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Population Balance Modeling of Polydispersed Bubbly Flow in Continuous-Casting Using Multiple-Size-Group Approach

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Abstract

A population balance model based on the multiple-size-group (MUSIG) approach has been developed to investigate the polydispersed bubbly flow inside the slab continuous-casting mold and bubble behavior including volume fraction, breakup, coalescence, and size distribution. The Eulerian–Eulerian approach is used to describe the equations of motion of the two-phase flow. All the non-drag forces (lift force, virtual mass force, wall lubrication force, and turbulent dispersion force) and drag force are incorporated in this model. Sato and Sekiguchi model is used to account for the bubble-induced turbulence. Luo and Svendsen model and Prince and Blanch model are used to describe the bubbles breakup and coalescence behavior, respectively. A 1/4th water model of the slab continuous-casting mold was applied to investigate the distribution and size of bubbles by injecting air through a circumferential inlet chamber which was made of the specially-coated samples of mullite porous brick, which is used for the actual upper nozzle. Against experimental data, numerical results showed good agreement for the gas volume fraction and local bubble Sauter mean diameter. The bubble Sauter mean diameter in the upper recirculation zone decreases with increasing water flow rate and increases with increasing gas flow rate. The distribution of bubble Sauter mean diameter along the width direction of the upper mold increases first, and then gradually decreases from the SEN to the narrow wall. Close agreements between the predictions and measurements demonstrate the capability of the MUSIG model in modeling bubbly flow inside the continuous-casting mold.

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Abbreviations

B B :

Birth rate due to breakup into smaller bubbles

B C :

Birth rate due to coalescence of smaller bubbles

C D :

Drag coefficient

C L :

Lift force model constant

C VM :

Virtual mass force model constant

C WL :

Wall lubrication force model constant

C TD :

Turbulent dispersion model constant

C μ,BI :

Sato and Sekiguchi model constant

C μ,T :

Model constant

d :

Bubble diameter

d ij :

Equivalent bubble diameter

D B :

Death rate due to breakup into smaller bubbles

D C :

Death rate due to coalescence of smaller bubbles

D S :

Sauter mean bubble diameter

f BV :

Stochastic breakage volume fraction

f g,i :

Fraction of dispersed phase volume fraction in group-i

G k :

Rate of production of turbulent kinetic energy

F B :

Breakage calibration factor

F C :

Coalescence calibration factor

F h, F lg, F gl :

Interfacial forces between the two phases

\( F_{\lg }^{D} \) :

Drag force

\( F_{\lg }^{L} \) :

Lift force

\( F_{\lg }^{\text{VM}} \) :

Virtual mass force

\( F_{\lg }^{WL} \) :

Wall lubrication force

\( F_{\lg }^{\text{TD}} \) :

Turbulent dispersion force

\( \vec{g} \) :

Gravity acceleration vector

\( h_{0} \) :

Initial film thickness

\( h_{\text{f}} \) :

Critical film thickness

I:

Unity tensor

k :

Turbulent kinetic energy

n i, n j :

Number density of group-i/j bubbles

\( \vec{n}_{\omega } \) :

Outward vector normal to the wall

P :

Static pressure

P B :

Production rate due to breakup

P C :

Production rate due to coalescence

t :

Physical time

\( \vec{u} \) :

Velocity

\( \vec{u}_{t} \) :

Turbulence velocity

v :

Volume

\( \alpha \) :

Volume fraction

ρ :

Density

\( \varepsilon \) :

Turbulent kinetic energy dissipation

μ eff :

Effective viscosity

μ L :

Molecular viscosity

μ T :

Turbulence viscosity

μ BI :

Bubble-induced viscosity

σ :

Surface tension

σ t :

Turbulent Schmidt number

τ :

Stress

\( \xi \) :

Size ratio

Reb :

Bubble Reynolds number

\( \Omega \) :

Bubble breakup rate

\( \upsilon_{\text{t}} \) :

Turbulence kinematic viscosity

\( \chi_{ij} \) :

Turbulent random coalescence rate

\( \eta_{\text{jki}} \) :

Transfer coefficient between bubble groups arising from bubble breakup

\( \varPi_{\text{k}} ,\prod_{\varepsilon } \) :

Influence of dispersed phase on continuous phase

g :

Gas phase

i, j, k :

Index of bubble

l :

Liquid phase

s :

Sauter model

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Acknowledgments

Authors are grateful to the National Natural Science Foundation of China for support of this research, Grant No. 51210007.

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Authors and Affiliations

  1. School of Materials and Metallurgy, Northeastern University, Shenyang, 110819, P.R. China

    Zhongqiu Liu (Ph.D. Candidate), Linmin Li (Ph.D. Candidate), Fengsheng Qi (Lecturer), Baokuan Li (Professor) & Maofa Jiang (Professor)

  2. Department of Advanced Materials, Graduate School of Frontier Science, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8561, Japan

    Fumitaka Tsukihashi (Professor)

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  1. Zhongqiu Liu
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Correspondence to Baokuan Li.

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Manuscript submitted October 8, 2013.

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Liu, Z., Li, L., Qi, F. et al. Population Balance Modeling of Polydispersed Bubbly Flow in Continuous-Casting Using Multiple-Size-Group Approach. Metall Mater Trans B 46, 406–420 (2015). https://doi.org/10.1007/s11663-014-0192-y

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