Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup

JMRTEC-1281; No. of Pages 12

ARTICLE IN PRESS j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

Available online at www.sciencedirect.com

www.jmrt.com.br

Original Article

Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup Gujun Chen a , Shengping He b,∗ a b

College of Materials Science and Engineering, Yangtze Normal University, Fuling 408100, China College of Materials Science and Engineering, Chongqing University, Chongqing 400044, China

a r t i c l e

i n f o

a b s t r a c t

Article history:

The single snorkel refining furnace (SSRF) is widely used in secondary refining for the

Received 8 December 2019

miniaturization of special steel. In this study, a computational fluid dynamics (CFD) model,

Accepted 8 January 2020

coupled with population balance equations (PBEs), is built to describe the argon–molten steel

Available online xxx

two-phase flow in an industrial SSRF. In this simulation, bubble expansion due to the sharp variation in hydrostatic pressure and bubble coalescence and breakup are considered for the

Keywords:

first time. The numerical results are basically consistent with the experimental observations

CFD

and calculated values published in the literature for the mixing behavior and local velocity

PBEs

in the physical simulation and the Sauter mean bubble diameter and circulation flow rate in

SSRF

the industrial SSRF. The simulated results indicate that the width of bubble plume increases

Coalescence

as the bubbles rise, and larger bubbles are formed in the center of the plume, while smaller

Breakup

bubbles are generated at its outer boundary. Meanwhile, the Sauter mean bubble diameter

Bubble

decreases gradually with rise height until reaching an equilibrium value. In addition, the circulation flow rate of molten steel is relatively independent of the initial bubble diameter. Finally, in the range of explored argon flow rates, the circulation flow rate and the refining efficiency can be enhanced as the argon flow rate is increased. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.

Introduction

Many metallurgical reactors, such as ladle [1], Ruhrstahl–Heraeus [2] and tundish [3,4], are involved to remove the non-metallic inclusions and the dissolved carbon, impurities and gases from the molten steel as well as

∗

adjust the alloy components and temperature of molten steel during the secondary refining and casting processes. The Ruhrstahl–Heraeus equipment has been extensively applied for the removal of carbon in the refining of ultra-low carbon steel [2]. The Ruhrstahl–Heraeus consists of a ladle, a vacuum chamber, and two snorkels, and the argon is blown to enable the molten steel to circulate under vacuum conditions, as illustrated in Fig. 1(a). Although large Ruhrstahl–Heraeus equipments have achieved great success, Ruhrstahl–Heraeus less than 80 tons have suffered many problems, such as low refining efficiency, severe erosion, and slag bonding [5]. With

Corresponding author. E-mail: [email protected] (S. He). https://doi.org/10.1016/j.jmrt.2020.01.026 2238-7854/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

JMRTEC-1281; No. of Pages 12

2

ARTICLE IN PRESS j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

Nomenclature q A facet area vector (m2 ) Coa Bre Bi , Bi birth rate of bubbles due to the coalescence and breakup (1/(m3 ·s)) coalescence rate (m3 /s) cij cf increase coefficient of the surface area drag coefficient CD C␮ , C1␧ , C2␧ , C3␧ constants tracer concentration C dimensionless constant C1 minimum bubble diameter (m) C2 d0 , d32 , de initial diameter, Sauter mean diameter, and equilibrium diameter of bubbles (m) bubble diameter, and plug diameter (m) di , dp Dt turbulent dispersion coefficient Bre , D death rate of bubble due to the coalescence DCoa i i and breakup (1/(m3 ·s) ) F D , F TD drag force and turbulent dispersion force (N/m3 ) production of turbulent kinetic energy (m2 /s3 ) Gk,l turbulent kinetic energy of liquid (m2 /s2 ) kl M number of bubble size classes molecular weight of argon Mg number concentration of bubbles (1/m3 ) ni pressure, pressure of vacuum chamber (N/m2 ) p, Pvac coalescence efficiency Pij Qg , Ql argon flow rate, circulation flow rate of molten steel (kg/s) R ratio factor gas constant (J/(mol·K) ) R Reynolds number Re argon temperature (K) Tg l g, u velocity of gas, and liquid (m/s) u u¯ ij characteristic velocity (m/s) bubble volume (m3 ) Vi Wekj Weber number Greek Letters ˛l , ˛g volume fraction of liquid and gas daughter bubble size distribution function ˇij turbulent kinetic energy dissipation rate of liqεl uid (m2 /s3 ) Kolmogorov lengthscale (m)  size of turbulent eddy (m)  l , t molecular viscosity and turbulent viscosity of molten steel (kg/(m·s)) dimensionless eddy size  g , l density of gas and liquid (kg/m3 ) dispersion Prandtl number  l surface tension of molten steel (N/m) constants k, ␧ breakup frequency (1/s) ˝ij kj bubble redistribution coefficient collision frequency (m3 /s) ωij Subscripts G gas phase liquid phase l

