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Extension of the EMMS Model to Gas-Liquid Systems

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From Multiscale Modeling to Meso-Science

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Abstract

The Dual-Bubble-Size (DBS) model is an extension of the energy minimization multiscale (EMMS) approach for gas-liquid systems. The system is resolved into a liquid phase, small bubbles and large bubbles, and is jointly dominated by two movement tendencies; i.e., those of the small and large bubbles. A stability condition is formulated to reflect the compromise between these dominant mechanisms, offering another constraint in addition to mass and momentum conservation equations. The DBS model can theoretically predict the regime transition in bubble columns and physically explain the macro-scale evolution of flow structures through the jump change in the global minimum of the micro-scale energy dissipation changing from one point to another within the model space of the structure parameters. The DBS model is found to be an intrinsic model for gas-liquid systems in contrast to the models for single, triple, and multiple classes of bubble. A new model for the ratio of drag coefficient to bubble diameter, that is, the EMMS drag, is then integrated into the Eulerian-Eulerian computational fluid dynamics (CFD) models. The resulting improved prediction demonstrates the ability of the DBS model to reveal the multiscale nature and complexity of gas-liquid systems.

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Abbreviations

C D :

Drag coefficient for a bubble in a swarm, dimensionless

C D0 :

Drag coefficient for a bubble in a quiescent liquid, dimensionless

C D,p :

Drag coefficient for a particle in multi-particle systems, dimensionless

c f :

Coefficient of surface area increase, \( c_{\text{f}} = f_{\rm BV}^{2/3} + (1 - f_{\rm BV} )^{2/3} - 1 \), dimensionless

d b :

Bubble diameter, m

d L :

Bubble diameter of large bubbles, m

d S :

Bubble diameter of small bubbles, m

Eo :

Eötvos number, dimensionless

f b :

Volume fraction of gas phase, dimensionless

f L :

Volume fraction of large bubbles, dimensionless

f S :

Volume fraction of small bubbles, dimensionless

f BV :

Breakup ratio of daughter bubble to its mother bubble, dimensionless

g :

Gravitational acceleration, m/s2

N break :

Rate of energy consumption due to bubble breakage and coalescence per unit mass, m2/s3

N surf :

Rate of energy dissipation due to bubble oscillation per unit mass, m2/s3

N turb :

Rate of energy dissipation in turbulent liquid phase per unit mass, m2/s3

N st :

Rate of energy dissipation for suspending and transporting particles per unit mass, m2/s3

N T :

Total rate of energy dissipation

P b :

Bubble breakup probability, dimensionless

U g :

Superficial gas velocity, m/s

U g,L :

Superficial gas velocity for large bubbles, m/s

U g,S :

Superficial gas velocity for small bubbles, m/s

U l :

Superficial liquid velocity, m/s

V rel :

Relative velocity between gas and liquid, m/s

ε l :

Volume fraction of liquid, dimensionless

λ :

Character size of eddy, m

μ :

Viscosity, Pa·s

ρ:

Density, kg/m3

σ :

Surface tension, N/m

ω :

Collision frequency, 1/s

b :

Bubble

g :

Gas

l :

Liquid

L :

Large bubble

p :

Particle

S :

Small bubble

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Authors and Affiliations

  1. Institute of Process Engineering, Chinese Academy of Sciences, Zhongguancun North Second Street 1, Beijing, 100190, People’s Republic of China

    Jinghai Li, Wei Ge, Wei Wang, Ning Yang, Xinhua Liu, Limin Wang, Xianfeng He, Xiaowei Wang, Junwu Wang & Mooson Kwauk

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  1. Jinghai Li
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  2. Wei Ge
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  3. Wei Wang
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  5. Xinhua Liu
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  10. Mooson Kwauk
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Correspondence to Jinghai Li .

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Li, J. et al. (2013). Extension of the EMMS Model to Gas-Liquid Systems. In: From Multiscale Modeling to Meso-Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35189-1_4

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  • DOI: https://doi.org/10.1007/978-3-642-35189-1_4

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