Transport in porous media 3MT130 II.pdf · Typical intrusion experiment . ... • Sublimination •...

Transport in Permeable Media

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Leo Pel, Henk Huinink, David Smeulders, Thomas Arends, Hans van Duijn

Faculty of Applied Physics Mechanical Engineering

Eindhoven University of Technology The Netherlands

[email protected]

5 ECTS 2018

Examination : Oral

Transport in porous media 3MT130


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Course + Lectures notes + additional info

www.phys.tue.nl/nfcmr/college/college.html

Examination : oral

3 days (to be determined)

http://www.phys.tue.nl/nfcmr/college/college.html


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Week 1: Intro tu 24-4-2018 13:45 15:30 dr.ir. L. Pel Introduction + porosity th 26-4-2018 08:45 10:30 dr.ir. L. Pel Capillary forces I Week 2: Capillary forces + Darcy tu 1-5-2018 13:45 15:30 dr.ir. L. Pel Capillary forces II th 3-5-2018 08:45 10:30 dr.ir. L. Pel Darcy + Dupuit Week 3 tu 8-5-2017 13:45 15:30 dr.ir. L. Pel Unsaturated absorption th 10-5-2017 08:45 10:30 Public holiday Week 4 tu 15-5-2018 13:45 15:30 dr.ir. H.Huinink Multiphase transport th 17-5-2018 08:45 10:30 No course due to conference visit Week 5 tu 22-5-2018 13:45 15:30 No course due to conference visit th 24-5-2018 08:45 10:30 No course due to conference visit Special subjects Week 6 tu 29-5-2018 13:45 15:30 T. Arends Moisture transport in wood th 31-6-2018 08:45 10:30 dr.ir. L. Pel Drying + Fire spalling Week 7 tu 4-6-2018 13:45 15:30 prof. H. van Duijn Density driven flow th 7-6-2018 08:45 10:30 dr.ir. L. Pel Component transport


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Surface tensions

Curved surface

Pressure difference


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wnwwnc rpppp γ2

=−=−=

rgh

ργ2

max =

Single pore/capillary

Capillary pressure


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Can surface tension really bring water from the roots up to the top?

Sequoia ~ 100 m tall

Xylem~30μm, γ= 73 dyne/cm


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Bundle of various capillaries

Look at various heights


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• We describe the soil as a bundle of capillary tubes of various sizes

rp wn

cθγ cos2

−=

Capillary pressure

Porous materials bundle of capillaries


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θ = n

Pc = 0

moisture content

Pres

sure

= s

uctio

n


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θ > 0

Pc = average

moisture content

Pres

sure

= s

uctio

n


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θ > 0

Pc = average

moisture content

Pres

sure

= s

uctio

n


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θ = 0

Pc = high

moisture content

Pres

sure

= s

uctio

n


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θ = 0 θ > 0 θ = n

Pc = high Pc = average Pc = 0

)(θcc pp =

Macroscopic capillary pressure

is function moisture content


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Decreasing

Pressure


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)(θcc pp =

Capillary pressure (Pa)

General convention (Hydrology)

Suction (m)

ρθψ

gpc )(

=

(practical use in soil)


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Capillary pressure

High

Moderate

Low

Soil

moisture content

Suc

tion

(m)


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Temperature dependence

ρθψ

gpc )(

=ργ

0-60 oC

γ~15%

ρ~2%


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Local curvature ~ local capillary pressure


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Porous material : macro coefficient

wwnc pppp −=−=

Macro coef => volume averages

REV


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Water content

Suct

ion

Pote

ntia

l, h,

tens

ion,

etc


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Different regions

Water content

Suc

tion

Pote

ntia

l, h,

tens

ion,

etc

Wet

Middle

Dry


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Wet region

θ

h

Wet

Pore only drains if: Big enough Not isolated Air can get to it

g hr

w cos2

ραγ

≥

Air entry Air access Structural pores


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A model porous medium being drained

Drainage allowed:

Pore radius:

Big

Small


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Pore radius:

Big

Small

Drainage allowed:



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Pore radius:

