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Mass transfer from a Taylor bubble to the surrounding flowing liquid at the micro-scale: a numerical approach

  • Mónica C. F. Silva
  • João M. Miranda
  • João B. L. M. Campos
  • José D. P. AraújoEmail author
Research Paper

Abstract

Gas–liquid slug flow is characterized by complex and intermittent hydrodynamic features that offer an efficient alternative to promote biofilm control. In the present work, the mechanism of transferring a gaseous solute into a co-current liquid in a micro-scale slug flow system was inspected in detail. Specifically, the gas–liquid mass transfer from an individual Taylor bubble filled with oxygen was numerically studied using CFD techniques. To accurately describe the referred phenomenon, the hydrodynamic and concentration fields were simultaneously solved. Furthermore, the interface capturing based on the VOF methodology was also coupled to this solution approach. Three sub-categories within slug flow pattern were identified based on the flow behavior in the liquid phase: no liquid in recirculation (Case A); closed wake below the bubble tail (Case B); and recirculation ahead and below bubble (Case C). Regarding the solute distribution, in Case A the solute is dispersed only backwards, it accumulates in the closed wake structure in Case B, and it reaches the wall within the film region in Case C. Local and average mass transfer coefficients were also estimated for the different cases. The influence of the two most relevant dimensionless groups (Reynolds and Capillary numbers) was also briefly analyzed. Global mass transfer coefficients results confirmed that the penetration theory can provide reasonable estimations for systems like Case C.

Keywords

Mass transfer Micro-scale CFD VOF Oxygen 

List of symbols

\(a\)

Interfacial area per unit cell volume (m2/m3)

\(A\)

Interfacial area (m2)

\({B_i}\)

Mass source defined by user (kg m−3 s−1)

\(c\)

Concentration (kg m−3)

\({c^*}\)

Saturation concentration (kg m−3)

\({c_{{\text{dis}}}}\)

Concentration on the cell center (kg m−3)

\(D\)

Tube diameter (m)

\({D_{\text{b}}}\)

Bubble diameter (m)

\({D_{\text{L}}}\)

Coefficient of diffusion (m2 s−1)

\(\vec {F}\)

External body force (N m−3)

\(g\)

Gravitational acceleration (m2 s−1)

\(h\)

Distance along the normal to the gas interface (m)

\({\vec {J}_i}\)

Diffusive flux of specie i (kg m−2 s−1)

\({k_{\text{L}}}\)

Mass transfer coefficient for the liquid phase (m s−1)

\({\bar {k}_{\text{L}}}\)

Average/global mass transfer coefficient for the liquid phase (m s−1)

\({k_{\text{L}}}a\)

Volumetric liquid side mass transfer coefficient (s−1)

\(l\)

Distance from the nose tip to the liquid film fully developed (m)

\({L_{\text{l}}}\)

Liquid slug length (m)

\({L_{\text{s}}}\)

Slug length (m)

\({L_{{\text{UC}}}}\)

Unit cell length (m)

\(\vec {n}\)

Normal vector

\({N_{\text{A}}}\)

Mass transfer rate (kg s−1)

\(p\)

Static pressure (Pa)

\({P_{\text{c}}}\)

Cell center position

\({P_{\text{i}}}\)

Initial position

\({P_{\text{n}}}\)

Computed position

\(r\)

Radial coordinate (m)

\({R_{\text{i}}}\)

Net rate production of specie i (kg m−3 s−1)

\(s\)

Coordinate along the bubble surface (m)

\({s_{\text{t}}}\)

Bubble perimeter (m)

\(t\)

Time (s)

\({t_{\text{c}}}\)

Gas–liquid contact time (s)

\(u\)

Velocity at the bubble interface (m s−1)

\(\vec {u}\)

Velocity vector (m s−1)

\(u\)

Average velocity (m s−1)

\(u\)

Liquid velocity at the interface (m s−1)

\({U_\infty }\)

Bubble rising velocity through stagnant liquid (m s−1)

\(~{U_{\text{b}}}\)

Bubble rising velocity (m s−1)

\({U_{\text{G}}}\)

Superficial gas velocity (m s−1)

\({U_{\text{f}}}\)

Average velocity at the film (m s−1)

\({U_{\text{L}}}\)

Superficial liquid velocity (m s−1)

\({\bar {U}_L}\)

Average liquid velocity (m s−1)

\({y_i}\)

Local mass fraction of specie i (–)

\(z\)

Axial coordinate (m)

\({z_{\text{o}}}\)

Axial coordinate at the nose tip (m)

\({z_{\text{t}}}\)

Axial coordinate at the tail tip (m)

Greek letters

\(a\)

Volume fraction (–)

\({\delta _{\text{h}}}\)

Film thickness (m)

\({\delta _{\text{c}}}\)

Thickness of the concentration boundary layer (m)

\({\varepsilon _{\text{G}}}\)

Gas hold-up

\(\mu\)

Viscosity (Pa s)

\(\rho\)

Density (kg m−3)

\(\sigma\)

Surface tension (N m−1)

\(\overline{\overline {\tau }}\)

Stress–strain tensor (Pa)

Dimensionless groups

Ca

Capillary number

CaB

Capillary number considering the bubble velocity

Eo

Eötvös number

Re

Reynolds number

ReB

Reynolds number considering the bubble velocity

Sh

Sherwood number

Sc

Schmidt number

WeB

Webber number considering the bubble velocity

Notes

Acknowledgements

The authors acknowledge the support of FEDER funds through COMPETE2020—Operational Programme for Competitiveness Factors (POCI) and National Funds (PIDDAC) through FCT under projects PEst-OE/EME/UI0532 and POCI-01-0145-FEDER-031758. M.C.F. Silva also acknowledges the financial support provided by FCT through the PhD Grant PD/BD/ 52622/2014.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Centro de Estudos de Fenómenos de Transporte, Departamento de Engenharia QuímicaFaculdade de Engenharia da Universidade do PortoPortoPortugal

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