Denny Vitasari , Paul Grassia - PowerPoint Presentation

Slide1

Denny

Vitasari

, Paul Grassia, Peter Martin

Surfactant transport onto a foam lamella

1

Material point method simulation

Annual Manchester SIAM Student Chapter Conference 2013

20 May 2013

Slide2

Background – Foam fractionation

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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.

Slide3

Foam structure – Dry foam

3

Lamella: thin film separating the air bubbles within foam.

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

Slide4

2D illustration of a foam lamella

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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.

Slide5

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. 5

Slide6

Film velocity profile

6

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

Film drainage

Marangoni effect

Surfactant mass balance:

viscosity

Gibbs-

Marangoni

parameter

Slide7

Rate of film drainage

Mobile interface (

Breward-Howell, 20021)Rigid interface (Reynolds, 18862)71. 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

Slide8

Analytical solution

Case

no film drainageSolution of surfactant mass balance equation: Complementary error functionShift 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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Slide9

Analytical solution Case

no film drainage

Solution of surfactant mass balance equation: Fourier seriesFourier series obtained from method of separation variable: Surfactant mass balance: Boundary conditions:

Fourier series: 9

Slide10

Numerical simulation

of surfactant concentration (

)Material point method310The 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

Slide11

both sides move to the left

Bookkeeping operation

11both sides move to the rightleft side moves to the rightright side moves to the leftEvery 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.

Slide12

Analytical solutionCase

with film drainage

12(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 borderSolution:

Slide13

Parameters for simulation

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ParameterSymbol ValueUnitCharacteristic `Marangoni’ time scaleL2/(G0)3.12510-2

sCharacteristic thinning time scale (mobile)0/(d/dt)0

1.48

10-3s

Characteristic

thinning time scale (rigid)



0

/(d/

dt

)

0

2.08

s

Initial half lamella thickness



0

20

10

-6

m

Half lamella length

L

510-3

mLiquid viscosity



110

-3

Pa s

Curvature

r

adius of the Plateau border

a

5

10

-4

m

Surfactant surface concentration at PB



Pb

210

-6

mol m

-2

Initial surface concentration at film



film

1

10

-6

mol m

-2

Surface tension of solution

at PB



Pb

45

10

-3

N m

-1

Gibbs-

Marangoni

parameter

G

40

10

-3

N m

-1

Slide14

Results

dimensionless

form14

Slide15

Surface excess profile (



vs x)no film drainage15tSurface excess of surfactant () increases

with time

Slide16

Verification of numerical resultCase

no film drainage

16Complementary error functiont’ = 0.005Simulation at early time  fewer reflection terms needed. Numerical simulation fits well with the analytical solution.

Slide17

Verification of numerical resultCase

no film drainage

17Fourier seriest’ = 2Simulation 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.

Slide18

Surface excess profile (



vs x)film drainage: rigid interface18tSurface excess of surfactant ()

increases with time but slightly more slowly than in case with no film drainage.

Slide19

Surface excess profile (



vs x)film drainage: mobile interface19tSurface excess of surfactant ()

decreases with time: surfactant washed off film by film drainage.

Slide20

Spatially-averaged surface excess 

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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.

Slide21

(Quasi) steady state solutionFilm drainage:

rigid interface

21The (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.

Slide22

Asymptotic boundary-layer solution

Film drainage:

mobile interface22Inner 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.

Slide23

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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Slide24

Thank you

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