubbles grow into larger bubbles with dynamic equilibrium size due to coalescence and ascend immediately through the upper roll to the top surface. Other bubbles are driven by the drag forces along the jet further into the liquid cavity toward the narrow wall. Due to the shear forces of the water jet, these bubbles may breakup into smaller bubbles in this process. Meanwhile, additional coalescence-growing phenomena may take place and the grown bubbles would float toward the top surface in the center region between the SEN and narrow wall. In this model, the turbulence is the primary mechanism responsible for bubble breakage and coalescence. Fig. 12 shows the contour maps of liquid turbulent dissipation of kinetic energy inside the SEN and mold. One can clearly observe that the turbulent dissipation of kinetic energy is of maximum value at the bottom of the SEN and the core of fluid jet while relatively lower value is located elsewhere. A clear swirling jet was also found at the bottom of the SEN which is in agreement with the observations from the water model experiment (also shown in Fig. 11). The swirling jet is composed of various small eddies which can break the bubbles. And the bubble breakup rate is considered to depend on the frequency of collisions between bubbles and turbulent eddies. The region of strong turbulent dissipation of kinetic energy can be divided into two parts: one part is along the

Fig. 12. Liquid turbulent dissipation distribution inside the SEN and mold.

separated fluid jet due to the buoyancy of bubbles moved up toward the top surface after leaving the SEN port, and another part along the primary fluid jet impinges on the narrow wall. This is corresponding to the result of water model experiment which was shown in Fig. 6(c). Fig. 13 shows the process of bubble coalescence phenomenon observed inside the mold with the water and gas flow rate of 19 L/min and 1.6 L/min respectively. In the first figure, at the beginning of the process, there were two bubbles in the circle position. The two bubbles then move towards each other gradually, as shown in Fig. 13(b) and (c). At 50 ms, as shown in Fig. 13(d), the two bubbles merged together into a larger bubble. However, all the above fundamental experimental observations have not been well modeled in previous studies. Therefore, a successful and specific development of the population balance approach for poly-dispersed bubbly flow in continuous casting mold should be developed. This is the major objective of the current work. Bubble size distribution The predicted bubble Sauter mean diameters distribution inside the SEN and mold using the MUSIG model is shown in Fig. 14, where

Please cite this article as: Z.Q. Liu et al., Modeling of bubble behaviors and size distribution in a slab continuous casting mold, International Journal of Multiphase Flow (2015), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2015.07.009

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Fig. 13. Process of bubbles coalescence in the water model experiment.

Fig. 14. Predicted bubble Sauter mean diameter distribution inside the mold.

the water and gas flow rates were as 19 L/min and 1.6 L/min respectively. From the figure, it can be seen that the bubble diameter at the center of SEN is larger than 2 mm. These large bubbles then breakup into various smaller bubbles at the bottom of the SEN, corresponding to the above results of Fig. 11. According to the adopted bubble breakup model of Luo and Svendsen, bubble breakup can be expected in regions with high turbulent kinetic energy. The predicted maximum values of the turbulent kinetic energy are located at the bottom of the SEN, as shown in Fig. 12. Furthermore, almost all bubbles in front of SEN port are less than 1.2 mm; indicating that the behavior of bubble breakage is dominated at the bottom of the SEN. After escaping from the SEN port, large bubbles rise closer to the SEN due to enhanced buoyancy and drag forces, while smaller bubbles are carried deeper into the mold by the fluid jet. Along the bubbles transport from the SEN port to the top surface, due to the effect of hydrostatic pressure and bubble coalescence, the diameters of bubbles would increase gradually. However, during the process of bubbles transport from the SEN port to the narrow wall of the mold, high shearing force of fluid jet decreasing the sizes of bubbles. In summary, the coalescence of bubbles is dominated during the bubbles rising to the top surface; and the breakage of bubbles is dominated during the bubbles transporting with the jet to the narrow wall. Fig. 15 illustrates the predicted by the MUSIG model and measured by water model bubble mean diameter distributions at a constant gas injection rate (1.6 L/min) with different water flow rates (15, 19 and 21 L/min) along the centerline at 25 mm below the top surface. These results show a satisfactory agreement between predicted bubble Sauter mean diameters and measured mean bubble sizes. Large bubbles are formed close to the SEN where highly concentrated bubbles having greater possibility to coalescence forming bigger bubbles. Smaller bubbles are migrating toward to the narrow wall of the mold, corresponding to the results of water model exper-

