Modeling Dynamics of Agglomeration, Transport, and Removal of

Article pubs.acs.org/IECR

Modeling Dynamics of Agglomeration, Transport, and Removal of Al2O3 Clusters in the Rheinsahl−Heraeus Reactor Based on the Coupled Computational Fluid Dynamics-Population Balance Method Model Gujun Chen, Shengping He,* Yugang Li, and Qian Wang College of Materials Science and Engineering, Chongqing University, Chongqing 404100 China ABSTRACT: A coupled 3-D computational fluid dynamics−population balance model (CFD−PBM) is developed to simulate the behaviors of Al2O3 inclusions in a Rheinsahl−Heraeus (RH) reactor. This work attempts to integrate the transport phenomena including steel flow as well as the aggregation and removal of Al2O3 clusters. The inhomogeneous multiple size group (inhomogeneous MUSIG) model is first employed to investigate the inclusion behaviors in steel. Compared to experimental data, numerical results show good agreement for the number density, Sauter diameter, size distribution, and mass fraction of inclusions in steel. Meanwhile, the application of the frequently used homogeneous MUSIG model in inclusion behaviors is also evaluated, and it is observed that this method may not be suitable for modeling the inclusion behaviors in steel where large and small inclusions are likely to segregate because of different momentum fields.

1. INTRODUCTION In the production of (ultra-)low-carbon steels, a Rheinsahl− Heraeus (RH) reactor plays a crucial role in removing deoxidation inclusions from the molten steel.1 As shown in Figure 1,

Therefore, there is a great need to minimize the mass fraction of inclusions and effectively remove macroinclusions in steel during RH refining.1,2,7 Obviously, deoxidation of molten steel is a complicated process, which involves gas−liquid−solid multiphase flow and the nucleation, growth, agglomeration, and removal of oxide inclusions. Nowadays, the population balance equation (PBE) has been widely applied in chemical engineering (such as crystallization, fluidized bed reactors, aerosol reactors, liquid membrane systems, microbial cultures, and gas−liquid, solid−liquid, and liquid−liquid dispersions),8−11 as well as in metallurgical engineering (such as bubbly plume flow and inclusion behaviors in metallurgical reactors).2,3,12 In the early studies, on the basis of the mean processing parameters, the static PBE was successfully applied to describe the inclusion behaviors in molten steel, such as aggregation, removal, etc.2,12−14 However, in a real refining process, variables describing the aggregation and removal of inclusions, such as turbulent energy dissipation rate, are functions of time and space. At the same time, the size distribution of inclusions should vary with spatial location due to the transport of inclusions by flow field. Recently, the PBE has been integrated into computational fluid dynamics (CFD) code,15−22 and the dynamic coupled computational fluid dynamics-population balance method (CFD−PBM) model has been employed to investigate the transport phenomena

Figure 1. Schematic diagram of a RH reactor.

during the RH treatment, the steel recirculates between a vacuum vessel and a ladle due to the drive of injected argon under a vacuum condition. Normally, aluminum is added from the vacuum vessel to react with the dissolved [O] in the steel, which gives rise to a large amount of Al2O3 inclusions.2,3 Al2O3 is usually considered to be one of the most harmful inclusions in relation to both the quality of the final steel product and the castability of steel because of its high melting point and great hardness.4−6 © 2016 American Chemical Society

Received: Revised: Accepted: Published: 7030

February 12, 2016 June 1, 2016 June 2, 2016 June 2, 2016 DOI: 10.1021/acs.iecr.6b00586 Ind. Eng. Chem. Res. 2016, 55, 7030−7042

Article

Industrial & Engineering Chemistry Research Table 1. Operating Conditions of RH Process total mass of molten steel, t up/down diameter of ladle, m diameter of vacuum vessel, m length/diameter of snorkel, m distance between snorkels, m pressure of vacuum vessel, Pa

210 3.72/3.2 2.15 1.65/0.6 1.5 200

density of molten steel, kg/m3 viscosity of molten steel, Pa·s temperature of molten steel, K density of Al2O3, kg/m3 argon flow rate, Nm3/h dissolved [O] in steel before Al addition, ppm

including fluid flow as well as the aggregation and removal of inclusions in metallurgical reactors such as a ladle reactor23−32 and a tundish,33,34 occasionally in a RH reactor,35 which has produced very promising results in terms of the inclusion behaviors. Several numerical techniques are available to solve the dynamic PBM, such as the class method (CM),36,37 method of moments (MOM),38 standard method of moment (SMM),39 quadrature method of moment (QMOM),40 sectional quadrature method of moments (SQMOM),41 direct quadrature method of moments (DQMOM),42 and the kinetic theory approach with size resolution (KTAWSR).43 Detailed descriptions can be found in these studies.36−43 Research studies based on the homogeneous multiple size group (homogeneous MUSIG) model by Hounslow and co-workers,36,37 Jonsson and co-workers,23,24 Bellot and co-workers,27−29,31 Ling and co-workers,33,34 and Lou and Zhu30,32 typified the application of the CM in simulations of the inclusion behaviors in molten steel. In addition to the CM, the QMOM25−27,35 has also been applied to solve the dynamic PBM for the inclusion behaviors in metallurgical engineering. However, many shortcomings still remain to be improved. For example, in nearly all the investigations, the shape of the inclusions was assumed to be an equivalent solid sphere, which might not reasonably describe the aggregation of the inclusions in a steel flow.23,24,27−29,31,34 At the same time, empirical formulas were employed for the initial inclusion size distribution,23,24,27−34 and the simulation results were seldom verified by experiments because it was quite expensive and difficult to obtain data from an actual plant.23−35 Actually, compared with the previously mentioned shortcomings, the most serious problem is that all of the previously mentioned studies using the CM23,24,27−34 and QMOM25−27,35 were based on the assumption that an inclusion phase with a wide size distribution shared the same velocity field. This may be unsuitable for modeling the inclusion behaviors in molten steel where large and small inclusions are likely to segregate as a result of different momentum fields. During the refining process, the larger inclusions are easily floated and absorbed by the molten slag, while the small inclusions tend to remain in the molten steel. In order to eliminate the above drawback, the inhomogeneous MUSIG model was proposed by Frank et al.44 and Krepper et al.45 In this method, the inclusion phase can be divided into N velocity groups, where each velocity group is characterized by its own velocity field. In addition, each velocity group can be subdivided into Mn subsize classes, which all have the same velocity. Currently, the inhomogeneous MUSIG model has received greater attention in chemical engineering.44−46 In the present work, a 3-D coupled CFD−PBM model for the description of transport, agglomeration, and removal of Al2O3 clusters in a RH reactor is developed. The inhomogeneous MUSIG model is first employed to investigate the inclusion behaviors in steel. Simulation results are validated by comparing the predicted the number density, Sauter diameter, size distribution, and mass fraction of inclusions in steel with those of the samples taken during the industrial RH trials. Meanwhile, the application of the frequently used homogeneous MUSIG model in inclusion behaviors is also