demand increasing for the miniaturization of special steel production in China at present, the single snorkel refining furnace (SSRF) has received considerable attention owing to the high refining efficiency and long-life cycle of vacuum chambers [6,7]. As illustrated in Fig. 1(b), an SSRF is equipped with only one large snorkel, and argon is blown through the bottom of the ladle. The SSRF has been used to eliminate the carbon, sulfur and non-metallic inclusions from molten steel [8]. The implementation of these refining functions depends on the multiphase flow in the SSRF. To improve the refining efficiency, it is essential to elucidate the mechanism of the flow and mixing behaviors during the SSRF process. Because it is neither economical nor practical to study these behaviors through industrial testing, computational fluid dynamics (CFD) models are usually adopted to understand them in the SSRF, as summarized in Table 1. Studies based on the mixture [9,10] and Eulerian [11–14] models focus on the argon–molten steel twophase flow, while studies based on the coupling of volume of fluid (VOF) and discrete phase model (DPM) [15,16] prefer to focus on the sharp interface among phases. Using these established models, the effects of gas flow rate, gas injection position, snorkel diameter and snorkel immersing depth on multiphase flow were investigated, and the related parameters were optimized. Obviously, the simulated results were very useful for understanding the flow and mixing behaviors in the SSRF. In the SSRF, argon bubbles expand owing to the sharp drop in static pressure as the bubbles rise in the molten steel. Simultaneously, the collisions between the bubbles and the interaction between the bubbles and the turbulent molten steel lead to bubble coalescence and breakup. These physical phenomena result in a wide bubble size distribution (BSD) and thereby determine the flow of molten steel. Nonetheless, in the works of Geng et al. [9,10], Mondal et al. [11], Yang et al. [12], Qi et al. [13], and Chen et al. [14], only a single bubble size was employed without considering any of the physical bubble phenomena mentioned above; in the works of Dai et al. [15,16], only the expansion of bubbles owing to static pressure drop was considered. That is, the coalescence and breakup of the bubbles were completely ignored in all the previous models available in the literature. Therefore, detailed information about the bubble expansion, breakup and coalescence should be considered simultaneously, and more in-depth studies are essential to increase the understanding of the BSD in the industrial SSRF. Recently, a three-dimensional CFD–population balance model (PBM) coupling received considerable attention for modeling the gas−liquid two-phase flow in the chemical, petrochemical, mineral, and metallurgical industries [17–20]. This coupled model can successfully describe the BSD and thereby the flow of liquid. Therefore, in this study, aiming to remedy the drawback of previous models, a CFD−PBM coupled model is developed to investigate the argon–molten steel two-phase flow in the SSRF by simultaneously considering the expansion, coalescence, and breakup of bubbles. Then, based on the established mathematical model, the BSD and the flow of molten steel in the SSRF are examined, and the effects of initial bubble diameter and argon flow rate are investigated.

Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

ARTICLE IN PRESS

JMRTEC-1281; No. of Pages 12

3

j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

Fig. 1 – Schematic of the (a) Ruhrstahl–Heraeus, and (b) SSRF.

Table 1 – CFD-based studies of the flow and mixing behaviors in the SSRF. Year

First author

2012 2012 2015,2017 2019 2019 2020

2.

Mondal Yang Geng Qi Dai Chen

CFD models

Bubble behaviors

Eulerian model Eulerian model Mixture model Eulerian model VOF − DPM model Eulerian model

In the combination of CFD and PBM, the CFD was used to couple the molten steel (liquid phase) and the argon bubbles (gas phase), and the PBM was used to take the bubble expansion, coalescence, and breakup into consideration. The assumptions were summarized as follows. 1) The top slag phase was not considered, and free surfaces of the liquid were presumed to be flat [9,10]. 2) The temperatures of the liquid and gas phases were kept at 1873 K [13,14]. 3) The liquid phase was treated as an incompressible fluid, while gas phase was supposed to be an ideal gas. In accordance with the ideal gas law, the gas density, g , was defined to consider the bubble expansion phenomenon as a result of the reduction in liquid static pressure [15,16].

g =

[Pvac + l g (H − z)] Mg RTg

(1)

where Pvac is the vacuum chamber pressure, Tg is the gas temperature, l is the liquid density, R is the gas constant, and Mg is the gas molecular weight. H and z are the distances from the bottom of the ladle to the surface of the vacuum chamber and local grid cell, respectively.

Coalescence and breakup

× × × × √

× × × × × ×

×

2.1.

Multifluid model description

Expansion

Reference

[11] [12] [9,10] [13] [15,16] [14]

CFD model

The mass and momentum conservation equations were solved separately for liquid phase (m = l) and gas phase (m = g) using a two-fluid model (Table 2) [21]. The drag and turbulence dispersion forces were considered for the interfacial momentum transfer. The k-ε dispersed turbulence model was used for the prediction of the liquid turbulence, and the gas turbulence is obtained using the Tchen theory [22]. This turbulence model is applicable when there is clearly one primary continuous phase and the rest are dispersed secondary phases, and it has been widely selected by many researchers to simulate the argon–molten steel two-phase flow inside the metallurgical reactor [12,14].

2.2.

PBM

The bubble population behaviors are characterized by the population balance equations (PBEs), which reflect the macroscopic bubble transport phenomenon and the bubble coalescence and breakup phenomena.

  ∂n (Vi )  g n (Vi ) = S (Vi ) +∇ · u ∂t

(2)

One can observe that the PBE is given as a function of bubble number density, n(Vi ), where Vi is the bubble volume. S(Vi ) is the source term that depends on taking the coalescence and breakup mechanisms into consideration.

Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

ARTICLE IN PRESS

JMRTEC-1281; No. of Pages 12

4

j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

Table 2 – . Equations of CFD used in the SSRF. Continuity and Momentum Conservation Equations m) = (˛m m ) + ∇ · (˛m m u 0 ∂  m ) + ∇ · (˛m m u mu  )= (˛m m u ∂t   m   m + (∇ u  m )T −˛m ∇p + ∇ · ˛m (l + t ) ∇ u + ˛m m g + F m F l = −F g = F D + F TD

 m are volume fraction, density and velocity of ˛m (˛l +˛g = 1), m and u liquid (m=l) and gas (m=g), respectively.

∂ ∂t

l and t are the molecular viscosity and turbulent viscosity, respectively. F m is the interfacial force between two phases.

