Microfluidics and Nanofluidics

, Volume 16, Issue 6, pp 1057–1067 | Cite as

Holdup characteristics of two-phase parallel microflows

  • Anil B. Vir
  • Shekhar R. Kulkarni
  • J. R. Picardo
  • Avinash Sahu
  • S. Pushpavanam
Research Paper
First Online:
Received:
Accepted:

Abstract

Two-phase parallel microflows, i.e., stratified flow and core-annular flow, have many applications in lab-on-chip devices. These include transport and reaction processes, such as liquid–liquid extraction and phase transfer catalysis. The phase holdup (fraction of the microchannel volume occupied by a specified phase) is a key parameter of these flow systems. In this work, mathematical models based on fundamental principles are used to predict the phase holdup in stratified flow and core-annular flow. For stratified flow, a two-dimensional model of flow in a rectangular channel of arbitrary aspect ratio is considered. A simpler one-dimensional model of stratified flow between infinite parallel plates is also analyzed. In the case of core-annular flow, axisymmetry is assumed in the model. The results of the models agree well with published experimental results. The dependence of phase holdup on the flow-rate fraction (the primary operating variable which can be controlled experimentally) is studied in detail. The nature of this relationship varies with the ratio of fluid viscosities and the channel’s aspect ratio (in the case of stratified flow). In the literature, the holdup is sometimes erroneously assumed to be identical to the flow-rate fraction. It is shown that this is not possible in the case of core-annular flow, while in stratified flow it is true only for a unique critical flow-rate. This critical flow-rate is viscosity dependent. The aspect ratio of the channel is found to have a considerable influence on the holdup in stratified flow when the fluids have different viscosities. However, even in such cases, there exists a point of geometric invariance at which the holdup is independent of the aspect ratio. At this point, the simple one-dimensional model of stratified flow can predict the holdup with complete accuracy.

Keywords

Holdup Flow-rate fraction Stratified flow Core-annular flow Microflows 

List of symbols

D

Depth of channel

H

Width of channel

h

Interface position

\(\frac{dp}{dz}\)

Constant pressure gradient in the direction of flow

Qi

Volumetric flow-rate of ith phase

vi

Velocity of ith phase

Greek letters

αca

Holdup of the core fluid in core-annular flow

αs

Holdup of phase 1 in stratified flow

αs*

Holdup at the point of geometric invariance \(\left( { \approx \frac{1}{1 + \mu }} \right)\)

λ

Aspect ratio \(\left( { = \frac{H}{D}} \right)\)

μi

Viscosity of ith phase

µ

Ratio of the viscosity of phase 1 (core fluid) to the viscosity of phase 2 (annular fluid) in stratified (core-annular) flow

Φca

Flow-rate fraction of the core fluid in core-annular flow

Φs

Flow-rate fraction of phase 1 in stratified flow

Φcrit,ca

Critical flow-rate fraction of the core fluid in core-annular flow (shown to be a physically unrealistic mathematical quantity)

Φcrit,s

Critical flow-rate fraction of phase 1 in stratified flow

Φs*

Flow-rate fraction of phase 1 in stratified flow at a holdup of \(\left( {\frac{1}{1 + \mu }} \right)\) which approximates the point of geometric invariance (α s * )

Subscripts

i = 1, 2

Phase 1, phase 2 in stratified flow

i = a, c

Annular fluid and core fluid in core-annular flow

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Notes

Acknowledgments

The authors are grateful to Council of Scientific and Industrial Research (CSIR) [CHE/1011/098/CSIR/SPUS] for supporting this work. S.R.K. dedicates this work to his sister and parents. The authors thank the editor and the anonymous referees for a valuable review which helped to improve the overall quality of the manuscript.

Supplementary material

10404_2013_1269_MOESM1_ESM.pdf (100 kb)
Supplementary material 1 (PDF 101 kb)
10404_2013_1269_MOESM2_ESM.nb (35 kb)
Supplementary material 2 (NB 35 kb)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Anil B. Vir
    • 1
  • Shekhar R. Kulkarni
    • 1
  • J. R. Picardo
    • 1
  • Avinash Sahu
    • 1
  • S. Pushpavanam
    • 1
  1. 1.Department of Chemical EngineeringIndian Institute of Technology Madras (IIT Madras)ChennaiIndia

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