7020 0.006 1868−1884 3960 150 284

evaluated. Then, the importance of various mechanisms for the agglomeration and removal of inclusions is discussed. The characteristic of the inclusions distribution in the RH reactor is presented.

2. INDUSTRIAL TESTS Steel samples were obtained from the ladle of a RH reactor during deoxidation of low-carbon steel. Some of the operating conditions are listed in Table 1. 2.1. Steel Sampling. The sampling point was located about 35 cm below the slag surface near the center of the ladle. At the beginning of deoxidation, aluminum alloy was added to react with the 284 ppm of dissolved [O] in the molten steel, and almost all of the dissolved [O] precipitated in the form of Al2O3 inclusions. Then, lollypop samples of the molten steel were taken every 1.0−2.0 min until the end of the deoxidation treatment (Figure 2). The samples were quickly cooled in water. Therefore,

Figure 2. Lollypop samples of the steel and the processed cylinders for the analysis of the total oxygen.

the inclusions in the sample were representative of the melt at the moment of sampling.47 The entire circulation process after Al addition lasted 12 to 15 min for each furnace. 2.2. Measurement of Mass Fraction of Inclusions in Steel. In the steel, almost all of the oxygen existed in the form of Al2O3 inclusions. Thus, the total oxygen could be used to characterize the Al2O3 content in the steel. Each sample was processed into three or four cylinders with a diameter of 5 mm using the wire cut electrical discharge machining (Figure 2). Then, these cylinders were used for the analysis of the total oxygen with an oxygen/nitrogen analyzer (Eltra NO 900). 2.3. Inclusion Extraction for Morphology Observations. The method for extracting the inclusions from the iron matrix was based on that developed by Dekkers et al.47 This method is appropriate in the case of the acid-insoluble inclusions such as Al2O3. Steel (3.0 g) was dissolved in 50 mL of HCl (1:1) and heated to 80−100 °C in a water bath. Once the steel was completely dissolved, the iron solution was vacuum filtered, and the Al2O3 inclusions were collected on an acid-resistant Nuclepore polycarbonate membrane with a diameter of 25 mm and pore size of 0.2 μm. After this, the membrane was washed 7031

DOI: 10.1021/acs.iecr.6b00586 Ind. Eng. Chem. Res. 2016, 55, 7030−7042

Article

Industrial & Engineering Chemistry Research

The number of fine particles making up the cluster is related to the diameter of an equivalent solid sphere with the same mass as the cluster.

with hot deionized water and hot diluted HCl (50:1). Then, the membrane that contained Al2O3 was coated with gold and used to obtain the morphology using a high-resolution scanning electron microscope (SEM, TESCAN VEGA 3 LMH), equipped with an energy dispersive X-ray spectrometer (EDS). It should be noted that inclusions at the cross-section of each steel sample were detected and analyzed directly by SEM-EDS to determine the chemical composition of inclusions and nearly all the oxide inclusions were found to be the Al2O3. Therefore, it is reasonable to use the aforementioned method for extracting the oxide inclusions in the present work. 2.4. Inclusion Extraction for Size Distribution. Al2O3 inclusions were extracted from a steel sample of approximately 40−80 g using the previously described method. The cleaned inclusions were placed in a 500 mL beaker with deionized water and used to measure the size distribution with the laser diffraction scattering method (Malvern Mastersizer 2000 particle size analyzer). The diameter was measured as the equivalent solid sphere. Thus, the number densities of inclusions with different sizes in the steel could be obtained by combining the measured size distribution with the mass fraction of inclusions in the steel.

(ds,e)3 = Ns,p(ds,p)3

(2)

Thus, the diameter of a cluster can be calculated from the measured diameter of an equivalent sphere.

(ds,c)Df =

(ds,e)3 (ds,p)3 − Df

(3)

The diameters of most of the fine particles fell in the 1−2 μm range, with an average value of about 1.5 μm, based on the measurement. The structure of a cluster would be a solid sphere when Df reaches three. 3.2. Population Balance Model. 3.2.1. Inhomogeneous MUSIG Model. In Fluent, for the inhomogeneous MUSIG model,44,45 the PBM is expressed in terms of the volume fraction and size fraction of inclusion size class i, i∈[1, Mn], of velocity group n, n∈[1, N], as follows: ∂ Agg Rem (ραs,nfs,n,i ) + ∇·(ρs us,n ⃗ αs,nfs,n,i ) = Ss,n,i + Ss,n,i ∂t s

3. COUPLED CFD−PBM MODEL A 3-D two-fluid model based on the Eulerian−Eulerian approach is used to describe the gas−liquid−solid three-phase flow in a RH reactor. The kinetic theory of granular flow (KTGF) is employed to consider the conservation of solid fluctuation energy. The inhomogeneous MUSIG model is used to solve the PBE. The transport, aggregation, and removal of Al2O3 clusters are integrated. 3.1. Three-Dimensional Fractal Structure Model of Al2O3 Clusters. Similar to the observations by Doo et al.48 and Tozawa et al.,12 most of the Al2O3 extracted from the samples in this study exhibited a cluster shape, as shown in Figure 3a, which

