Schmidt

Fluid Flow in Absorbing Porous Media Presented at the Marie Curie Workshop for Flow and Transport in Industrial Porous Media Nov. 12 – 16, 2007 University Utrecht, The Netherlands

Dr. Mattias Schmidt Research Fellow – Victor Mills Society, Procter & Gamble - Baby Care R&D

Procter & Gamble © 2007

ABOUT THE PRESENTER

Dr. Mattias Schmidt is a Research Fellow and member of the Victor Mills Society. He joined P&G with a PhD in Physics from University of Leipzig and has worked in the area of fluid flow, absorbent core design and superabsorbent b b t polymer l d development l t ffor more th than 16 years.


1

Outline • P&G at a glance • Importance of Modeling & Simulation for R&D • Fluid Flow in Hygiene Products – – – – –

Introduction Some FemCare examples Impact of hysteresis on fluid (re-)distribution Introduction to Diaper Cores and AGM Impact of AGM swelling on fluid flow

• Typical challenges and opportunities for collaboration Procter & Gamble © 2007

It began with Soap & Candles… James Gamble Soap Maker

FOUNDED IN 1837 •Fifth Oldest Company on the Fortune 50…


William Procter Candle Maker


2

P&G In a Glance P&G at a glance – 170 years old –•Sales AnnualofSales of more than $ 70 $ 68.2 billions billion 300brands brands in more –•Nearly Nearly 300 in more than 160 160 countries than countries – 22 global brands with sales of •22 global brands with sales over $ 1 billion over $ 1 workforce billion of 140,000 –ofWorldwide –•Workforce 140 plants and 25 R&D centers of 140.000 globally billions people –•3Spend more than $touched 2 billion a year on R&D by P&G products everyday

•Spends more than $5 million a day on R&D Procter & Gamble © 2007

22

Billion-Dollar Brands Procter & Gamble © 2007

3

How do we define Innovation? Innovation is the blend of “What’s Needed” with “What’s Possible”

What’s Needed? •Consumer •Customer •Competition

What’s Possible? Consumer Delight

•Technology

Leading Edge Innovation Procter & Gamble © 2007

‘Innovation is our Lifeblood’ •Set up first product research lab in U.S. in 1890

•Currently have 24,000 active Patents, receive 3800 per y year. •Invest over $2.0 Billion per year in R&D


4

Importance of Modeling & Simulation for R&D


Typical Challenges Products must perform when used … But face Fundamental Engineering Contradictions. •Materials … strong but soft—even wet, stretch not break, breath but contain, break…not tear/selectively tear. •Liquids … mixtures can’t separate, must stay where applied…but dispense easily. •Packages … design is key, be strong but light, never leak but open easily. Procter & Gamble © 2007

5

Scales of Modeling Industrial Eng/ Operations Resrch (Statistical, Discrete Event, Agent Based)

days

Computational Chemistry y

hrs

MechEng/ChE (Closed Form Equations)

sec

Time

Continuum or Finite Difference Finite Element

ms

Coarse Grain or Mesoscale Modeling – Polymers

ns

Quantum Chemistry - subatomic

fs

angstroms

Molecular Mechanics -atoms, molecules

Computer Aided Engineering (CAE) nm

mm

microns

m

km

Distance Procter & Gamble © 2007

‘Moore’s Law’ Computing Hardware Performance ORNL ‘peta-Scale’ 1015

10

12

Petaflops

SNL ‘Red Storm’ LLNL ASCI ‘White’

Teraflops

LANL “Blue Mountain”

Gigaflops

U.S. DOE Leadership ‘Leadership’ Class Machines

Megaflops Kilaflops

1990

2000

2010

P&G’s 1st, 2nd, & 3rd Generations


6

Thermal Performance


Current Body Scans

Current Full Body Scans


7

Validating the Fit Model Full Body Scan

Current Model


Bone Modeling

Women’s Bone Health…

Based on these results, three factors are critical for addressing bone strength in osteoporosis: (1) reduction in the stress risers, (2) targeted increase of bone volume, (3) preservation of architecture. Procter & Gamble © 2007

