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Denny Vitasari, Paul Grassia, Peter Martin

Material point method simulation

Annual Manchester SIAM Student Chapter Conference 201320 May 2013

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Background – Foam fractionation

Transport of surfactant onto the film interface determines the efficiency of a foam fractionation column.

• Foam fractionation: Separation of surface active material using rising column of foam.

• Foam fractionation column with reflux:Some of the top product is returned to the column.

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Foam structure – Dry foam

Lamella: thin film separating the air bubbles within foam.

Plateau border: three lamellae meet at 120 to form an edge.

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2D illustration of a foam lamella

• Due to reflux, the surface tension at the Plateau border (Pb) is lower than that at the lamella (F) transport of surfactant from the surface of Plateau

border to the surface of film Marangoni effect.

• Pressure in the Plateau border is lower due to curvature (Young-Laplace law) liquid is sucked to the Plateau border film drainage.

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Assumptions

• The lamella is always flat and has a uniform thickness along the length.

• At initial time the surface concentration (F0) of surfactant along the film is uniform.

• The surface concentration (Pb) of surfactant at the

Plateau border interface is fixed.

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Film velocity profile

The equation for velocity profile of liquid on the lamella surface:

Film drainage Marangoni effect

𝑢𝑠 = −𝑥𝛿d𝛿d𝑡 + 13𝛿𝜇𝜕𝛾𝜕𝑥

Surfactant mass balance: 𝜕Γ𝜕𝑡 = ൬𝑥𝜕Γ𝜕𝑥+ Γ൰1𝛿d𝛿d𝑡 + 𝐺3𝜇𝜕2Γ𝜕𝑥2

viscosity

Gibbs-Marangoni parameter

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Rate of film drainage

• Mobile interface (Breward-Howell, 20021)

• Rigid interface (Reynolds, 18862)

1. Breward, CJW and Howell, PD, Journal of Fluid Mechanics, 458:379-406, 20022. Reynolds, O, Philosophical Transaction of the Royal Society of London, 177:157-234, 1886

d𝛿d𝑡 = −38𝛾𝑃𝑏𝛿3 2Τ𝜇𝐿ξ𝑎

d𝛿d𝑡 = − 𝛾𝑃𝑏𝛿33𝜇𝐿2𝑎

Analytical solution Case no film drainageSolution of surfactant mass balance equation: Complementary error function• Shift one boundary condition to - and solve the equation analytically to result in a

complementary error function.• Reflection method to correct the boundary condition.• Violation of boundary condition due to reflection method.• Improving accuracy using additional reflections.

Surfactant mass balance:

Boundary conditions:

Complementary error function (1 reflection):

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Analytical solution Case no film drainageSolution of surfactant mass balance equation: Fourier seriesFourier series obtained from method of separation variable:

Surfactant mass balance:

Boundary conditions:

Fourier series:

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Numerical simulation of surfactant concentration ()

Material point method3

• The surface velocity (us) applies on every material point material point change its position.

• Surface excess () averages between two material points.

• Surfactant is conserved same area of the rectangle.

3. Embley, B and Grassia, P, Colloids and Surfaces A, 382: 8-17, 2011

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both sides move to the left

Bookkeeping operation

both sides move to the right left side moves to the rightright side moves to the left

• Every time step: material points change their positions uneven spatial interval over time.

• The spatial interval (Δx) is restored and the value of is corrected.• Take a weighted average of in the restored interval as the new

value.

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Analytical solutionCase with film drainage

(Quasi) steady state (rigid interface): us = 0 (no surfactant flux on the surface)

Solution:

Asymptotic solution (mobile interface): Uniform inner solution inner pulls boundary layer near the Plateau border

Solution:

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Parameters for simulation

Parameter Symbol Value UnitCharacteristic `Marangoni’ time scale L2/(G0) 3.12510-2 s

Characteristic thinning time scale (mobile) 0/(d/dt)0 1.4810-3 s

Characteristic thinning time scale (rigid) 0/(d/dt)0 2.08 s

Initial half lamella thickness 0 2010-6 m

Half lamella length L 510-3 mLiquid viscosity 110-3 Pa sCurvature radius of the Plateau border a 510-4 mSurfactant surface concentration at PB Pb 210-6 mol m-2

Initial surface concentration at film film 110-6 mol m-2

Surface tension of solution at PB Pb 4510-3 N m-1

Gibbs-Marangoni parameter G 4010-3 N m-1

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Resultsdimensionless form

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Surface excess profile ( vs x)no film drainage

t

Surface excess of surfactant () increases with time

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Verification of numerical resultCase no film drainage

Complementary error function

• t’ = 0.005• Simulation at early

time fewer reflection terms needed.

• Numerical simulation fits well with the analytical solution.

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Verification of numerical resultCase no film drainage

Fourier series

• t’ = 2• Simulation at later

time fewer Fourier terms needed.

• Numerical simulation fits well with the analytical solution.

• Accuracy of the numerical result increases with more grid elements.

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Surface excess profile ( vs x)film drainage: rigid interface

t

Surface excess of surfactant () increases with time but slightly more slowly than in case with no film drainage.

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Surface excess profile ( vs x)film drainage: mobile interface

t

Surface excess of surfactant () decreases with time: surfactant washed off film by film drainage.

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Spatially-averaged surface excess

• Surface excess increases with time for the case of no drainage and draining film with a rigid interface.

• Surface excess decreases with time for the case of draining film with a mobile interface.

• Surface excess of draining film with a rigid interface is slightly lower than that of no drainage.

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(Quasi) steady state solutionFilm drainage: rigid interface

• The (quasi) static solution has weak spatial variation in .

• Quasi static solution does not significantly change with time.

• Agreement between numerical and (quasi) static solution only possible at reasonably long time.

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Asymptotic boundary-layer solutionFilm drainage: mobile interface

• Inner region (near the film centre) and outer region (near the Plateau border).

• Inner region: Marangoni effect is negligible.

• Outer region: Marangoni effect is retained.

• Agreement between numerical and asymptotic analytical results.

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Conclusions• The equations of surfactant transport onto a foam lamella can be solved

numerically using a material point method followed by a bookkeeping operation.

• The numerical simulation is validated by analytical solution obtained from complementary error function and Fourier series in the case of no film drainage.

• In a foam fractionation column with reflux, when Marangoni flow dominates the film drainage, the surface concentration of surfactant increases with time.

• When film drainage dominates the Marangoni effect such as in a film with mobile interface, surfactant is washed away to Plateau border and its concentration decreases with time.

• Quasi steady state solution agreed with the numerical simulation for the case of a film with a rigid interface, in the limit of long times.

• Asymptotic boundary-layer solution agreed with the numerical simulation for the case of a film with a mobile interface.

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Thank you