(4)

with additional relations given by Mn

αs,n =

Mn

N

∑ αs,n,i ∑ fs,n,i i=1

= 1 αs =

i=1

N

Mn

∑ αs,n = ∑ ∑ αs,n,i n=1

n=1 i=1

(5)

where αs, n and u⃗s, n are the volume fraction and velocity of velocity group n, respectively. fs, n, i is the size fraction of size class i in velocity group n. SRem s, n, i is the source in terms of the inclusion removals of size class i in velocity group n due to wall adhesion and slag absorption. SAgg s, n, i is the source term for aggregation between inclusions and can be further written as follows: Agg Agg Agg Ss,n,i = Bs,n,i − Ds,n,i

BAgg s, n, i

(6)

DAgg s, n, i

and are the birth and death rates of the inclusions of size class i due to aggregation, respectively, which can be defined as follows:44,45 ms,k + ms,l 1 Agg Bs,n,i = (ρα )2 ∑ ∑ βkl fs,k fs,l ξkl s s 2 k≤i l≤i ms,k ms,l (7) Agg Ds,n,i = (ρα )2 s s

Figure 3. Structure of an alumina cluster: (a) typical SEM image of a cluster, (b) schematic structure of a cluster, and (c) equivalent solid sphere of a cluster.

k

1 ms,k

(8)

where ξkl is a factor projecting the corresponding part of birth into the size class i. ms, i represents the mass of a single inclusion of size class i. 3.2.2. Aggregation. Stokes collision and turbulent collision are usually considered to determine the agglomeration of micrometer-scale inclusions.25,26,35 Stokes Collision. The Stokes collision kernel takes the rising velocity difference between small and large inclusions into consideration.49

consisted of many fine Al2O3 particles. Figure 3b shows a schematic structure of an Al2O3 cluster, where ds, c is the diameter of a cluster and ds, p is the diameter of a fine particle. Figure 3c shows the equivalent solid sphere with a diameter of ds, e, which is measured using the laser diffraction scattering method. According to the fractal theory, the relations between the diameter of the cluster and the 3-D distribution of fine particles can be expressed quantitatively as follows:12 ⎛ d ⎞ Df s,c ⎟⎟ Ns,p = ⎜⎜ ⎝ ds,p ⎠

∑ βik fs,i fs,k

βijStokes =

1 Rising Rising π (ds,c,i + ds,c,j)2 |Vs,c,i − Vs,c,j | 4

VRising s, c, i

(9)

VRising s, c, j

where ds, c, i and ds, c, j are the cluster diameters. and are the rising velocities of clusters. Turbulent Collisions. Zaichik et al.50 recently developed a collision kernel for homogeneous isotropic turbulence, and this

(1)

where Ns, p is the number of fine particles in a cluster and Df is the 3-D fractal dimension. 7032


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Figure 4. (a) Relationship between diameters of equivalent solid sphere and cluster according to fractal theory. (b) Comparison of different aggregation rates with and without considering the actual shape of clusters.

kernel is valid for an arbitrary ratio between the particle size and fluid turbulent length scale, as well as values of the particleto-fluid density and particle inertia parameter. ⎛ ds,c,i + ds,c,j ⎞ = 4 π⎜ ⎟ Vt 2 ⎠ ⎝

Thus, the aggregation rate is the sum of these two aggregation rates. βij = ςijStokesβijStokes + ςijTurbulentβijTurbulent

2

βijTurbulent

As previously mentioned, the diameter of a cluster, ds, c, is much larger than the diameter, ds, e, of the equivalent solid sphere, which depends on the 3-D fractal dimension, Df, as calculated in Figure 4a.The value of Df is 1.98 for Al2O3 clusters measured by Doo et al.48 Figure 4b shows the behavior and relative intensity of these aggregation rates. The turbulent collision governs the small and medium scales, whereas both collisions are dominant at the larger scales. In addition, as seen in Figure 4b, the consideration of a cluster can significantly enhance the aggregation rate. Obviously, if the diameter of the equivalent solid sphere with Df = 3.0 is used, the aggregation rate will be underestimated. Conversely, if the cluster diameter with Df = 1.98 is used, the aggregation rate will be overestimated because the number density of the particles in a cluster is very small toward the outer side (Figure 3a). A reasonable value for the fractal dimension is believed to be between 3.0 and 1.98, which will be determined by the calibration of the model with experimental measurements in this study. 3.2.3. Inclusion Removal. In the RH reactor, all of the argon is discharged from the vacuum vessel, and there is no slag. The inclusions reaching the free surface in the vacuum vessel will become trapped again in the molten steel. Therefore, bubble attachment has no effect on inclusion removal. In the present work, wall adhesion and slag absorption for inclusion removals are considered. Wall Adhesion. In metallurgical reactors, when the inclusions reach the wall surface, if the property of the inclusions is similar to that of the solid surface, they may adhere to it. The transport of inclusions to the refractory surface is referred to as a diffusion process, which can be calculated by13

(10)

where Vt denotes the total turbulent collision, as can be found in the work of Zaichik et al.50 Collision Efficiency. Higashitani et al.51 introduced a collision efficiency factor, ςTurbulent , for a turbulent collision to estimate the ij probability that two particles stick together when they collide. ςijTurbulent

⎛ ⎞0.242 5 ⎜ = 0.732⎜ Turbulent ⎟⎟ ⎝ Nij ⎠

(11)

where NTurbulent is the ratio between the viscous force and the van ij der Waals force. NijTurbulent = 12