8

Fluid Flow in Absorbent Hygiene Products


Typical Liquid Handling Processes Macroscopic Liquid Handling • Absorption • Release • Distribution / Redistribution • Storage / Retention (i.e. stop flow) • Provide Barrier (i.e. stop flow) Microscopic Liquid Handling • Contact Angle & Wetting • Deposition / Spreading • Dewatering • Condensation • Filtration

~mm ~cm, ~m

µm


9

Typical Liquid Handling Tasks in BabyCare

DIAPERS

WIPES

Urine

absorb & retain

remove (from skin)

BM

contain & absorb

remove (from skin)

Lotion

melt, release & deposit (spread)(onto skin)

release & deposit (spread) (onto skin)


Typical BabyCare Liquids Viscosity

Surf. Tension

Key ingredients

Urine

~1 cP

~50 ... 65 mN/m

water, salts, urea, surfactants

BM

~106 ... 1015 cP ~30 ... 60 mN/m

water, bacteria, fibers, mucins, surfactants, salts

Lotion solid at RT (Diaper)

low (temp dep.)

petrolatum, stearyl alcohol, aloe

Lotion ~1 ... 500 cP (Wipes)

~30 ... 65 mN/m

Water, silicone (2-3%), polymer (stabylen), preservatives, surfactant


10

Example – BM viscosity varies widely BM distribution

All BM’s Runny BM’s

Urine

10-3

100

103

106

109

1012

Viscosity Vi it [P [Pa s]] (at zero shear)

For comparison: Viscosity range of some Other products Procter & Gamble © 2007

Porous Media Flow • Simulation of fluid flow in porous structures applied to product design of „paper products“ e.g. Diapers, Feminine Pads, Towels, Swiffer, Wipes, Make-up-Applications, Olay Facial Wipes ...

• Being used to develop new designs and new materials • Richards equation is a typical formulation


11

Examples of Fluid Flow in Porous Media for Aborbent Hygiene Products


Absorbent Core Modeling

Optimization O ti i ti Model Output: • Acquisition time • Liquid distribution / partitioning • Capillary Pressure* • Flow Patterns*

Model Input: • Core design • Raw material prop´s • Test protocol

Simulation

* Additional insights that can experimentally not easily detected.

Validation

Experimental (Lab): • Acquisition time • Liquid distribution / partitioning


12

Example: X-Ray Analysis j=0 .. 10

Front

Load ding Distribution – Loadi,j

Weigh diaper – total load L

i=0 .. 40

0

X-R Ray Image – Intensityi,j

ction, MD) i (x-direc

j (y-direction, CD)