Interphase Forces F D = 

 →u − →u  (u − u )  g l g l CD =   0.687

CD is drag coefficient, which is calculated from the model of Schiller and Naumann [23]. Re is Reynolds number for gas bubbles. d32 is the Sauter mean diameter of gas bubbles, as calculated by Eq. 20 in Section 2.3. Dt is turbulent dispersion coefficient [24].  is dispersion Prandtl number, can be set 0.75 [25].

3˛g ˛l l 4d32

CD

max 24 1 + 0.15Re F TD =  −

CD

 ∇˛g

3˛g ˛l l 4d32 ∇˛l ˛l

−

˛g



/Re, 0.44

  →u − →u  Dt  g l 



·

Turbulence Model k2

t = l C εl l ∂  k)= (˛l l kl ) + ∇ · (˛l l u ∂t    l l t ∇ · ˛l l + ∇kl + ˛l Gk,l − ˛l l εl + ˛l l ˘kl k ∂  ε)= (˛l l εl ) + ∇ · (˛l l u ∂t    l l εl t ∇ · ˛l l + ∇εl + ˛l (C1ε Gk,l − C2ε l εl ) + ˛l l ˘εl ε kl Species Transport Equation ∂ ul l C) = ∂t (l C) + ∇ · ( ∇·

 +t l Sct

kl and εl are the turbulence kinetic energy and dissipation rate of liquid, respectively. Gk,l is the production of turbulent kinetic energy due to the mean velocity gradient of liquid [26,27]. ˘kl [28] and˘εl [29] denotes the influence of gas turbulence on liquid turbulence. Constants: C␮ = 0.09, C1␧ = 1.44, C2␧ = 1.92, C3␧ = 1.2,  k = 1.0,  ␧ = 1.3 [27].



Sct is turbulent Schmidt number, which is set to 0.7. C is the concentration of the tracer.

∇(C)

In this work, the PBE is solved using the discrete method developed by Hounslow et al. [30], which is based on representing the continuous BSD in terms of a set of discrete size classes. In Fluent software, the PBE is written in terms of the volume fraction of bubble size class i, ˛g,i , i ∈ [1, M].



 Bre

∂  g ˛g, i ) = g Vi BCoa − DCoa + BBre − Di (g ˛g, i ) + ∇ · (g u i i i ∂t

= DCoa i

M

cij ni nj

(7)

˝i ˇij nj

(8)

j=1

BBre = i

M j=i

(3)

DBre = ˝i ni i where ˛g,i is the volume fraction of bubble size class i, which is calculated as



Vi+1

˛g, i = n (Vi ) Vi = Vi

n(V)dV

where ıkj in Eq. 6 is set to 0 (i = / j) or 1 (i = j). The bubble redistribution coefficient, jk , in Eq. 6 can be calculated by

(4)

Vi

The sum of all the ˛g,i equals to the total gas volume fraction, ˛g .

˛g =

M

(5)

˛g, i

i=1

In Eq. 3,BCoa and DCoa are the bubble birth rate and death i i rate as a result of coalescence, respectively. BBre and DBre are i i the bubble birth rate and death rate of bubbles a result of breakup, respectively.

BCoa = i

M M  k=1 j=1

1−



1 c n n ı 2 kj kj kj k j

(6)

(9)

kj =

2.2.1.

⎧ (Vk + Vj ) − Vi-1 ⎪ Vi−1 < Vk + Vj < Vi ⎪ ⎪ Vi − Vi-1 ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

Vi+1 − (Vk + Vj ) Vi+1 − Vi

0

Vi < Vk + Vj < Vi+1

(10)

otherwise

Coalescence

The general coalescence rate, ckj , in Eq. 6 indicates the bubble formation rate owing to the collisions between bubbles of volume Vk and volume Vj, and it is calculated as ckj = ωkj Pkj

(11)

where ωij and Pij are the collision frequency and coalescence efficiency, respectively. As reviewed by Liao et al. [31], there are five kinds of mechanisms that facilitate collisions among the bubbles in a

Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

ARTICLE IN PRESS

JMRTEC-1281; No. of Pages 12

5

j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

gas-stirred turbulent flow. In gas-stirred secondary steelmaking furnaces, the bubble size [32] is evidently larger than the



1/4

) [33]. In this case, the Kolmogorov microscale ( = l3 /εl collision is mainly induced by the turbulent random motion of bubbles according to Wang et al. [34] Therefore, the expression proposed by Luo [35], which is based on the theory of isotropic turbulence, is adopted to calculate the collision frequency in Eq. 11 as follows.

2

 dk + dj nk nj u¯ kj 4

ωkj =

(12) Fig. 2 – Solution process of the combination of CFD and PBM.

where dk and dj are the bubble diameter of size class k and j, respectively, and the characteristic velocity of collision of two bubbles, u¯ kj , can be calculated as

u¯ kj =

1/3 1.43εl



2/3 dk

2/3 + dj

(13)

where the dimensionless eddy size =/di and  min =min /di . The daughter BSD function, ˇij , in Eq. 8 is given by

1

⎧  ⎫   2    3  ⎪ ⎪ ⎪ ⎪ 0.75 1 + d /d 1 + d /d k j k j ⎨ ⎬  Pkj = exp − We  kj  3 ⎪ ⎪ ⎪ ⎪ g /l + 1/2 1 + dk /dj ⎩ ⎭ (14)

where the Weber number, Wekj , is defined as



Wekj =

l dk u¯ kj

ˇij =

2/3 5/3 11/3 d  l i

2.047l ε

 exp

d (17)

 −

12cf l 2.047l ε

2/3 5/3 11/3 d  l i

ddf

where cf is defined as the coefficient of increase in surface area given by (18)

where f = Vj/Vi .