3 π μ l ⎛ ds,c,i + ds,c,j ⎞ Turbulent ⎟ γij̇ ⎜ 15 Ha ⎝ 2 ⎠

(12)

where Ha is the Hamaker constant and is assigned a value of 2.3 × 10−20 for the collision between alumina inclusions in molten steel.52 μl is the dynamic viscosity of the steel. γ̇ Turbulent is ij the strain rate for turbulent flows, which is given by γijTurbulent = ̇

εl νl

(13)

where εl and νl are the turbulent kinetic energy dissipation rate and kinematic viscosity of the steel, respectively. Recently, the efficiency factor was also extended for use with Stokes kernel by Rimbert and co-workers.25,26,35 The associated strain rate was related to the difference in terminal rising velocity. γijStokes = ̇

Ss,iWall = −

Rising Rising |Vs,c,i − Vs,c,j |

0.5(ds,c,i + ds,c,j)

=

4g (ρl − ρs ) ds,e 3 45μ l

ds,c

2 0.0062εl 3/4 Acell ⎛ ds,e,i ⎞ ρ Vs,ins,i ⎜ ⎟ Vcell ⎝ 2 ⎠ s νl 5/4

(17)

where Acell and Vcell are the local interface area and cell volume, respectively. Vs, i and ns, i are the volume of a single inclusion and number concentration of size class i, respectively. Slag Absorption. When an inclusion is transported to the mixing zone of steel and slag by the steel flow, where slag is the dominating phase, it is separated from the steel.23,24 In the present work, it is assumed that inclusions in the top-layer grid adjacent to the slag phase are removed ideally. Thus, the mass of the inclusions with size class i per unit volume

(14)

The terminal rising velocity of an Al2O3 cluster was deduced on the basis of the measured drag coefficient, CD, s, c, for a cluster by Tozawa et al.12 Rising Vs,c

(16)

(15) 7033


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Industrial & Engineering Chemistry Research Table 2. Continuity Equation, Momentum Conservation Equation, and KTGF description

equation

Eulerian−Eulerian two fluid model continuity equation (gas)

∂ (αgρg ) + ∇·(αgρg ug⃗ ) = 0 ∂t

continuity equation (liquid)

∂ (αlρl ) + ∇·(αlρl ul⃗ ) = 0 ∂t

continuity equation (solid)

∂ Rem Agg (αs,nρs ) + ∇·(αs,nρs us,n ⃗ ) = Ss,n + Ss,n ∂t

momentum balance equation (gas)

∂ (αgρg ug⃗ ) + ∇·(αgρg uu⃗ ⃗) = − αg∇p + ∇·τg + αgρg g ⃗ + Kl ‐ g(ul⃗ − ug⃗ ) + Fg⃗ ∂t ∂ (αlρl ul⃗ ) + ∇·(αlρl ul⃗ ul⃗ ) = − αl∇p + ∇·τl + αlρl g ⃗ + Kl ‐ g(ug⃗ − ul⃗ ) ∂t

momentum balance equation (liquid)

N

+

∑ Kl‐s,n(us,n ⃗ − ul⃗ ) + Fl⃗ n=1

momentum balance equation (solid)

∂ (αs,nρs us,n ⃗ ) + ∇·(αs,nρs us,n ⃗ us,n ⃗ ) ∂t = − αs,n∇p − ∇ps,n + ∇·τs,n + αs,nρs g ⃗ + Kl ‐ s,n(ul⃗ − us,n ⃗ )

2 αgμ (∇ug⃗ )I 3 g

stress tensor (gas)

τg = αgμg (∇ug⃗ + (∇ug⃗ )T ) −

stress tensor (liquid)

τl = αlμ l (∇ul⃗ + (∇ul⃗ )T ) −

stress tensor (solid)

⎛ 2 ⎞ T τs,n = αs,nμs,n (∇us,n ⃗ + (∇us,n ⃗ ) ) − αs,n⎜λs,n − μs,n ⎟∇·us,n ⃗ I ⎝ 3 ⎠

2 αlμ (∇ul⃗ )I 3 l

kinetic theory of granular flow59 solid pressure

ps,n = αs,nρs Θs,n + 2ρs (1 + ess)(αs,n)2 g0,ssΘs,n

solid bulk viscosity

λs,n =

radial distribution function59

−1 ⎡ ⎛ αs,n ⎞1/3⎤ ⎟⎟ ⎥ g0,ss,n = ⎢1 − ⎜⎜ ⎢ ⎝ αs,max ⎠ ⎥⎦ ⎣

granular temperature60

Θs,n =

Θs,n 4 (αs,n)2 ρs ds,eg0,ss,n(1 + ess) 3 π

1 2 (us,n ⃗′ ) 3

⎤ 3⎡ ∂ ⃗ )⎥ = (− ps,n I + τs,n) ⎢ (αs,nρs Θs,n) + ∇·(αs,nρs Θs,nus,n ⎦ 2 ⎣ ∂t : ∇us,n ⃗ + ∇·(k Θs,n∇Θs,n) − γΘ + Φ ls,n s,n

150ρs ds,e (Θs,nπ ) ⎡ ⎤2 6 2 ⎢1 + αs,ng0,ss,n(1 + ess)⎥⎦ + 2(αs,n) ρs ⎣ 384(1 + ess)g0,ss,n 5

k Θs,n = diffusion coefficient61

⎛ Θs,n ⎞ ds,e(1 + ess)g0,ss,n ⎜ ⎟ ⎝ π ⎠ 12(1 − (ess)2 )g0,ss,n

collision dissipation of energy59

γΘ =

transfer of kinetic energy62

Φ ls,n = − 3Kls,n Θs,n

dynamic viscosity61,62

ds,e π

s,n

μs,n =

4(αs,n)2 ρs ds,eg0,ss,n(1 + ess)

Θs,n

5

π

+

ps,n sin φ 2 I2D

+

αs,nρs ds,e Θs,nπ 6(3 − ess)

⎡ ⎤ 2 ⎢⎣1 + (1 + ess)(3ess − 1)αs,ng0,ss,n ⎥⎦ 5

The removal of inclusions with different size classes can be calculated as the sum of the two mechanisms.