1

2

3

4

5

6

7

8

9

10

0

0.058 0.056 0.058 0.057

0.059 0.059 0.063 0.058 0.057 0.058 0.061

1

0.058 0.066 0.069 0.073

0.081 0.091 0.103 0.084 0.078 0.081 0.067

2

0.057

3

0.058 0.117 0.138 0.175

0.164

4

0.067 0.153 0.261 0.353

0.07

0.09 0.106

0.448 0.578 0.553 0.367 0.336 0.271 0.087

0.104 0.104 0.138 0.114 0.113 0.16 0.205

0.2

0.13 0.075

0.19 0.174 0.128

5

0.059 0.103 0.345 0.632

0.772 0.856 0.844 0.727 0.567 0.323 0.076

6

0.056 0.207 0.624 0.901

1.061 1.153 1.157 1.052 0.835 0.414

7

0.09

0 0.298 0.888 1.054

1.25 1.371 1.395 1.295 1.126 0.592 0.062

8

0.056 0.324 1.058 1.249

1.409 1.636 1.666 1.453 1.101 0.453 0.057

9

0.058 0.304 1.153 1.331

10 0.062 0.218 1.149 1.465 11

0.06 0.127 0.998 1.441

12 0.061 0.125 1.061 1.459

1.579 1.768

1.75 1.681 1.376 0.406 0.062

1.723 1.838 1.871 1.873 1.575 0.228 0.056 1.721 1.809 1.867 1.851 1.546 0.141

0

1.624 1.725 1.771 1.729 1.063 0.075 0.054

13 0.063 0.165 1.086 1.456

1.563 1.562 1.542

14 0.063 0.172 0.927 1.214

1.197 1.241 1.142 0.819 0.562 0.066 0.057

1.51 0.803 0.061

0

15 0.063 0.116 0.561 0.629

0.733 0.833 0.763

16 0.062 0.069 0.262 0.827

1.054 1.061 1.059 0.959 0.199 0.062 0.055

17

0.06 0.065 0.444 1.007

1.087 1.256 1.208 1.128 0.555 0.063 0.055

18

0.06 0.104 0.777 1.172

1.291 1.236 1.232 1.153 0.579 0.063 0.055

19

0.06 0.138 0.941 1.405

1.704 1.765 1.526 1.346 0.699 0.072 0.055

Lo ad M = FRAME 20 0.061 0.252 1.081 1.525 1.813 1.843 1.679 21 0.061 0.337 1.205 1.607

0.74 0.206 0.063 0.057

1.44 1.015 0.117 0.057

1.859 1.935 1.867 1.647

1.19 0.349 0.056

22 0.062 0.405 1.272 1.622

1.818 1.962 1.843 1.554 1.126 0.326 0.056

23 0.063 0.472 1.199 1.531

1.745 1.886 1.721 1.419 0.912 0.213 0.057

24 0.064 0.526 25 0.083

1.05 1.332

0.5 0.857 0.999

26 0.134 0.619 0.865 0.975

1.56 1.617 1.487

1.21 0.875 0.193 0.057

1.124 1.145 1.069 1.052

0.72 0.143 0.055

1.058 1.074 1.034 0.965 0.613

0.14 0.054

27 0.118 0.642 0.859 0.989

1.01 1.054 0.937 0.814 0.398 0.094 0.054

28 0.062 0.516 0.843 1.025

1.023 0.885 0.686 0.512 0.264 0.186 0.061

29

0.06 0.367 0.788 0.974

0.949 0.649 0.419 0.203 0.177

0.13 0.055

30 0.059 0.139 0.591 0.692

0.619 0.349 0.165 0.131 0.117 0.118 0.059

31 0.062 0.086 0.089 0.135

0.172 0.109 0.134 0.115 0.088 0.075

32 0.059 0.083 0.082 33 0.056 0.062

0

0.07 0.086 0.119 0.122 0.098 0.087 0.072 0.055

0.08 0.083

0.132 0.186 0.128 0.084 0.075 0.078 0.054

34 0.061 0.064 0.072 0.083

0.109 0.155 0.118 0.079 0.067 0.068 0.054

35 0.057 0.063 0.068 0.069

0.087 0.124 0.107 0.075 0.062 0.063 0.055

36 0.055

0.06 0.066 0.072

0.089 0.129 0.107

0.07 0.064

0.06 0.054

37 0.056 0.058 0.067 0.071

0.086

38 0.055 0.061 0.067 0.071

0.087 0.104

39 0.058 0.062 0.071 0.072

0.073 0.089 0.084 0.072 0.069 0.063 0.058

40 0.055

0

0

0

0

0.11 0.096 0.069 0.068 0.061

0

0

0.1 0.074 0.063 0.061 0.055 0

0

0

0 0.055

Back

Quantitative determination of load distribution Procter & Gamble © 2007

Examples from FemCare* Dual layer structure Insult Upper Core Lower Core

* Not limited to FemCare – could be other dual layer structure


13

3D simulations – fluid generated at corner

2 layer homogeneous structure

1 layer homogeneous structure

Top – high permeability, low Pcap Bottom – low permeability, high Pcap


Fluid generated at corner No hysteresis, gravity included

Hysteresis, gravity included

Key learning: • for limited fluid amounts we need to include hysteresis to describe product behavior


14

Why does liquid not spread over an entire fabric?