Breakup

Liao and Lucas [36] summarized four kinds of bubble breakup mechanisms in turbulent dispersions, and the turbulent fluctuation and collision mechanism played the dominant effect on bubble breakup. Therefore, only the breakup model developed by Luo et al. [37], which was also based on the theory of isotropic turbulence, was used. Acording to this model, the breakup takes place when the turbulent eddy  with sufficient energy hits a bubble of diameter di , and the breakup frequency, ˝i , in Eqs. 8 and 9 was modeled as follows.



˝i = 0.9238 1 − ˛g

−

(1+)2  11/3

12cf l

(15)

where  l is the surface tension of the molten steel.

exp

min

1 1

 −

cf = f 2/3 + (1 − f )2/3 − 1

2

l



 exp

0 min

2.3.

2.2.2.

(1+)2  11/3

2

The coalescence efficiency in Eq. 11 is described as [35]





εl

1/3 1

d2i

12cf l 2/3 5/3 11/3 di 

2.047l εl

(1 + )  11/3

The solution process of the CFD and PBM combination developed in the present study is schematically structured in Fig. 2. The volume fraction and velocity of gas phase and the turbulent energy dissipation rate and eddy size of the liquid phase are first calculated by CFD at given initial conditions. Then, these variables are delivered to PBM to describe the coalescence and breakup of bubbles. After the PBEs are solved, the Sauter mean diameter of the bubbles is returned to CFD to modify the drag and turbulence dispersion forces and update the related variables for the next iteration. In the discrete method, the volume coordinates for the bubbles are discretized into M size classes using a geometric factor suggested by Lister et al. [38] Vi+1 /Vi = 2r

2

min



d

CFD and PBM combination

(16)

(19)

where r is the ratio factor. The Sauter mean diameter of bubbles, d32 , in the equations of drag and turbulence dispersion forces is an average of bubble size. It is defined as the mean diameter with the same

Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

ARTICLE IN PRESS

JMRTEC-1281; No. of Pages 12

6

j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

Table 3 – Dimensions, material properties and other parameters of the SSRF. Parameters

Water model [15]

Actual SSRF [16]

Mass of liquid, ton Up/down diameter of ladle, m Height of ladle, m Inside diameter of vacuum chamber, m Inside diameter of snorkel, m Length of snorkel, m Snorkel immersed depth, m Diameter of plug, m Pressure of vacuum chamber, Pa Density of liquid/gas at 300 K and 0.1 MPa, kg/m3 Viscosity of liquid, Pa·s Interfacial tension between liquid and gas, N/m Temperature, K

0.37 0.817/0.728

70 2.45/2.184

1.02 0.39

3.06 1.3

0.3

1.0

0.533 0.13

1.6 0.4

0.033 96600

0.1 2000

998.2/1.25

7020/1.623

8.5 × 10−4 0.072

0.006 1.823

300

1873

20, and the flow courant number and relaxation factors were taken carefully to ensure convergence. The definition of the circulation flow rate of the SSRF in the present study was the same as that used in the work of Dai et al. [15,16] Both the upflow and downflow of molten steel coexisted in the large snorkel. The mass flow rate through Up the cross section of the snorkel bottom with positive, Ql , Down , vertical steel velocity was calculated as or negative, Ql follows.

⎧ N-Up   ⎪ Up ⎪ q  l,q · A ⎪ Q = l u ⎪ l ⎨ q=1

N−Down  ⎪  ⎪ Down ⎪ q  l,q · A Q = l u ⎪ ⎩ l

M

ni (di )3 /

i=1

2.4.

ni (di )2

(20)

The dimensions, material properties and other parameters of the SSRF are listed in Table 3. Fig. 3(a) illustrates the mesh and the boundary conditions of the simulation related to the mass flow inlet for argon injection, free surface of the molten steel, no-slip wall at the refractory, and symmetry. Fig. 3(b) shows the points and lines for monitoring the fluid flow. The initial diameter of the argon bubbles, d0 , at the plug exit point was calculated using an experimental formula from Sano and Mori [39]. This formula was applicable for describing the bubble size in molten steel for both low and high gas flow rates and hence was widely used in argon–molten steel flow [40,41].

d0 =

6l dp l g

2



+ 0.0248



0.867 Qg2 dp

1/6 (21)

where dp is the plug diameter and Qg is the argon flow rate.

2.5.

 Up   Down  Q  + Q  /2 l

l

3.

Results and discussion

3.1.

Independence analysis

(23)

i=1

Boundary and initial conditions



<0

 q is  l,q is the facet velocity vector of molten steel, A where u the the facet area vector, N-Up and N-Down are the number of faces with positive or negative vertical steel velocity, respectively. The circulation flow rate of the molten steel in the SSRF, Ql , is defined as

ratio of volume to surface area as the entire ensemble, and it is calculated as M

(22)  Zl,q u

q=1

Ql =

d32 =

 Zl,q > 0 u

Computational procedure

The computations based on the CFD − PBM combination were carried out with the CFD software ANSYS Fluent. The pressure-based Coupled algorithm was selected to couple the pressure and velocity to improve the convergence. The maximum iterations per time step was set to a default value of