and unit time that is separated to the slag can be expressed as follows: Slag Ss,i = −ρs Vs,ins,i /Δt

ρs (αs,n)2 (Θs,n)3/2

Rem Slag Ss,n,i = Ss,iWall + Ss,i

(18)

(19)

3.3. CFD Model. Numerical simulations are based on the Eulerian−Eulerian approach, together with the KTGF and the

where Δt is the time step. 7034


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Industrial & Engineering Chemistry Research k-ε dispersed turbulence model,53 as listed in Table 2. In Table 2, M M Rem Rem Agg Agg Ss,n = ∑i =n1 Ss,n,i = ∑i =n1 Ss,n,i and Ss,n are the sources of removal and aggregation of inclusions for velocity group n, respectively. F⃗g(= −F⃗l) is the sum of the nondrag forces acting on the gas phase.54 Kl‑g is the gas−liquid transfer coefficient.54 The liquid−solid transfer coefficient for velocity group n, Kl‑s, n, can be obtained by the drag model of McKeen and Pugsley55

solve the PBE to simulate the inclusion behaviors in the same RH reactor. The simulations were performed in a platform of a Core eight-core processor 3.4 GHz machine with 24 GB RAM.

4. RESULTS AND DISCUSSION 4.1. Hydrodynamic Prediction. 4.1.1. Grid Independency for Gas−Liquid Flow. Gambit 2.4.6 is adopted to create the geometry and full-hexahedral grids with the Cooper method.53 Only half of the geometric model is constructed as a computational domain because of the symmetry of the flow. In order to ensure that the solutions are independent of the grids, simulations are performed using different numbers of grids, as shown in Figure 5. The steel circulation rate is defined as the quality of

⎛ 17.3 ⎞ ρ −1.8 l + 0.336⎟⎟ |us,n Kl ‐ s,n = C ⎜⎜ ⃗ − ul⃗ |αs,n(αl) Re d ⎝ s,c,n ⎠ s,c,n,32 (20)

ds,c,n,32 =

∑ ns,i(ds,c,i)3 ∑ ns,i(ds,c,i)2

(21)

where Res, c, n is the Reynolds number based on the Sauter diameter, ds, c, n, 32, of Al2O3 clusters for velocity group n. C is a scale factor, which considers the effect of the interparticle forces leading to the agglomeration of micrometer-scale particles. A C range of 0.1−0.3 is suggested by many investigators.11,55,56 During the RH deoxidation process, the cohesive forces (mainly the cavity bridge force caused by the poor wettability with molten steel) play an important role in the aggregation of Al2O3 inclusions,57,58 and these should be considered. In the present work, the value of C is assumed to be 0.2. 3.4. Simulation Conditions and Modeling Method. The coupled CFD−PBM model was solved using the commercial computational fluid dynamics software Fluent with a user defined function (UDF) in the double precision mode. The phase-coupled SIMPLE algorithm was used to couple the pressure and velocity. The dimensions of the RH reactor and other parameters are listed in Table 1. The model assumes a uniform temperature equal to 1873 K.54 The walls of the ladle, snorkels, and vacuum vessel were considered to be no-slip conditions. The standard wall function was employed for the turbulent characteristic in the near-wall regions. A free liquid surface was assumed at the liquid surfaces of the ladle and vacuum vessel. The mass flow inlet boundary was used for gas injection from the nozzles. The measured diameters, ds, e, of the equivalent solid sphere of Al2O3 clusters in the present work ranged from 1.4 μm to about 50 μm, in which the number of size classes was determined by Vs, i+1/Vs, i = 21/q.37 The complete mixing time in the present RH reactor, which corresponded to the time needed for the complete homogenization of ferro-alloys, was about 70 s when the argon flow rate was 150 N m3/h, as calculated in our recent work.63 At the same time, the nucleation of Al2O3 inclusions and growth due to diffusion lasted only for a few seconds,49 and both processes were ignored in the present simulation. That is, the collision and removal would determine the inclusion size distribution after the complete homogenization. Therefore, it was reasonable that the inclusion size distribution measured at about 60 s was employed for initialization in the present work, and the inclusions were assumed to be uniformly distributed in the RH reactor at the initial time. In this work, the approach adopted for the RH reactor modeling was divided into two stages. First, the steady state simulation of a gas−steel two-phase flow was implemented for the 3-D geometry of the RH reactor, and a strong coupling was obtained between the steel flow and the gas injection. This stage was briefly described here, because the detailed modeling has already been presented in our recent publication.54 In the second stage, the inhomogeneous MUSIG model was introduced to

Figure 5. Effect of the number of grids on steel circulation rate at different argon rates in the RH reactor.

molten steel through the up- or down-snorkel per unit time. It can be seen that the refined grids are required in the gas−liquid zone (zone 1), while the coarse grids are enough in the other zones (zones 2, 3, 4, and 5). If the number of grids in gas−liquid zone is less than 85 656, the steel circulation rate will be overestimated; decreasing the number of grids in zones 2, 3, 4, and 5 from 209 497, 93 686, 210 029, and 1 485 809 to 90 690, 34 503, 94 581, and 419 551 has negligible effect on steel circulation rate. Therefore, a total of 724 981 grids is sufficiently fine for obtaining reasonably grid independent results, and it is selected for the prediction of gas−liquid flow in the RH reactor. 4.1.2. Flow Pattern and Turbulent Kinetic Energy Dissipation Rate. The calculated steel velocity vectors in the RH reactor are shown in Figure 6. As seen in Figure 6, the molten steel in the ladle enters the vacuum vessel through the up-snorkel as a result of the drive of the argon injected from the nozzles and spreads throughout the vacuum vessel. After this, it flows into the ladle through the down-snorkel and immediately impinges the bottom of the ladle with a high velocity. Then, it generates a large recirculation zone around the downward stream in the ladle. Subsequently, the bulk of the steel circuitously flows toward the entrance of the up-snorkel and restarts the next circulation. Figure 7 illustrates the predicted turbulent kinetic energy dissipation rate of the steel in the RH reactor. It can be seen that the maximum turbulent kinetic energy dissipation rate is primarily concentrated in the vacuum vessel, snorkels, and ladle under the down-snorkel. Generally, a larger dissipation rate can promote the collision, aggregation, and removal of inclusions from the steel. 4.2. Inclusion Behaviors. The operations in the RH reactor should be improved to minimize the mass fraction of inclusions 7035


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Figure 6. Predicted steel velocity vectors in the RH reactor (Qg = 150 N m3/h). Section (A-A) is the symmetry plane (y = 0 m plane), and the positions of other sections are indicated in section (A-A).

