Pressure drop of three-phase liquid–liquid–gas slug flow in round microchannels

  • Agnieszka Ładosz
  • Eugen Rigger
  • Philipp Rudolf von RohrEmail author
Research Paper


In this paper we present a model for the calculation of pressure drop of three-phase liquid–liquid–gas slug flow in microcapillaries of a circular cross section. Introduced models consist of terms attributing for frictional and interfacial pressure drop, incorporating the presence of a stagnant thin film at the wall of the channel. Different formulations of the interfacial pressure drop equation were employed, using expressions developed by Bretherton (J Fluid Mech 10:166–188, 1961), Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989). Models were validated experimentally using oleic acid–water–nitrogen and heptane–water–nitrogen three-phase flows in round Teflon or Radel R microchannels of 254- and 508-µm nominal inner diameter, for capillary numbers Ca b between 10−4 and 4.9 × 10−1 and Reynolds numbers Re between 0.095 and 300. Best agreement between measured and calculated values of pressure drop, with relative error between −22 and 19 % or −20 and 16 %, is reached for Warnier’s or Ratulowski and Chang’s interfacial pressure drop equation, respectively. The results prove that three-phase slug flow pressure drop can be successfully predicted by extending existing two-phase slug flow correlations. Good agreement of Bretherton’s equation was reached only at lower Ca numbers, indicating that an extension of the interfacial pressure drop equation as performed by Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989) for higher capillary numbers is necessary. Additionally it was demonstrated that pressure drop increases substantially if dry slug flow occurs or if microchannels with significant surface roughness are employed. Those influences were not accounted for in the models presented.


Multiphase flow Three-phase flow Liquid–liquid–gas Slug flow Microchannel Pressure drop Surface roughness 

List of symbols


Area (m2)


Constant in Bretherton’s pressure drop equation, c = 9.04 in this work (–)


Capillary number (–)


Capillary number based on bubble velocity at the tube outlet and gas–liquid surface tension (–)


Capillary number based on bubble velocity at the tube outlet and liquid–liquid surface tension (–)


Diameter (m)


Diameter of the test tube (m)


Slug frequency (1/s)


Film thickness (m)


Length (m)


Pressure drop (Pa)


Radius (m)


Reynolds number (–)


Surface roughness parameter: absolute peak to valley distance (m)


Superficial velocity (m/s)


Velocity of bubble or droplet (m/s)


Weber number (–)


Distance in the z-direction (m)

Greek symbols


Correction of the slug length due to bubble or droplet cap curvature (m)


Viscosity (Pa s)


Surface tension (N/m)





Continuous phase








Unit cell



The authors are grateful to Roger Wallimann from Transport Processes and Reactions Laboratory at ETH for providing SEM images of channel surfaces, group of Prof. Mazza at ETH for access to the confocal laser scanning microscope and the Scientific Center for Optical and Electron Microscopy ScopeM at the ETH for access to the SEM facility and help with the analysis of CLSM images.

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Agnieszka Ładosz
    • 1
  • Eugen Rigger
    • 1
  • Philipp Rudolf von Rohr
    • 1
    Email author
  1. 1.Institute of Process EngineeringETH ZurichZurichSwitzerland

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