The computational grid and time step have an important effect on bubble expansion, breakup, and coalescence and thereby on the transient flow of molten steel in the SSRF. Computations were carried out using various numbers of grids and time steps to find a solution independent of the grid and time step. Fig. 4(a) illustrates the effect of grid numbers on the velocity of molten steel. It is observed that the calculated velocities for 26,000 and 82,000 grids deviate from that for 205,000 grids, while the refinement from 205,000 to 690,000 grids produces almost the same results. Therefore, a total of 205,000 grids was used in subsequent simulations to save the CPU time. Fig. 4(b) illustrates the effect of time steps on the velocity of molten steel with 205,000 grids. It is seen that the calculated velocities for 0.01, 0.05, and 0.5 s deviate from those calculated for 0.001 and 0.005 s, and that the predicted velocity does not depend on the time step when the time step is 0.005 s or small. Considering the CPU time, 0.005 s was selected for the rest of the study. The discretization of bubble volume coordinates has an important influence on the predicted results, and the discretization is determined by the ratio factor, r, in Eq.19. Evidently, the number of bubble size classes increases with the decrease in the ratio factor. In the study of Huang et al. [34], Bart et al. [42], and Sattar et al. [17], the value of the ratio factor was set to 0.6, 0.5, and 1.0, respectively, for different gas–liquid systems. Therefore, the influence of the ratio factor on simulated results should be investigated for the argon–molten steel system. Fig. 5 illustrates the influence of the ratio factor on the velocity of molten steel along the vertical line (X = 0.25 m) marked in Fig. 3(b). It is seen that decreasing the ratio factor

Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

JMRTEC-1281; No. of Pages 12

ARTICLE IN PRESS j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

7

Fig. 3 – (a) Mesh and boundary conditions, and (b) specified points and lines on the symmetry plane (Y = 0 m) for monitoring.

Fig. 4 – Influence of (a) grid numbers and (b) time step on the calculated velocity of molten steel along the vertical line (X = 0.25 m) marked in Fig. 3(b) (Qg = 100 NL/min). from 1.5 to 0.5 has no influence on the simulated result. The ratio factor should be set to 1.5 to save computational time because fewer PBEs are solved at this higher ratio. However, the initial bubble diameter is heavily dependent on the argon flow rate according to Eq. 21. Thus, the number of bubble size classes should be high enough to cover all the initial diameters at different argon flow rates. Therefore, the ratio factor was set to 0.5 in the present model.

3.2.

Comparison with water model experiment

As the operation of an industrial SSRF is complex and dangerous, measuring the velocity of molten steel or the diameter of argon bubbles is difficult to achieve. Therefore, all the mathematical models reported in the literature, without exception, are validated by the velocity or mixing time of water measured in a water model experiment. Fig. 6 illustrates the change of the tracer concentration ratio, C(t)/C(∞), with time predicted in the present model, accompanied by the result measured by Dai et al. [15] C(t) is the tracer concentration at the monitoring point under different time, and C(∞) is the tracer concentration after sufficient mixing. The injection location of the tracer and the monitoring point are shown in Fig. 3(b). The tracer dimensionless concentration increases suddenly to a peak, and then decreases gradually. After that, this oscilla-

tion phenomenon slightly repeats several times. Evidently, the simulated results of response time, peak time, and the variation tendency of the dimensionless concentration are consistent with those of the experiments. In addition, it is worth mentioning that the measured dimensionless concentration peak is only 70% of the predicted value, as shown in Fig. 6. It is probably due to that the complete addition of tracer may take some time in the experiment, while it is added instantaneously through the initialization during the simulating process. Dai et al. [15] measured the local velocity of water via a digital differential manometer when the argon flow rate was 10 and 20 N L/min. The measured velocities at points A and B are given in Fig. 7, along with the velocity contours on the symmetry plane (Y = 0 m) calculated in the present model. It can be clearly observed that the calculated velocity agrees well with the measured velocity.

3.3.

Flow field

When bubbles rise, the expansion due to the pressure reduction, the coalescence and breakup phenomena due to the collision between bubbles, and the interaction between bubbles and the molten steel turbulence jointly cause variation in

Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

JMRTEC-1281; No. of Pages 12

8

ARTICLE IN PRESS j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

Fig. 5 – Influence of ratio factor on calculated velocity of molten steel (Qg = 100 NL/min).

Fig. 6 – Variation of tracer concentration ratio with time (Qg = 15 NL/min).

the BSD and thereby determine the transient flow of molten steel. Fig. 8 shows the calculated volume fraction and Sauter mean diameter of bubbles on the symmetry plane of the SSRF when the argon flow rate is 300 N L/min. It can be seen in Fig. 8(a) that the bubble plume slightly inclines to the right during the initial rising of the bubbles, and then it arrives at the vacuum chamber along the right wall of the immerged snorkel. Meanwhile, the width of the bubble plume area increases markedly as the bubbles rise owing to the influence of the turbulent dissipation of the molten steel. The shape of bubble plume calculated in this work is similar to the results reported by Qi et al. [13] and Chen et al. [14] But there also exists a different feature: the volume fraction of bubbles continues to reduce all the way to the surface of molten steel in their studies [13,14], while it reduces initially and then increases near the vacuum chamber. And this is due to that the bubble expansion caused by sharp variation in hydrostatic pressure is considered in the present model, while it is ignored in their studies. According to Eq. 21, larger bubbles will be produced at a high argon flow rate. At 300 N L/min, the initial bubble diam-

eter at the point of exit of the plug is 0.085 m, which is much larger in diameter than that can be existed in the fully developed bubble plume above the plug exit. Thus, bubble breakup is more dominant than coalescence and expansion during the initial stage of rising, leading to the sharp reduction in bubble mean diameter, as can be seen in Fig. 8(b). Then, the mean diameter continues to decrease slightly and finally trends to be an equilibrium value (about 0.03 m), and this is attributed to the fact that the breakup rate is close to the sum of the coalescence and expansion rates at the later stage of rising. The variation in bubble mean diameter with vertical position predicted in the present model is the same as the experimental observation made by Anagbo et al. [43], in which the mean diameter continuously reduced as the bubbles rise in the liquid. In addition, it also can be observed in Fig. 8(b) that, larger bubbles are formed in the center region of the plume as a result of a large number of collisions between the bubbles, which promotes coalescence. At the outer boundary of the plume, the bubble density is small, and hence collisions between the bubbles are few. Thus, the bubbles at the outer boundary tend to break up due to the turbulence of molten steel, resulting in smaller bubbles.

