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Front Tracking simulations on liquid-liquid systems; an investigation of the drag force on droplets Ivo Roghair, Wouter Dijkhuizen, Martin van Sint Annaland and Hans Kuipers Fundamentals of Chemical Reaction Engineering CFD2008-071 June

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Ivo Roghair, Wouter Dijkhuizen, Martin van Sint Annaland and Hans Kuipers CFD2008-071 – June 12th, 2008

Front Tracking simulations on liquid-liquid systems; an investigation of the drag force on droplets

Fundamentals of Chemical Reaction Engineering

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Contents

  • Introduction
  • Objectives
  • Numerical simulations

– Grid dependency study – Drag force study

  • Conclusions and outlook
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Introduction

Multi-level modelling strategy for multiphase flow

Direct numerical simulations Discrete element model Multi-fluid continuum model Closures for:

  • Drag, lift, virtual mass
  • Swarm effects
  • Mass transfer coefficients

Medium scale structures Large scale structures

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Introduction

Direct Numerical Simulations (DNS)

  • Fully resolved

– Based only on fundamental equations for fluid flow

  • Navier-Stokes + continuity equation for incompressible

flow

– Can be used to derive closures for forces on

  • Bubbles
  • Droplets
  • Particles
  • Only valid when grid independence can be

shown!

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Front tracking

  • Incompressible fluids
  • Fixed Eulerian grid
  • Interface consists of Lagrangian marker

points that build up a triangular mesh

– Points are moved with the interpolated fluid flow – Straightforward surface tension force calculation

  • Advantages

– Calculation of surface tension force with sub-grid accuracy. – No numerical coalescence of dispersed phase elements

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Front tracking

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Drag force

FD FG FP FL FVM Droplet velocity Σ F FD

Forces acting on a droplet Stationary force balance in the rise direction

mb  dvb dt =  FG  F P  F D  F L  F VM=∑  F c−dg  6 d eq

3 −1

2 C Dc  4 d eq

2  

ud , z−  uc , z

2=0

CD= 4 c−d  g d eq 3c  ud , z−  uc , z

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Drag force

  • Determine drag force coefficient by

different averaging procedures

– Average rise velocity, then determine CD – Determine CD as a function of time, average this value → No difference

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Drag force

CD= 24 Re CD= 16 Re 1 2 1 16 Re3.315 Re

0.5 

CD=max[ min[ 16 Re

10.15Re

0.687, 48

Re], 8 3 Eo Eo4] Re=c  ud d eq c Eo=

c−d  g d eq

2

 CD=max[ 24 Re

10.15Re

0.687 , 8

3 Eo Eo4]

Correlations from literature (bubbly flow)

Rigid sphere: Mei et al. (1994): Tomiyama (1998):

– Pure – Contaminated

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Drag force

  • Experiments and simulations on drag

force for bubbly flow

From: Wouter Dijkhuizen, PhD thesis, University of Twente, 2008

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Objectives

  • Investigate the behavior of the Front

Tracking model for liquid-liquid systems

  • Simulate droplets in an infinite quiescent

liquid to derive drag force closures

  • Investigate the relation between gas-

liquid and liquid-liquid drag force and their dependencies

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Grid dependency

  • Vary resolution in droplet, domain 5 times droplet size
  • Vary resolution in droplet, keep domain at 1003 cells
  • Keep resolution in droplet at 20 cells, vary domain size

Simulation parameters:

ρc = 1000 kg/m3, μc = 10-3 Pa·s ρd = 800 kg/m3, μd = 10-1 Pa·s σ = 52.9 mN/m, deq = 1 mm tend = 1 s dt = 10-5 s

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Grid dependency

  • Vary resolution in droplet, domain 5 times droplet size
  • Vary resolution in droplet, keep domain at 1003 cells
  • Keep resolution in droplet at 20 cells, vary domain size

30 100 20 6

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Grid dependency

  • Vary resolution in droplet, domain 5 times droplet size
  • Vary resolution in droplet, keep domain at 1003 cells
  • Keep resolution in droplet at 20 cells, vary domain size

100 100 20 8

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Grid dependency

  • Vary resolution in droplet, domain 5 times droplet size
  • Vary resolution in droplet, keep domain at 1003 cells
  • Keep resolution in droplet at 20 cells, vary domain size

50 100 20 20

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Drag force simulations

  • Used settings:

– 20 grid cells in droplet diameter – 1003 grid cells in domain

  • Variation of continuous phase viscosity

between 0.001 - 0.2 Pa·s

  • Variation of equivalent droplet diameter

between 0.2 – 5 mm

  • “Dodecane droplet in water” system:

– ρc = 1000 kg/m3; – ρd = 746 kg/m3; μd = 1.34·10-3 Pa·s – σ = 0.0529 N/m;

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Drag force simulations

  • Variation of continuous phase viscosity
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Drag force simulations

  • Variation of dispersed phase viscosity

between 10-3 – 10-1 Pa·s

  • Variation of equivalent droplet diameter

between 0.2 – 7 mm

  • Physical properties

– ρc = 1000 kg/m3; μc = 10-1 Pa·s – ρd = 800 kg/m3; – σ = 0.0529 N/m;

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Drag force simulations

  • Variation of dispersed phase viscosity
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Drag force simulations

  • Due to volume losses more detailed

simulations:

– Computational grid 1503 cells – 30 cells within droplet diameter – Higher surface tension

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Drag force simulations

Simulation parameters: ρc = 1000 kg/m3; μc = 10-3 Pa·s ρd = 800 kg/m3; μd = 10-1 Pa·s σ = 0.1 N/m; deq = 0.5 - 7 mm

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Drag force simulations

Simulation parameters: ρc = 1000 kg/m3; μc = 10-3 Pa·s ρd = 800 kg/m3; μd = 10-3 - 0.5 Pa·s σ = 0.1 N/m; deq = 1 mm

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Conclusions and outlook

  • Front tracking model can simulate dispersed liquid

phases but a high resolution is required

  • Volume loss strongly depending on droplet resolution
  • Correlations of Mei et al. and Tomiyama for bubbly

flow are well predicted – Some overshoot due to wall effects

  • Transition of free-slip to no-slip condition as a function
  • f μd shown
  • Outlook:

– Eo dependence of drag force coefficient – Droplet and bubble swarms

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Thank you

Thank you for your attention

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Front tracking

 Surface tension is

mapped from the interface mesh to the Eulerian grid.

a b c m

Fc Fb Fa na nb nc tm,a tm,c tm,b

 F a =σ  t m,a× na  F b=σ  t m,b× nb  F c=σ  t m,c× nc 