... Capillary hysteresis stops wicking ... How can we control stain size?

time Procter & Gamble © 2007

TRI/Autoporosimeter (PVD) PC Control Gas

Pc Measure

m(t) Sample

reservoir scale

Frit/Membrane

- Capillary pressure as a function of saturation (measure saturation as a function of pressure) - Data used to generate: -Pore volume distribution -Cumulative volume -Saturation vs. Pressure

Fluids used include Hexadecane, H2O/Surfactant, Salt Solutions Typically 2 steps: 1) absorption with dry material 2) drainage 3) absorption with wetted material Procter & Gamble © 2007

15

Inputdata & Measurements full hysteresis 1 absorption measurement desorption measurement absorption fit desorption fit

saturation

0.8 0.6

desorption

0.4 absorption

0.2 0 0

20

40

60

80

100

pressure (kPa)


saturatio on

Hysteresis Scanning 1 0.9 0.8 0.7 06 0.6 0.5 0.4 0.3 0.2 0.1 0

absorption measurement desorption measurement absorption fit desorption fit incomplete cycle i incomplete l t fit

0

2

4

6

8

10

pressure (kPa)

p( S ) = p des

p0 − pabs ( S max ) p0

(( SS

1

)

− mdes

max

1

−1)

n des

+ pabs ( S max )

[1] Parameterization of the complete cycles used is from: M. T. van Genuchten, “A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J.., 44, 892-898, 1980. Pabs,P0, pdes, ndes, mdes, are van-Genuchten parameter


16

Wicking Experiment and Simulation

n

∂S ( x , t ) ∂ k ( S ) ∂ = ⋅ Pc [ x , S , S& ] ∂t ∂x μ ∂x

liquid reservoir

sample

plastic foil

Simulation done in FORTRAN/NAG Library Procter & Gamble © 2007

Results - development in time


17

Results - development in time


Results - experiment/simulation measurements vs. simulation 1 3 h data mean 1 h data mean 1 h simulation (test8) 3 h simulation (test8)

saturation

0.8 0.6 0.4 0.2 0 0

5

10

15

20

25

30

35

x (cm)


18

Predictions - different load different percentages of maximum load simulation 3 hours

1

8.25 %

saturation

0.8

reference 17.9 % 0.6

30.90%

0.4 0.2 0 0

5

10

15

20

25

30

35

x (cm )


Predictions - reduced hysteresis reduced hysteresis simulation 3 hours 1 1/2 hysteresis y reference full hysteresis 1/4 hysteresis zero hysteresis

saturation

0.8 0.6 0.4 0.2 0 0

5

10

15

20

25

30

35

x (cm)


19

Predictions - permeability

k = k0 ⋅ S b

different permeability

b ~ 4.5 for reference

1

0.5 k0 reference 2 k0 0.5 b (b=2)

saturation

0.8 0.6 0.4 0.2 0 0

5

10

15

20

25

30

35

x (cm )


Key Learnings • Hysteresis leads to a stop of the liquid front, the higher the load the less impact of hysteresis • Decrease of b greatly improves liquid distribution • Increase of k0 improves liquid distribution • Reduction to zero hysteresis shows good distribution but reducing hysteresis has essentially no impact


20

Diaper Core Technology

Diaper (from Side ) Lotion Stripes

Waist feature

Topsheet

Acquisition Patch

Back of Nappy ppy

Front of Nappy

Backsheet

Inside Surface

Dusting Layer

Storage Core (homogenous blend of AGM and cellulose)

Outside Surface


How does a diaper work? Liquid Handling Tasks of diaper Cores

Acquisition & Distribution

Liquid

•Nonwoven Layer •Modified Cellulose Fibers

Storage Core: AGM + Cellulose


21

How does a diaper work? Liquid Handling Tasks of diaper Cores

Final Absorption

• AGM can absorb about 30 times of its own weight, whereas cellulose can absorb only around 4 times of its own weight. Pampers Core AGM 70%

Airfelt 30%


What is AGM? AGM = absorbent gelling material = superabsorber, supersorber, superabsorbent polymer (SAP), = superabsorbent material (SAM) (SAM), = hydrogels, hydrogel-forming polymers •

White granulate powder with particles ranging from 45 mm to 850 mm.

•

A cross-linked poly-acrylate, about 75% neutralized with Na+ - Ions.

•

Enables ssuper-thin per thin diaper designs at improved impro ed leakage and dryness dr ness performance (move from 100% pulp cores to ~30% pulp cores today).