Fig. 7 – Comparison of calculated velocity contours with results measured in the water model: (a) 10 N L/min, and (b) 20 N L/min. Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

JMRTEC-1281; No. of Pages 12

ARTICLE IN PRESS j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

9

Fig. 8 – Predicted volume fraction and Sauter mean diameter of bubbles on the symmetry plane of the SSRF (Qg = 300 N L/min).

Fig. 9 – Predicted (a) mean diameter of bubbles, and (b) volume fraction of bubbles and turbulent kinetic energy dissipation rate of molten steel along the vertical line (X = 0.25 m) marked in Fig. 3(b) (Qg = 300 N L/min).

In the literature, many correlations have been suggested to calculate the equilibrium diameter (local mean diameter) of bubbles, de , that are dispersed in a turbulent flow field [44], and the most frequently used correlation is that given by Calderbank [45].

de = C1

 (l /l )0.6 ˛g

(εl )

0.4

+ C2

(24)

where C1 has a constant value of 4.15 (dimensionless), while C2 refers to the minimum bubble diameter (about 0.012 m). The equilibrium diameter is the diameter a bubble will achieve if it resides sufficiently long under the same flow conditions. The calculated equilibrium diameter of bubbles according to Eq. 24 is shown in Fig. 9(a) (blue squares), accompanied by the simulated Sauter mean diameter in the present work (black triangles). The volume fraction of bubbles, ˛g , and turbulent kinetic energy dissipation rate of molten steel, εl , in Eq. 24 are shown in Fig. 9(b). It can be noticed from Fig. 9 that the flow conditions (˛g , εl , etc.) vary dramatically at the initial stage of rising, and the residence time of the bubbles is not sufficient

to achieve an equilibrium value for bubble diameter. Thus, the Sauter mean bubble diameter calculated in the present work is much larger than its equilibrium value calculated in Eq. 24. At a later stage of rising, the flow conditions change slightly and the bubbles gradually approach their equilibrium states. Thus, the discrepancy of mean bubble diameter calculated in this work and that predicted by Eq. 24 is quite small. The above comparison also indicates the validity of the present model. Fig. 10 shows the calculated velocity vectors of molten steel when the argon flow rate is 300 N L/min. Clearly, the molten steel is dragged into the vacuum chamber by bubbles along the right wall of the snorkel. Then, bulk of the molten steel returns to the ladle along the left wall of the snorkel and begins to circulate again.

3.4.

Effect of initial bubble diameter

Eq. 21, suggested by Sano and Mori [39], was used for the calculation of initial bubble diameter in this study, while it is often debated in the literature. Therefore, the influence of initial bubble diameter on the Sauter mean bubble diameter and

Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

JMRTEC-1281; No. of Pages 12

10

ARTICLE IN PRESS j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

Fig. 12 – Variation in circulation flow rate of molten steel with initial bubble diameter in the SSRF (Qg = 300 N L/min).

Fig. 10 – Calculated velocity of molten steel on the symmetry plane of the SSRF (Qg = 300 N L/min).

process. Fig. 12 illustrates the effect of initial bubble diameter on predicted circulation flow rate. Evidently, the circulation flow rate of molten steel reduces slightly with increasing the initial bubble diameter from 0.015 to 0.108 m.

3.5. the velocity of molten steel were investigated, as shown in Fig. 11. As seen in Fig. 11(a) that the bubbles with a large initial diameter tend to break up, while the bubbles with small initial diameter tend to coalesce as they rise. As a result, the Sauter mean diameter of all cases tends toward the equilibrium range of 0.03–0.04 m. In addition, it is seen in Fig. 11(b) that there are differences in the velocity of molten steel when the vertical height is less than about 2.0 m as a result of the large differences in Sauter mean bubble diameter. When the vertical height is greater than about 2.0 m, varying the initial bubble diameter from 0.015 to 0.108 m has no influence on the velocity of the molten steel due to the slight difference in Sauter mean bubble diameter. It is generally recognized that refining efficiency increases with the increase of the circulation flow rate during the SSRF

Effect of Argon flow rate

The argon flow rate is usually adjusted to improve refining efficiency during the SSRF process. Fig. 13 shows the change of Sauter mean bubble diameter at various argon flow rates. According to Eq. 21, the initial bubble diameter increases with increasing the argon flow rate. When the argon flow rate is high, the turbulence of the molten steel is correspondingly stronger, facilitating the breakup of larger bubbles, as found by Zhang and Taniguchi [46]. Therefore, the rate of reduction in bubble diameter at a high argon flow rate is much faster than that for a low argon flow rate during the initial stage of rising. At a later stage of rising, the equilibrium bubble diameter for a high argon flow rate is less than that for low argon flow rate. Fig. 14 shows the variation of circulation flow rate with argon flow rate calculated in the present model, along with the results calculated by Dai et al. [16] Clearly, the circulation flow

Fig. 11 – Effect of initial bubble diameter on (a) Sauter mean diameter, and (b) steel velocity along the vertical line (X = 0.25 m) marked in Fig. 3(b) (Qg = 300 N L/min). Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

JMRTEC-1281; No. of Pages 12

ARTICLE IN PRESS j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

Fig. 13 – Predicted Sauter mean bubble diameter along the vertical line (X = 0.25 m) marked in Fig. 3(b) for various argon flow rates.