Figure 7. Predicted turbulent kinetic energy dissipation rate of steel in the RH reactor (Qg = 150 N m3/h).

Figure 8. Effect of the number of grids (a) and time steps (b) on inclusion number densities of different sizes at 120 s in the RH reactor (Qg = 150 N m3/h).

and effectively remove macroinclusions.1,2 In the present work, ave the local mass fraction, wlocal s , and average mass fraction, ws , of the inclusions in the whole RH reactor or at a sampling position are defined, respectively, as follows:

wslocal = αsρs /ρl

(22)

wsave = (∑ wslocalVcell)/(∑ Vcell)

(23)

4.2.1. Grid Independency and Time Step Analysis for Inclusion Behaviors. To find the grid independent results of inclusion behaviors, simulations are carried out using a different number of grids in the zones except zone 1, and the number of grids in zone 1 (85 656) remains unchanged to make the steel circulation rate constant, as shown in Figure 8a. It can be seen that all results show similar number densities of different sizes in steel, although slight differences are observed when 127 691 grids are used in the zones except zone 1. In the present work, 209 129 grids (the zones except zone 1) will be used for time step analysis and the simulations of inclusion behaviors in the steel. The inclusion number densities in steel are investigated at different time steps with 20 iterations per each time step, as shown

The local and average number density and Sauter diameter are defined in the same way as the local and average mass fraction of inclusions, respectively. 7036


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Figure 9. Influence of number of size classes on calculated inclusion size distribution at 120 s in the RH reactor.

Figure 10. Predicted and measured values at sampling position: (a) evolution of average number density and (b) evolutions of average mass fraction and Sauter diameter of equivalent solid sphere of inclusions in steel.

Figure 11. Predicted and measured values at 750 s at sampling position: (a) number densities of different sizes in steel and (b) mass percent of inclusions of different sizes in total inclusions.

greatly deviates from those of 11, 16, and 21 classes, and the result of 11 classes slightly deviates from those for 16 and 21 classes, whereas the refinement from 16 to 21 classes produces almost the same values. Therefore, 16 classes are sufficient for the description of the problem and will be employed in the following simulations. The measured and predicted evolutions of the average number density, Sauter diameter, and mass fraction of the Al2O3 inclusions in steel at a sampling position are illustrated in Figure 10. The calculated and measured inclusion number densities and mass percent of inclusions of different sizes in total inclusions at 750 s are illustrated in Figure 11. It can be seen in Figure 10 that the simulation results of average number density, Sauter diameter, and mass fraction for three velocity groups deviate slightly from those for five and seven velocity groups, whereas the

in Figure 8b. It can be seen that increasing the time step from 0.01 to 0.5 s has no significant influence on the simulation results. Taking CPU time and convergence into account, the time step of 0.25 s is applied in the rest of the Article. The convergence criteria are that the standardized root-mean-square residual for all variables are less than 10−3. 4.2.2. Comparison with the Measured Values. The simulations are carried out with various size classes, and each velocity group always contains an approximately equal number of size classes when N ≥ 2. Figure 9 shows the effect of the number of size classes on the calculated inclusion size distribution in the RH reactor when N = 1 and N = 5, respectively. It should be noted that the inhomogeneous MUSIG model is actually the frequently used homogeneous MUSIG model36,37 when N = 1. It can be seen that the calculated cumulative volume for 6 classes 7037


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Industrial & Engineering Chemistry Research results obtained for five and seven velocity groups are almost always the same. However, compared to results with five and seven velocity groups, relatively large deviations are observed in the predictions of number density and mass fraction of different diameters with three velocity groups, as shown in Figure 11. In general, compared to the simulation results of three velocity groups, the results of five and seven velocity groups are found to yield comparatively better predictions. Therefore, in the following simulations, the inhomogeneous MUSIG model was used with five velocity groups as a trade-off, although three velocity groups would probably be enough, especially since we mainly focus on the correct predictions. Encouraging predictions by the inhomogeneous MUSIG model clearly demonstrates its valuable application in inclusion behaviors in the metallurgical reactor. In this section, let us return to the comparison between the presently used inhomogeneous MUSIG model (N ≥ 2) and frequently used homogeneous MUSIG model (N = 1). It can be observed in Figures 10 and 11 that the calculated results with the inhomogeneous MUSIG model nearly catch up to the measured values, whereas unacceptable deviations are observed with the homogeneous MUSIG model. Specifically, the average number density for the homogeneous MUSIG model decreases sharply with the refining time compared to the inhomogeneous MUSIG model, whereas the average mass fractions show little difference between the two methods (Figure 10). In addition, the Sauter diameter in the homogeneous MUSIG model is significantly overestimated (Figure 10b). These results can be explained according to Figures 11, 12, and 13. Figure 12 shows the change in the inclusion number densities of different sizes in steel over time. Figure 13 shows the variations