22

AGM Chemistry

COO

CO2H

CO2H

COO Na+

Na+

O R - C - Et Na+

Na+

O C=O

CO2H

Na+ CO2H

• AGM = absorbent gelling material • white powder made of lightly crosslinked polymer networks • based on water soluble hydrophilic polymer • crosslinked to connect the chains of the polymer: “elastic springs” • liquid is absorbed via diffusion … forms a hydrogel much like gelatin Procter & Gamble © 2007

Function of AGM

• Take up urine and lock it away – as much as possible! (Storage Capacity) • Transport urine within itself, e.g. within a swollen gel bed. (Permeability) y fast. • Work reasonably (Speed)


23

What is „Gel Blocking“?

Uncontrolled Swelling / Gel Blocking Urine

Urine

Controlled Swelling / High Permeability

Poor permeability causes under-utilized AGM


Gel-Blocking Demo


24

Surface Crosslinking

Higher Crosslinked Shell

Shell creates tangential forces „balloon effect“

L Lower Crosslinked C li k d Bulk B lk

• surface crosslinking maintains particle shape during swelling • improves permeability and swelling capacity Procter & Gamble © 2007

Fluid Flow in AGM-containing cores • AGM absorbs liquid „away from the pores between AGM particles and fibers“ – – –

Acts like a „sink term“ term in the fluid flow equation of the pore structure Liquid absorbed into each AGM particle by diffusion / osmotic process Swelling of AGM changes the pore structure Fluid absorbed by

• Described as „two types of liquid“ – m1: „mobile fluid“ in the pore structure – m2: „immobile“ fluid in the gel

swellable material

m2

• Requires modified equation system m1 – Richards equation extended by „sink term“ – Key properties of pore structure now depend on m2 Fluid in pores of the swellable material – Additional equation for absorption of liquid into the gel phase


25

Example Equations (1D, horizontal) ∂m1 ( x, t ) ∂ ⎛ ∂m ∂m ⎞ + ⎜ D1 (m1 ( x, t ), m 2 ( x, t )) 1 + D2 (m1 ( x, t ), m 2 ( x, t )) 2 ⎟ = ∂t ∂x ⎝ ∂x ∂x ⎠ = − f ( S (m1 ( x, t ), m2 ( x, t )) ⋅ τ ⋅ C AGM ⋅ A0 ⋅

m max − m 2 ( x, t ) m max

m − m 2 ( x, t ) ∂m2 ( x, t ) = + f ( S (m1 ( x, t ), m2 ( x, t )) ⋅ τ ⋅ C AGM ⋅ A0 ⋅ max mmax ∂t • Calculates liquid movement through absorbent core and AGM • Predict liquid handling properties of absorbent cores. Î optimize AGM properties for specific core designs Î optimize core designs for specific AGMs • This also applies in 2D/3D Procter & Gamble © 2007

Example: Typical Assumptions

1) The fluid-swellable composite material comprises fluid-swellable particulate material and may comprise voids between the particles of said material; liquid is either in said voids or inside the fluid-swellable particulate material. 2) Liquid movement in one dimension only (e.g. xx direction). 3) The fluid-swellable (composite) material swells only transverse (perpendicular) to the direction of liquid movement (swelling only in y or z direction). 4) Once liquid is inside the fluid-swellable material it remains inside. 5) The liquid does not move inside the fluid-swellable material (e.g. liquid that enters the fluid-swellable material at point x will always stay at point x). 6)

Liquid can distribute inside the voids; this distribution is governed by Darcy’s Darcy s law and liquid mass conservation.

7) The flow direction is horizontal, such that gravity can be neglected. The model may however also be applied to a two dimensional or three dimensional situation.


