Modeling the forces acting on water droplets falling in oil under the influence of an electric field

1
Modeling the forces acting on water droplets
falling in oil under the influence of an electric
field
M. Chiesa, G. Berg, J. A. Melheim, S.
Ingebrigtsen Dept. of Energy Processes and Power
Technology, SINTEF, Norway

• INTRODUCTION AND SCOPE OF THE WORK
• LAGRANGIAN FORMULATION
• MAIN RESULT
• FORCES ON EACH SINGLE DROPLET
• CONCLUSION

2
GENERAL INTRODUCTION

• The oil extracted from offshore reservoirs will
  normally contain a large and during the reservoir
  lifetime, increasing percentage of water in oil.
• When the water-oil mixture is passed through the
  pressure relief valve an emulsion with high
  percentage of small water droplets is formed.
• Before the oil is pumped onshore or into tankers
  it is desirable to extract the water from this
  emulsion.

3
GENERAL INTRODUCTION

• The separation tanks are mainly built or operated
  as gravity separator with low flow rates and long
  residence times.
• The residence time mainly depends on the
  sedimentation velocity of the smallest droplets
  (dlt100?m)
• The electric fields are to some extent use to
  help smaller droplets to coalesce in to larger
  droplets that sediment quicker.
• The sedimentation velocity increases
  proportionally to the square of the diameter and
  therefore one wishes to get the smallest water
  droplets to coalesce

4
SCOPE OF THE PRESENT WORK

• Assess how well the forces acting on two
  approaching droplets are modeled in our numerical
  simulator
• Compare the optical observations and the
  numerical results obtained simulating a drop of
  water falling in oil in the presence of an
  electrostatic field

5
Test case

• High-speed CMOS camera
• Droplet released from syringe
  
• Stationary drop
• Electric field 0-400 V/mm

6
Modeling the forces acting on water droplets
falling in oil under the influence of an electric
field
M. Chiesa, G. Berg, J. A. Melheim, S.
Ingebrigtsen Dept. of Energy Processes and Power
Technology, SINTEF, Norway

• MOTIVATION
• LAGRANGIAN FORMULATION
• MAIN RESULT
• FORCES ON EACH SINGLE DROPLET
• CONCLUSION

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LAGRANGIAN FORMULATION OF DROPLET KINEMATICS

• Each droplet is tracked
• The droplet paths are given by a set of ordinary
  differential equations
• The forces acting on each droplet determine its
  motion
• Fext (external forces on the droplet)
• Ffluid (forces from the fluid on the droplet)
• Fd-d (inter-droplets forces)

8
Modeling the forces acting on water droplets
falling in oil under the influence of an electric
field
M. Chiesa, G. Berg, J. A. Melheim, S.
Ingebrigtsen Dept. of Energy Processes and Power
Technology, SINTEF, Norway

• MOTIVATION
• LAGRANGIAN FORMULATION
• MAIN RESULT
• FORCES ON EACH SINGLE DROPLET
• CONCLUSION

9
Main result

• In the Lagrangian formulation of the droplet
  motion, the best results is obatined with

• Drag force of LeVan 1
• Film-thinning force of Vinogradova 2
• Analytical expression of the electric force of
  Davis 3

10
Modeling the forces acting on water droplets
falling in oil under the influence of an electric
field
M. Chiesa, G. Berg, J. A. Melheim, S.
Ingebrigtsen Dept. of Energy Processes and Power
Technology, SINTEF, Norway

• MOTIVATION
• LAGRANGIAN FORMULATION
• MAIN RESULT
• FORCES ON EACH SINGLE DROPLET
• CONCLUSION

11
Drag force

• The following expressions
• for were considered

1. The rigid sphere model

1. The Hadamard-Rybczynski formula

E 300 V/mm, r 110 µm

1. LeVan 1 formula

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Film-thinning

• The following expressions
• for were considered

1. The rigid sphere model

1. The model of Vinogradova 2

1. Barnocky and Davis 4

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Film-thinning
E 300 V/mm, r 110 µm,
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Dipole-dipole force