11

process and was verified by experimental data and calculated values from the literature. Bubble expansion, coalescence, and breakup phenomena were simultaneously considered. The BSD and fluid flow in the SSRF were presented, and the effects of the initial bubble diameter and argon flow rate were clarified. The conclusions can be summarized as follows: (1) Simulated results were basically consistent with the experimental data and calculated values in the literature, indicating that this coupled model can predict the argon–molten steel flow in the SSRF. (2) The spread of the bubble plume increases as the bubbles rise, and larger bubbles are formed in the center of the plume, while smaller bubbles are generated at its outer boundary. In addition, the Sauter mean diameter of the bubbles decreases gradually with rise height, and it eventually decreases to an equilibrium value. (3) The Sauter mean bubble diameter and the velocity of molten steel are closely related to the initial bubble diameter in the initial stage of rising, while they are irrelevant to the initial diameter in the later stage of rising. Meanwhile, the initial bubble diameter has little influence on the circulation flow rate of molten steel. (4) With the increase of the argon flow rate, the initial bubble diameter increases, while the ultimate Sauter mean diameter of the bubbles decreases. When the argon flow rate is high, the circulation flow rate of the molten steel is correspondingly high, which facilitates the refining process.

Conflicts of interest The authors declare no conflicts of interest.

Acknowledgments

Fig. 14 – Variation of circulation flow rate of molten steel with argon flow rate in the SSRF.

rate of the SSRF increases significantly with initially increasing the argon flow rate, while this tendency becomes weaker at a high argon flow rate. As can also be noticed from Fig. 14 that the circulation flow rate calculated in the present model is close to the value calculated by Dai et al. [16], with a deviation of 6–16%, possibly for the following reasons. First, in the work of Dai et al. [16], the DPM was adopted to couple the bubbles and molten steel. Thus, the influence of gas volume on the liquid flow was not considered because the gas bubbles did not occupy any fraction of the liquid in the DPM. Second, their work did not consider the coalescence and breakup phenomena, while they are considered in the present model.

4.

Conclusions

A CFD–PBM coupled method was established to describe the argon–molten steel two-phase flow during the industrial SSRF

The authors wish to express thanks to the financial supports of the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201801413), and the Science and Technology Program of Fuling Science and Technology Commission (Grant No. FLKJ, 2018BBA3043), and the Guangxi Natural Science Foundation under Grant (Grant No. 2017GXNSFBA198128).

references

[1] Mantripragada VT, Sarkar S. Wall stresses in dual bottom purged steel making ladles. Chem Eng Res Des 2018;139:335–45. [2] de Melo P, Peixoto J, Galante G, Loiola B, da Silva C, da Silva I, et al. The influence of flow asymmetry on refractory erosion in the vacuum chamber of a RH degasser. J Mater Res Technol 2019;8(5):3764–71. [3] de Sousa Rocha JR, de Souza EEB, Marcondes F, de Castro JA. Modeling and computational simulation of fluid flow, heat transfer and inclusions trajectories in a tundish of a steel continuous casting machine. J Mater Res Technol 2019;8(5):4209–20. [4] Fang Q, Zhang H, Luo R, Liu C, Wang Y, Ni H. Optimization of flow, heat transfer and inclusion removal behaviors in an odd multistrand bloom casting tundish. J Mater Res Technol 2019, http://dx.doi.org/10.1016/j.jmrt.2019.10.064.

Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026

JMRTEC-1281; No. of Pages 12

12

ARTICLE IN PRESS j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx

[5] Rui Q, Jiang F, Ma Z, You Z, Cheng G, Zhang J. Effect of elliptical snorkel on the decarburization rate in single snorkel refining furnace. Steel Res Int 2013;84(2):192–7. [6] Chang S, Wu L, Guo J, Pan Y, He F. Industrial investigation of decarburization and desulphurization behaviour of 120 t new single snorkel degasser. Ironmak Steelmak 2019, http://dx.doi.org/10.1080/03019233.2019.1580029. [7] You Z, Cheng G, Wang X, Qin Z, Tian J, Zhang J. Mathematical model for decarburization of ultra-low carbon steel in single snorkel refining furnace. Metall Mater Trans B 2015;46(1):459–72. [8] Duan J, Zhang Y, Yang X, Cheng G, Zhang J. Analysis of commercial production test on metallurgical functions of an 80 t single snorkel vacuum refining furnace. Spec Steel 2012;33(1):26–9. [9] Geng D, Zhang X, Liu X, Wang P, Liu H, Chen H, et al. Simulation on flow field and mixing phenomenon in single snorkel vacuum degasser. Steel Res Int 2015;86(7):724–31. [10] Geng D, Lei H, He J. Decarburization and Inclusion Removal Process in Single Snorkel Vacuum Degasser. High Temp Mater Process 2017;36(5):523–30. [11] Mondal M, Maruoka N, Kitamura S, Gupta G, Nogami H, Shibata H. Study of fluid flow and mixing behaviour of a vacuum degasser. T Indian I Metals 2012;65(3):321–31. [12] Yang X, Zhang M, Wang F, Duan J, Zhang J. Mathematical simulation of flow field for molten steel in an 80-ton single snorkel vacuum refining furnace. Steel Res Int 2012;83(1):55–82. [13] Qi F, Liu J, Liu Z, Cheung S, Li B. Characterization of the mixing flow structure of molten steel in a single snorkel vacuum refining furnace. Metals 2019;9(4):400. [14] Chen S, Lei H, Wang M, Yang B, Dou W, Chang L, et al. Ar-CO-Liquid Steel Flow with Decarburization Chemical Reaction in Single Snorkel Refining Furnace. Int J Heat Mass Transf - Theory Appl 2020;146:118857. [15] Dai W, Cheng G, Li S, Huang Y, Zhang G, Qiu Y, et al. Numerical simulation of multiphase flow and mixing behavior in an industrial single snorkel refining furnace (SSRF): the effect of gas injection position and snorkel diameter. ISIJ Int 2019;59(7):1214–23. [16] Dai W, Cheng G, Li S, Huang Y, Zhang G. Numerical simulation of multiphase flow and mixing behavior in an industrial single snorkel refining furnace: effect of bubble expansion and snorkel immersion depth. ISIJ Int 2019;59(12):2228–38. [17] Sattar MA, Naser J, Brooks G. Numerical simulation of two-phase flow with bubble break-up and coalescence coupled with population balance modeling. Chem Eng Process 2013;70(7):66–76. [18] Bao Y, Yang J, Chen L, Gao Z. Influence of the top impeller diameter on the gas dispersion in a sparged multi-impeller stirred tank. Ind Eng Chem Res 2012;51(38):12411–20. [19] Liu Z, Li B, Qi F, Cheung S. Population balance modeling of polydispersed bubbly flow in continuous casting using average bubble number density approach. Powder Technol 2017;319:139–47. [20] Zhan S, Wang Z, Yang J, Zhao R, Li C, Wang J, et al. 3D numerical simulations of gas-liquid two-phase flows in aluminum electrolysis cells with the coupled model of computational fluid dynamics-population balance model. Ind Eng Chem Res 2017;56(30):8649–62. ´ H. Simulation study of blast furnace drainage [21] Shao L, Saxen using a two-fluid model. Ind Eng Chem Res 2013;52(15):5479–88. [22] Hinze JO. Turbulence. New York: McGraw-Hill Publishing Co.; 1975. [23] Schiller L, Naumann A. A drag coefficient correlation. Z Ver Deutsch Ing 1935;77:318–20.