in the total inclusion numbers of different sizes caused by aggregation and direct removal in 750 s in the whole RH reactor. In the homogeneous MUSIG model (Figure 12a), it seems that large inclusions in steel are hard to remove, and their number densities still increase continuously as a result of aggregation at the early stages of the RH process. On the contrary, the number densities of small inclusions in steel decrease all the time, because both the aggregation and direct removal make great contributions to the reduction of small inclusions, as shown in Figure 13a. As a result, the number densities of large inclusions are predicted excessively, and the number densities of small inclusions are underpredicted severely in final steel, as shown in Figure 11a. This is because, in the homogeneous MUSIG model, all the size classes of the Al2O3 phase share the same velocity field, and the small inclusions will be removed as quickly as the large ones. Thus, the direct removal of small inclusions is seriously overestimated, whereas the removal of large inclusions is severely underestimated. Accordingly, the reduction of the average number density of inclusions (Figure 10a), the average Sauter diameter of inclusions (Figure 10b), and the number densities of large inclusions (Figure 11) in the homogeneous MUSIG model are seriously overestimated. In the inhomogeneous MUSIG model (Figure 12b), the number densities of large inclusions in steel decrease quickly all the time, although the large ones are constantly produced by aggregation. On the contrary, the number densities of small inclusions in steel decrease slightly all the time, because nearly all of the small inclusions are consumed by aggregation, and direct removal has little influence on their reduction, as shown in Figure 13b. As a result, size distribution of inclusions in steel can

Figure 12. Change in average number densities of different diameters over time in the RH reactor: (a) homogeneous MUSIG and (b) inhomogeneous MUSIG.

Figure 13. Variations in inclusion number caused by aggregation ((−): reduced; (+): produced) and direct removal as a function of diameters in 750 s in whole RH reactor: (a) homogeneous MUSIG and (b) inhomogeneous MUSIG. 7038


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the fractal dimension for the aggregation rate is believed to be between 3.0 and 1.98, which should be determined by calibrating the model using experimental measurements. As shown in Figure 14, the average mass fractions of the inclusions in steel decrease quickly with exponential curves, and the average mass fraction decreases obviously with a decrease in the fractal dimension from 2.7 to 2.3 at a given time. It seems that the calculated average mass fraction agrees reasonably well with the measurement for the whole refining process when the value of the fractal dimension is in the range of 2.4−2.5. For simplification, a cluster diameter of Df = 2.5 is used to calculate the aggregation rate in the rest of the Article. It should be noted that the actual fractal dimension of Al2O3 clusters (Df = 1.98) is still used to calculate the rising velocity of clusters. To clarify the effects of the different collision mechanisms on the aggregation of inclusions, the influence of various collision mechanisms on the evolution of the average number density of the inclusions in the RH reactor is calculated, as shown in Figure 15a. It can be observed that the aggregation of inclusions is mainly attributed to turbulent collision rather than Stokes collision. This can also be found in Figure 4b, where the turbulent aggregation rate is about ten times larger than the Stokes aggregation rate at the small and medium scales when the turbulent kinetic energy dissipation rate of steel is 0.0025 m2/s3. In fact, the dissipation rate in many regions of the RH reactor exceeds 0.0025 m2/s3 (Figure 7), where the turbulent collision will play an absolutely essential role in determining the aggregation of inclusions. Figure 15b shows the predicted number of inclusions removed by various mechanisms over 300 and 900 s. The model predictions indicate that about 99% of the inclusions with different sizes are separated to the top slag, whereas the inclusions separated to the refractory can be ignored. Obviously, the inclusion removal due to slag absorption dominates throughout the entire RH process. 4.2.4. Distribution of Inclusions in the RH Reactor. Figure 16 shows the comparison of the calculated values in the RH reactor and at the sampling position. It can be seen that the average mass fraction, Sauter diameter, and number densities of different diameters in the RH reactor and at the sampling position show no significant differences. Therefore, it is reasonable to use the measured size distribution as the initialization for the whole RH reactor, and the measured values can also be employed to reflect the information for the whole RH reactor.

be predicted appropriately (Figure 11a). This is because, in the inhomogeneous MUSIG model, the Al2O3 phase is divided into several velocity groups, and each velocity group has its own velocity field. Thus, the large inclusions in steel can be removed quickly, but the small inclusions tend to remain in the steel. This can also explain why, in the inhomogeneous MUSIG model, the average number density of inclusions in steel slowly decreases but the average mass fraction still decreases dramatically (Figure 10). It also can be observed in Figure 11b, in the steel after the refining process, that the relative mass of large inclusions is very small, and more than 70% of the mass comes from inclusions with equivalent diameters of less than 10 μm. However, in the homogeneous MUSIG model, most of the mass of inclusions in final steel comes from inclusions larger than 24 μm, and this is unacceptable to both the cleanliness and quality of steel, as reviewed by Zhang and Thomas.7 Therefore, it can be concluded that the differences between the simulation results with the two methods are attributable to the merit of splitting the velocity of the Al2O3 phase into several independent fields, which allow the model to recapture the separation of small and large inclusions caused by the different buoyancy force actuations. However, more CPU time is needed for the inhomogeneous MUSIG model. In the present work, it takes the CPU 4.6, 5.2, 7.8, and 11.2 days to obtain the real flow time of 15 min for one, three, five, and seven velocity groups, respectively, under the condition of parallel calculation with an eight-core processor. 4.2.3. Aggregation and Removal by Different Mechanisms. As previously mentioned in Section 3.2.2, a reasonable value of

Figure 14. Comparison between measured average mass fractions of inclusions and values calculated with different fractal dimensions at sampling position.

Figure 15. (a) Effects of various collision mechanisms on the evolution of an average number density of inclusions in the RH reactor. (b) Predicted numbers of inclusions removed in the RH reactor by various mechanisms. 7039


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Figure 16. Comparison of calculated values at sampling position and in the RH reactor: (a) evolution of average mass fraction and Sauter diameter of equivalent solid sphere and (b) average number density at 300 and 1500 s.