26

Example: Multilayer Core Structure incl. AGM

•

Short time after gush

„In equilibrium“ after gush

AGM load

Interstitial saturation I n

•

Pictures show snapshots of fluid in pores (~saturation) and fluid in AGM at two different times after a gush onto a pre-loaded core Illustrates how upper acquistition layers and interstitials are being dewatered and fluid is absorbed into the AGM


Challenges & Opportunities for Collaboration


27

Typical Challenges •

Designs are long / large area but very thin – Difficult to mesh properly (balance of speed / accuracy)

•

Product developers want the answers fast – Computation time, stability of simulations Î new algorithms – Need to be able to run lots of simulations

•

Extreme material properties – – – –

•

Typical porosities ~90% or higher Large parameter contrast in K(S) and Pc(S) and hysteresis Properties change during use (swelling, external pressure changes) Thin materials (see below)

Multiple physical effects – Porous Media Flow – Free Surface Flow – Mechanical Deformation

•

Liquids can be difficult – Surfactants – Surface tension as function of time, Surfactant Transport – Non-newtonian effects – most equations do not apply to high porosities Procter & Gamble © 2007

Opportunities for Collaboration • Fluid flow models in presence of – Thin layers – Hydrophobic layers / bridging

• • • •

Measurement M t off K(S)* Design materials with target K(S) Large parameter contrast and initially very dry materials Initial wetting behavior of very dry materials (analogy to soil) • Experimental characterization of surfactant release and transport p in p porous media • Micromodels to predict K(S, pext), Pc(S, pext), n(pext) • Multiple capillary pressure cycles (absorption / desorption) implementation in models * The relative permeability of materials as a function of capillary pressure and/or saturation is currently determined by comparing the virtual spatial map of saturation predicted in the virtual test simulations to the physical spatial map of saturation measured in the physical test environment and altering one or more of the absorbent-fluid interaction properties the absorbent used in the virtual test environment, until the spatial maps of saturation compare favorably. We are looking for more direct measurement techniques. Procter & Gamble © 2007

28

Thin Layers – Key Definitions • A „thin layer“ is characterized by typical pore dimensions that are similar (order-of-magnitude) to the thickness (or: the smallest dimension) of the layer. Î Only y few number of p pore layers y through g the caliper p Î 3D pore structure is influenced by interface / adjacent layers Î „bridging“ across the thin layer may be a dominating effect* Î Validity of Richards Equation?* Î Difficult to generate model input data Î Disconnected liquid / Instability effects may be present

• In diaper cores most of the „thin layers“ are non-wovens that also show h additional dditi l challenges: h ll Î Inhomogeneous pore structure (patterned embossing, laydown) Î Inhomogeneous hydrophilicity (surfactant coating) Î Change of the fluid in contact (surfactant wash-off) Î Hydrophobic bridging * This means that the flow may not be just driven by capillary pressure differential but also by absolute pressures / curvature of meniscus.


Thin Layers – Product Locations •

As NWCC (non-woven core cover, also called „core wrap“)

•

As TS (top-sheet): hydrophilic (spunbond) or hydrophobic (apertured TS)

•

As AQL (non-woven acquisition layer)

Top-sheet Acquisition System

NWCC

Storage Core

NWDL (not fluid flow relevant) Backsheet (not fluid flow relevant) Note: we are using 2 different systems 1. NW AQL / curly fibers combination (in most designs) 2. NW AQL only (in low-tier designs) Procter & Gamble © 2007

29

Pc(S): Comparison 1 layer and 10 layers

•

Thin NW layer

•

1 layer has more g/g uptake than 10 layers

•

Interfaces between sample and weight / frit form new pores

•

Diff Difference iin uptake t k especially at low pressures, i.e. large pore size (>0.3mm)


Thank you! Questions?


30

Additional Information – Terms and Abbreviations


Example Equations (1D, horizontal) ∂m1 ( x, t ) ∂ ⎛ ∂m ∂m ⎞ + ⎜ D1 (m1 ( x, t ), m 2 ( x, t )) 1 + D2 (m1 ( x, t ), m 2 ( x, t )) 2 ⎟ = ∂t ∂x ⎝ ∂x ∂x ⎠ = − f ( S (m1 ( x, t ), m2 ( x, t )) ⋅ τ ⋅ C AGM ⋅ A0 ⋅

m max − m 2 ( x, t ) m max

m − m 2 ( x, t ) ∂m2 ( x, t ) = + f ( S (m1 ( x, t ), m2 ( x, t )) ⋅ τ ⋅ C AGM ⋅ A0 ⋅ max mmax ∂t • Calculates liquid movement through absorbent core and AGM • Predict liquid handling properties of absorbent cores. Î optimize AGM properties for specific core designs Î optimize core designs for specific AGMs • This also applies in 2D/3D. Procter & Gamble © 2007