[24] Simonin C, Viollet P. Predictions of an oxygen droplet pulverization in a compressible subsonic coflowing hydrogen flow. Numer Methods Multiphase Flow 1990;91(2):65–82. [25] Simonin O, Viollet P. Phenomena in multiphase flows. Washington: Hemisphere publ. Corp; 1990. [26] Anil Kishan P, Dash S. Numerical and experimental study of circulation flow rate in a closed circuit due to gas jet impingement. Int J Numer Method H 2006;16(8):890–909. [27] Launder B, Spalding D. Lectures in mathematical models of turbulence. London, UK: Academic Press; 1972. [28] Bel Fdhila R, Simonin O. Eulerian prediction of a turbulent bubbly flow downstream of a sudden pipe expansion. In: In Proceedings 5th Workshop on Two-Phase Flow Predictions. 1992. p. 264–73. [29] Elghobashi S, Abou-Arab T. A two-equation turbulence model for two-phase flows. Phys Fluids 1983;26(4):931–8. [30] Hounslow M, Ryall R, Marshall V. A discretized population balance for nucleation, growth, and aggregation. AIChE J 1988;34(11):1821–32. [31] Liao Y, Lucas D. A literature review on mechanisms and models for the coalescence process of fluid particles. Chem Eng Sci 2010;65(10):2851–64. [32] Cloete S, Eksteen J, Bradshaw S. A numerical modelling investigation into design variables influencing mixing efficiency in full scale gas stirred ladles. Miner Eng 2013;46:16–24. [33] Lou W, Zhu M. Numerical simulations of inclusion behavior in Gas-Stirred Ladles. Metall Mater Trans B 2013;44(3):762–82. [34] Chen A, Yang W, Geng S, Gao F, He T, Wang Z, et al. Modeling of microbubble flow and coalescence behavior in the contact zone of a dissolved air flotation tank using a computational fluid dynamics-population balance model. Ind Eng Chem Res 2019;58(36):16989–7000. [35] Coalescence Luo H, Ph.D. Dissertation Breakup and liquid circulation in bubble column reactors. Trondheim: The Norwegian Institute of Technology; 1993. [36] Liao Y, Lucas D. A literature review of theoretical models for drop and bubble breakup in turbulent dispersions. Chem Eng Sci 2009;64(15):3389–406. [37] Luo H, Svendsen H. Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J 1996;42(5):1225–33. [38] Lister JD, Smit D, Hounslow M. Adjustable discretized population balance for growth and aggregation. AIChE J 1995;41(3):591–603. [39] Mori K, Sano M, Sato T. Size of bubbles formed at single nozzle immersed in molten Iron. Trans ISIJ 1979;19(9):553–8. [40] Conejo AN. Effect of nozzle diameter on mixing time during bottom-gas injection in metallurgical ladles. Metall Mater Trans B 2015;46(2):711–8. [41] Karouni F, Wynne BP, Talamantes-Silva J, Phillips S. Hydrogen degassing in a vacuum arc degasser using a three-phase eulerian method and discrete population balance model. Steel Res Int 2018;89(5):1700550. [42] Drumm C, Attarakih M, Bart H. Coupling of CFD with DPBM for an RDC extractor. Chem Eng Sci 2009;64(4):721–32. [43] Anagbo P, Brimacombe J. Plume characteristics and liquid circulation in gas injection through a porous plug. Metall Mater Trans B 1990;21(4):637–48. [44] Garcia-Ochoa F, Gomez E. Bioreactor Scale-Up and Oxygen Transfer Rate in Microbial Processes: an Overview. Biotechnol Adv 2009;27(2):153–76. [45] Calderbank P. The interfacial area in gas-liquid contacting with mechanical agitation. Process Saf Environ Prot 1958;36:443–63. [46] Zhang L, Taniguchi S. Fundamentals of inclusion removal from liquid steel by bubble flotation. Int Mater Rev 2000;45(2):59–82.

Please cite this article in press as: Chen G, He S. Numerical simulation of Argon–Molten steel two-phase flow in an industrial single snorkel refining furnace with bubble expansion, coalescence, and breakup. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.026