• Tested Models for the dipole-dipole force

1. The point dipole

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Dipole-dipole force

1. The analytical expression of Davis 3

where are complicated series
depending on and

1. Dipole induced dipole model (DID) of Siu et al.
   5

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Dipole-dipole force
E 300 V/mm, r 110 µm,
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The force account

• Drag force of LeVan 1
• Film-thinning force of Vinogradova 2
• Analytical expression of the electric force of
  Davis 3

E 300 V/mm, r 110 µm,
18
Modeling the forces acting on water droplets
falling in oil under the influence of an electric
field
M. Chiesa, G. Berg, J. A. Melheim, S.
Ingebrigtsen Dept. of Energy Processes and Power
Technology, SINTEF, Norway

• MOTIVATION
• LAGRANGIAN FORMULATION
• MAIN RESULT
• FORCES ON EACH SINGLE DROPLET
• CONCLUSION

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RIGID SPHERE SPHERICAL PARTICLES FALLING TOWARD
AN ELECTRODE
RESULTS OVERVIEW
Predicted and observed velocity versus normalized
particle surface distance. No electric field is
applied r1 82.5µm b10-7m
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RIGID SPHERE SPHERICAL PARTICLES FALLING TOWARD
AN ELECTRODE
RESULTS OVERVIEW
Different electric field magnitudes are applied
in the numerical calculations of the particle
kinematics Davis analytical expression are used
r1 70µm, b10-7m
Constant electric field magnitude is applied E
300 V/mm and different models for the induced
electric field are used in the numerical
calculation. r1 70µm, b10-7m
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FLUID DROPLET FALLING TOWARD AN ELECTRODE
RESULTS OVERVIEW
Different electric field magnitudes are applied
in the numerical calculations of the particle
kinematics Davis analytical expression are used
r1 110µm, b10-6m, ?110-5N/m
Constant electric field magnitude is applied E
300 V/mm and different models for the induced
electric field are used in the numerical
calculation. r1 110µm, b10-6m, ?110-5N/m
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FLUID DROPLET FALLING TOWARD AN ELECTRODE
RESULTS OVERVIEW
Different drag force models are compared. E
300 V/mm. Davis analytical expression,
Vinigradovas model with b10-6m, and an
interfacial tension variation of ?110-5N/m. r1
110µm
Different drainage models are compared. E 300
V/mm. In Vinogradovas model b10-6m. Davis
analytical expression are used. r1 110µm
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FLUID DROPLET FALLING TOWARD AN ELECTRODE
RESULTS OVERVIEW
Different droplet sizes r1110µm and r258µm.
E 300 V/mm. The slip distance in the
Vinigradovas model is b10-6m for r1 and b10-7m
r2. ?110-5N/m for r1 and ?1210-5N/m for r2
Simple versus best modelling at E 300. The
simple model strategy includes Point dipole,
rigid sphere drag, no film force. The best
modelling approach employs Davis model, LeVan and
Vinogradova
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Conclusions

• The interfacial tension gradient, caused by the
  electric forces, inhibit the internal circulation
  that influences the drag force and the
  film-thinning force.
• The dipole-dipole force is best modelled by the
  analaytic expression of Davis 3. The DID model
  is a numerical efficient and accurate
  alternative, as long as the inter-droplet
  distance is not too small. (h/r gt 0.5)

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References

• 1 D. M LeVan. Motion of droplets with a
  Newtonian interface. J. Colloid Interface Sci.
  83(1)11-17, 1981.
• 2 O. I. Vinogradova. Drainage of a thin liquid
  film confined between hydrophobic interfaces.
  Langmuir, 112213-2220, 1995.
• 3 M. H. Davis. Two charged spherical conductors
  in a uniform electric field Forces and field
  strength. Rand. Corp. Memorandum RM-3860-PR,
  1964.
• 4 G. Barnocky and R. H. Davis. The lubrication
  force between spherical drops, bubbles, and rigid
  particles in a viscous fluid. Int. J. Multiphase
  Flow, 15627-638, 1989.
• 5 Y. L. Siu, T. K. Wan Jones, and K. W. Yu.
  Interparticle force in polydisperse
  electrorheological fluid Beyond the dipole
  approximation. Comput. Phys. Commun.,
  142446-452, 2001.