Fig. 15. Predicted bubble Sauter mean diameter along line 2 with different water flow rates.

iments, as shown in Fig. 11. When the water flow rate is 19 L/min, the coalescence process of bubbles is prominent between w/t = 0.3 and w/t = 2.5 distance from the SEN. The reason may be that the bubble size of escaping from the SEN port is smaller than 1.2 mm, as shown in Fig. 12. The net effect of bubble coalescence encourages the formation of larger bubbles after flowing out of the SEN ports. As the water flow rate increases, for a given gas flow rate, the bubble sizes in the width of the mold are distributed more uniformly. The maximum sizes of bubble are reduced slightly, and the position of maximum size moves toward to the narrow wall of the mold. For two-phase flow inside the mold, another important operating parameter is the gas flow rate. Argon bubbles alter the flow pattern

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diameter against measurements of water model experiments. With the encouraging results, the MUSIG model can be considered as a practicable population balance approach for modeling the polydispersed bubbly flow inside the slab continuous-casting mold. Acknowledgments This work reported in this paper was funded partly by the major international (regional) cooperation projects of the National Natural Science Foundation of China (Grant no. 51210007); partly by Research Fund for International Young Scientists supported by National Natural Science Foundation of China, (Grant no. 51450110083). References

Fig. 16. Predicted bubble Sauter mean diameter along line 2 with different gas flow rates.

in the upper recirculation zone in proportion to increasing gas fraction, shifting the impingement point and recirculation zones upward. Fig. 16 illustrates the predicted and measured bubble mean diameter distributions at a constant water flow rate (21 L/min) with different gas flow rates (0.8, 1.6 and 2.4 L/min) along the centerline at 25 mm below the top surface. Stronger buoyancy due to the introduction of more bubbles not only changes the flow pattern to a larger extent, but also increases the probability of bubble coalescence. The maximum size of bubbles is located approximately at w/t = 1.25 distance from the SEN when the argon gas flow rate is of 0.8 L/min. As the gas flow rate increases, with the same water flow rate, the maximum sizes of bubble increase gradually while the position of maximum bubble sizes remains almost unchanged. Nevertheless, for all the cases with different water flow rates and gas flow rates, the MUSIG model tends to over-predict the bubble Sauter mean diameter closer to the SEN (about 20–50 mm away from the SEN). Further research works are needed to improve the capability of MUSIG model through developing new breakup and coalescence model. Advancement for the interfacial momentum transfer model may be also required to properly account the effect of surface waves at the top surface. Conclusions An Euler–Euler two-phase model coupled with population balance approach for polydispersed bubbly flow in a slab continuouscasting mold was presented in the current work. Bubbles were distributed into 10 diameter classes through the formulation of a MUSIG model; each of them experienced coalescence and breakage phenomena. An extra contribution in the effective viscosity for the turbulence induced by bubbles was taken into account using the Sato model. A 1/4th scaled water model was established to observe and test the bubble behavior and size distribution in the mold. Considering the expansion of argon gas when it was injected into the high temperature of molten steel, the modified Froude similarity number was used to calculate the cold air flow rate in the water model. Dyeinjection experiments without and with gas injection showed the different evolutions of the transient flow pattern in the liquid pool. Bubble breakage and coalescence behaviors in the water model were captured by a high speed video system. The mean bubble diameter distribution along the width of the mold was analyzed using a digital image processing technique of ImageJ. Predictions by the traditional MS model and MUSIG model are compared against the measurement data of polydispersed bubbly flow in the water model. Close agreements by MUSIG model were achieved for the gas volume fraction, liquid flow pattern, bubble breakage and coalescence phenomena, and local bubble Sauter mean

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