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Numerical Simulation of Gas and Liquid Two-Phase Flow in Gas-Stirred Systems Based on Euler–Euler Approach

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Abstract

Based on the Euler–Euler approach, a mathematical model is established to describe gas and liquid two-phase flow in the gas-stirred system for steelmaking, and the influences of the interphase force including turbulent dispersion force, drag force, and lift force are investigated. The modified k–ε model with extra source terms to account for the bubble-induced turbulence is adopted to model the turbulence in the system, and the simulation results of gas volume fraction, liquid velocity, and turbulent kinetic energy are compared with the measured data. The results show that the turbulent dispersion force dominates the bubbly plume shape and is responsible for successful prediction of the gas volume fraction. The bubble-induced turbulence has a significant influence on the liquid turbulence, and the conversion coefficient C b, which denotes the fraction of bubble-induced energy converted into liquid turbulence, should be in the range of 0.8 and 0.9. The drag force also strongly influences the bubbly plume dynamics, and the coefficient model proposed by Kolev performs the best for determining the drag force; however, the lift force and bubble diameter do not have much effect on the current bubbly plume system. For different gas flow rates, the current Euler–Euler approach predictions are more consistent with the measured data than the Euler–Lagrange approach and the early Euler–Euler model.

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Abbreviations

C A :

Virtual mass force coefficient (–)

C b :

Turbulent conversion coefficient (–)

C D :

Drag force coefficient (–)

C L :

Lift force coefficient (–)

D tgl :

Turbulent dispersion coefficient (–)

e R :

Total rate of pressure energy lost by the bubble (kg m2/s3)

e D :

Rate of energy converted into turbulence and spent in viscous dissipation (kg m2/s3)

Eo :

Eötvös number (–) \( Eo = {{g\left( {\rho_{\text{l}} - \rho_{\text{g}} } \right)d_{\text{g}}^{2} } \mathord{\left/ {\vphantom {{g\left( {\rho_{\text{l}} - \rho_{\text{g}} } \right)d_{\text{g}}^{2} } \sigma }} \right. \kern-0pt} \sigma } \)

F D :

Drag force per unit volume (N/m3)

F L :

Lift force per unit volume (N/m3)

F VM :

Virtual mass force per unit volume (N/m3)

F TD :

Turbulent dispersion force per unit volume (N/m3)

G k,l :

Production of turbulent kinetic energy due to liquid mean velocity gradients (m2/s3)

G b :

Bubble-induced turbulent kinetic energy (m2/s3)

d g :

Diameter of the bubbles (m)

d 0 :

Diameter of the nozzle (m)

dt i :

Residence time of the ith bubble in the control cell volume (s)

\( \bar{g} \) :

acceleration due to gravity (m2/s)

H :

Bath liquid height (m)

k l :

Liquid turbulent kinetic energy (m2/s2)

\( \overline{M}_{\text{k}} \) :

interphase momentum exchange term (N/m3)

M Wg :

Molecular weight of gas phase (kg/kmol)

N :

Number of bubbles particle released from the gas-inlet (–)

P :

Static pressure (Pa)

P op :

Operating pressure (Pa)

Q g :

Gas flow rate (m3/s)

R :

Universal gas constant (J/(mol K))

Re:

Local bubble Reynolds number (–) \( \text{Re} = {{\rho_{l} \left( {u_{g} - u_{l} } \right)d_{g} } \mathord{\left/ {\vphantom {{\rho_{l} \left( {u_{g} - u_{l} } \right)d_{g} } {\mu_{l} }}} \right. \kern-0pt} {\mu_{l} }} \)

\( \bar{u}_{\text{l}} \) :

liquid velocity (m/s)

\( \bar{u}_{g} \) :

gas velocity (m/s)

u drift :

D

drift velocity between liquid and gas (m/s)

V :

Liquid velocity magnitude (m/s)

V cell :

Grid cell volume (m3)

V bub,pi :

Volume of the ith bubble in the control cell volume (m3)

We :

Webber number (–) \( We = {{\rho_{l} \left( {u_{l} - u_{g} } \right)^{2} d_{g} } \mathord{\left/ {\vphantom {{\rho_{l} \left( {u_{l} - u_{g} } \right)^{2} d_{g} } \sigma }} \right. \kern-0pt} \sigma } \)

z :

Height far from the bath bottom (m)

α :

Volume fraction (–)

ρ l,ρ g :

Liquid and gas density (kg/m3)

μ l, μ t, μ eff :

Liquid molecular viscosity, turbulent viscosity, effective viscosity (kg/(m s))

σ :

Gas–liquid surface tension coefficient (N/m)

ω gl :

Dispersion Prandtl number (–)

τ tgl :

Bubble turbulent characteristic time (s)

τ Fgl :

Characteristic time of particle entrainment by the continuous fluid motion (s)

τ tl :

Characteristic time of the energetic turbulent eddies (s)

θ :

Angle between the mean particle velocity and the mean relative velocity (rad)

ɛ l :

Turbulent dissipation rate (m2/s3)

Π k,l , Π ɛ,l :

Influence of the dispersed phases turbulence on the continuous phase turbulence in Eqs. [25] and [26][50,51]

g:

Gas phase

l:

Liquid phase

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Acknowledgments

The authors wish to express thanks to the National Outstanding Young Scientist Foundation of China (Grant No. 50925415) and the National Natural Science Foundation of China (Grant No. 51134009) for supporting this work.

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Authors and Affiliations

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

    Wentao Lou (Ph.D. Candidate) & Miaoyong Zhu (Professor)

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  1. Wentao Lou
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Correspondence to Miaoyong Zhu.

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Manuscript submitted March 28, 2012.

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Lou, W., Zhu, M. Numerical Simulation of Gas and Liquid Two-Phase Flow in Gas-Stirred Systems Based on Euler–Euler Approach. Metall Mater Trans B 44, 1251–1263 (2013). https://doi.org/10.1007/s11663-013-9897-6

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