31

Example: Typical Assumptions

1) The fluid-swellable composite material comprises fluid-swellable particulate material and may comprise voids between the particles of said material; liquid is either in said voids or inside the fluid-swellable particulate material. 2) Liquid movement in one dimension only (e.g. xx direction). 3) The fluid-swellable (composite) material swells only transverse (perpendicular) to the direction of liquid movement (swelling only in y or z direction). 4) Once liquid is inside the fluid-swellable material it remains inside. 5) The liquid does not move inside the fluid-swellable material (e.g. liquid that enters the fluid-swellable material at point x will always stay at point x). 6)

Liquid can distribute inside the voids; this distribution is governed by Darcy’s Darcy s law and liquid mass conservation.

7) The flow direction is horizontal, such that gravity can be neglected. The model may however also be applied to a two dimensional or three dimensional situation.


where (a) (b) (c) (d) (e)

(f)

x is the space dimension t is the time m1 is the amount of liquid in voids per length. m2 is the amount of liquid in fluid-swellable material, e.g. particles, per length

∂f is the partial derivative of any variable f(x,t) in respect to time t, e.g. ∂t ∂m1 is the partial derivative of m1 in respect to time t ∂t ∂f is the partial derivative of any variable f(x,t) in respect to space x, e.g. ∂x ∂m1 is the partial derivative of m1 in respect to space x ∂x

(g) D1 = D1 (m1 ( x, t ), m 2 ( x, t )) is the diffusivity 1 defined as

D1 (m1 , m2 ) = ρ liq ⋅ A(m2 ) ⋅

k (m1 , m2 ) ∂Pc (m1 , m2 ) ⋅ μ ∂m1

(l)

Pc (m1 , m2 ) is the capillary pressure. This is in general a function of m1 and

(m)

m2. (see the method section below) μ is the viscosity of the liquid - (see the method section below)

(n) (o)

is the partial derivative of Pc in respect to m1 is the partial derivative of Pc in respect to m2

(p) τ is the swelling speed (see the method section below). In general τ is a function of m2 (q) mmax is the maximum capacity (see method section below) (r)

C AGM =

and m2.

k (m1 , m2 ) ∂Pc (m1 , m2 ) ⋅ μ ∂m2

Mass (dry fluid swellable material) is the fluid-swellable Volume (dry fluid swellable material)

material concentration, determined as ratio between mass and dry volume, where the mass is determined by weighing the fluid-swellable material, and the dry volume is calculated by determining caliper, length and width of the dry fluid-swellable composite material (s) S is the liquid saturation in the voids and can be expressed as function of m1

((h)) D 2 = D 2 ( m1 ( x, t ), m 2 ( x, t )) is the diffusivityy 2 defined as

D2 (m1 , m2 ) = ρ liq ⋅ A(m2 ) ⋅

∂Pc ∂m1 ∂Pc ∂m2

S (m1 , m2 ) =

m1

ρ liq ⋅ n(m2 ) ⋅ A(m2 )

f ( S (m1 ( x, t ), m2 ( x, t )) is an empirical function expressing the dependency of the swelling kinetics on saturation in the voids. This function can be approximated with several equations, an example is to assume f ( S ) = S (u) n is the porosity and is function of m2. (see method section below)

(t)

(i)

ρ liq

(j)

A(m2 ) is the cross section area. This is a function of m2 and porosity (n). A(m2 ) = A(m2 ( x, t ), n(m2 )) . From volume conservation it is possible to (1 − nmax ) m ( x, t ) express A( m2 ) as A( m 2 ) = ⋅A + 2 (1 − n(m2 ( x, t ) ) 0 ρ liq

(k)

k (m1 , m2 ) is the permeability. This is in general a function of m1 and m2. (see the methods section below)

is the density of the liquid

(v)

nmax is the value of porosity in dry conditions.


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