Figure 17. Predicted local distributions of inclusions in the RH reactor: (a) local mass fraction, (b) local number density, and (c) local Sauter diameter of equivalent solid sphere.

suitable for modeling the inclusion behaviors in molten steel where large and small inclusions are likely to segregate because of different momentum fields. (3) The inclusions are relatively uniformly distributed in the RH reactor as a whole, and the measured values can also reflect the information for the whole RH reactor, although a slight enrichment of inclusions exists in some regions. In the steel after the refining process, the relative mass of larger inclusions is very small, and more than 70% of the mass came from inclusions with equivalent diameters of less than 10 μm. (4) The calculated value agrees reasonably well with the measurement when the fractal dimension for the aggregation rate is in the range of 2.4−2.5. In the RH reactor, the aggregation of inclusions is mainly attributed to turbulent collision, and the impact of Stokes collision is limited; nearly all the inclusions are separated to the top slag, and the inclusions separated to the refractory can be ignored. It could be concluded that the present model is an interesting step in predicting the inclusion behaviors in the RH reactor. However, there are still some improvements to be made. In the modeling, the nucleation is ignored, and the measured value is used for the initialization. In the future, heterogeneous nucleation should be introduced. Another important drawback is the computational efficiency. Because computation with the inhomogeneous MUSIG model can take several months, recent advances could be used, such as SQMOM41 and KTAWSR.43

Figure 17 provides the predicted local distribution of inclusions in the RH reactor 900 s after the start of the refining process. As seen in Figure 17, the distributions of mass fraction, number density, and Sauter diameter of the inclusions are slightly large and annular at the recirculation zone around the downward stream in the ladle. This is because, at the recirculation zone, both large and small inclusions are difficult to remove, and some inclusions accumulate here. Despite these differences, the inclusions are still relatively uniformly distributed in the RH reactor as a whole because the RH reactor is essentially a great mixer.

5. CONCLUSION In this study, a 3-D coupled CFD−PBM model using an Eulerian−Eulerian approach was developed to simulate the transport, aggregation, and removal of Al2O3 clusters in a RH reactor. The inhomogeneous MUSIG model was first employed to solve the PBE for the inclusion behaviors in steel. Industrial tests were implemented to verify the simulation results. Meanwhile, the application of a commonly used homogeneous MUSIG model in inclusion behaviors was also evaluated. Then, the importance of various mechanisms for the aggregation and removal of inclusions was discussed. The distribution of the inclusions in the RH reactor was presented. The following conclusions can be drawn from this work: (1) The simulated results show that the mass fraction, number density, Sauter diameter, and size distribution of the inclusions in the RH reactor can be well predicted by the present coupled CFD−PBM model with the inhomogeneous MUSIG method, along with the appropriate dynamics of the transport, agglomeration, and removal of Al2O3 clusters. (2) The homogeneous MUSIG method may not be

■

AUTHOR INFORMATION

Corresponding Author

*Tel: +86-023-65102469. Fax: +86-023-65102469. E-mail: [email protected]. 7040


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Industrial & Engineering Chemistry Research Notes

The authors declare no competing financial interest.

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NOMENCLATURE Acell = local interface area, m2 BAgg s, n, i = birth rate of inclusion of size class i in velocity group n due to aggregation, kg/(m3·s) C = a scale factor ds, e, ds, c, ds, p, ds, c, n, 32 = equivalent diameter, cluster diameter, particle diameter, and Sauter diameter, m Df = three-dimensional fractal dimension DAgg s, n, i = death rate of inclusion of size class i in velocity group n due to aggregation, kg/(m3·s) ess = coefficient of restitution for particle collisions fs, n, i = mass fraction of solid of size class i in velocity group n F⃗g, F⃗l = sum of the nondrag forces, N/m3 g ⃗ = gravitational acceleration vector, m/s2 g0, ss = radial distribution function Ha = Hamaker constant kΘs = diffusion coefficient for the energy fluctuation, kg/(m·s) kl = turbulent kinetic energy of liquid, m2/s2 Kl‑g, Kl‑s, n = momentum transfer coefficient, kg/(m3·s) ms, i = mass of a single inclusion of size class i, kg ns, i = number concentration of inclusions of size class i, 1/m3 Ns, p = number of fine particles in a cluster, 1 NTurbulent = ratio between the viscous force and the van der ij Waals force p, ps, n = pressure, N/m2 Qg = argon flow rate, Nm3/h Res, c, n = Reynolds number of clusters for velocity group n Agg SAgg s, n , Ss, n, i = mass source term of inclusion due to aggregation, kg/(m3·s) Rem SRem s, n , Ss, n, i = mass source term of inclusion phase due to removals, kg/(m3·s) Slag SWall s, i , Ss, i = mass source term of inclusion due to wall and slag removal, kg/(m3·s) u⃗g, u⃗l, us⃗ , n = velocity of gas, liquid, and solid, m/s Rising VRising = rising velocity of solid of an equivalent solid s, e , Vs, c sphere and a cluster, m/s Vs, i = volume of a single inclusion of size class i, m3 Vt = total turbulent collision rate, m/s Vcell = local cell volume, m3

■

e = equivalent solid sphere g = gas phase l = liquid phase n = velocity group n s = solid phase

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Greek Letters

α = volume fraction βij, βStokes , βTurbulent = aggregation rate, m3/s ij ij Stokes Turbulent γ̇ij , γ̇ij = strain rate for Stokes collision and turbulent collision, 1/s εl = turbulent kinetic energy dissipation rate of steel, m2/s3 Θs, n = granular temperature, m2/s2 λs, n = solid bulk viscosity, kg/(m·s) μ = dynamic viscosity, kg/(m·s) νl = kinematic viscosity of liquid, m2/s ξkl = a factor projecting the corresponding part of the birth particle into the size class i ςStokes , ςTurbulent = collision efficiency factor ij ij ρ = density, kg/m3 τg , τl , τs, n = stress tensor for gas, liquid, and solid, N/m2 φ = angle of internal friction, deg Φls, n = transfer of kinetic energy of random fluctuations, kg/(m3·s) Subscripts

c = cluster 7041


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