Big

Small

Drainage allowed:



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Pore radius:

Big

Small

Drainage allowed:



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Pore radius:

Big

Small

Drainage allowed:



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Pc


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hysteresis


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Hysteresis in capillary pressure


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Try yourself


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MIP: Mercury Intrusion Porosimetry

Revisted

Mercury θ=140o, γ=500 10-3 Nm-1

BE AWARE INK BOTTLE EFFECT (overestimation) Overestimation of

small pores


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Typical intrusion experiment

Cum

ulat

ive

Intr

usio

n –

mL/

g

Diameter – micrometers

Extrusion Intrusion


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Have to know the complete history

Many ‘small’

hysteresis


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Vertical Zones of Subsurface Water

• Soil water zone: extends from the ground surface down through the major root zone, varies with soil type and vegetation but is usually a few feet in thickness

• Vadose zone (unsaturated zone): extends from the surface to the water table through the root zone, intermediate zone, and the capillary zone

• Capillary zone: extends from the water table up to the limit of capillary rise, which varies inversely with the pore size of the soil and directly with the surface tension


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Moisture Storage Function z What force is affecting on the

water inside the porous media? Air pressure

Air pressure

z What pressure is needed to force water out of a material?


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41

Moisture Storage Function z What force is affecting on the

water inside the porous media?

z What pressure is needed to force water out of a material?

Air pressure

0.1 bar


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Moisture Storage Function

z Capillary force is acting on the water inside the porous media?

z What pressure is needed to remove water from a material?

Air pressure

0.5 bar


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Air pressure

5 bar




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Pressure must be higher than the capillary pressure!

Air pressure

50 bar




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• We describe the soil as a bundle of capillary tubes of various sizes

Soil Capillary pressure

High

Moderate

Low

rp wn

cθγ cos2

−=

Capillary pressure

P


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Membrane method (hard materials)

sample

P

semi-permeable membrane

water drainage/wetting

Slow measurement (order weeks)

)(θcc pp =


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Measurement technique: pressure plate apparatus

up to 100 bar Pressure


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Empirical & phenomenological equations

Brooks & Corey:

θs θ at saturation θr θ at 1.5 MPa

(“residual”) hb bubbling pressure λ fitting (“pore size

distribution index”)

log θ

log

h

log θ

h

hb: Lowest pressure at which air can flow through the soil

<

=−− −

otherwise

for 1λ

θθθθ

hh

hh

b

b

rs

r

hb


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( )

m

nrs

r

h

+

=−−

≡Θαθθ

θθ1

1

θs θ at saturation θr θ at 1.5 MPa α 1/hb n, m fitting. Often, m ≡ 1-(1/n)

van Genuchten:

θs

hb

θr

θ

h

Empirical & phenomenological equations


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Tensiometer for Measuring Soil Water Potential

Porous Ceramic Tip Vacuum Gauge (0-100 centibar)

Water Reservoir Variable Tube Length (12 in- 48 in) Based on Root Zone Depth


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Water distribution???

Interface : capillary pressure continuous

suction continuous


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Θ =0.1 Θ =0.1

Material A Material B

WHAT HAPPENS IF WE BRING THEM IN CONTACT ???????


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Θ =0.1 Θ =0.1


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Θ =0.1 Θ =0.1


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Θ =0.15 Θ =0.08


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Θ =0.2 Θ =0.2


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Θ =0.25 Θ =0.05

capillary pressure is constant

=

jump in moisture content


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Towards poultice

coarse

fine

fine

coarse

airflow airflow

WHAT WILL BE DRYING BEHAVIOUR ????


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Over boundary capillary pressure constant

)()( rrll θψθψ =

)()( 11rrllll θψψθψψ −− =

)(1rrll θψψθ −=

)( rl f θθ =JUMP in moisture content


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Beach Large and small sand particles

Fine sand

Sand

Moisture content C

apilla

ry p

ress

ure


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Water distribution???

Interface : capillary pressure continous

suction continous


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Problem: different capillary pressures


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Phase

changes water

• Sublimination • Condensation –

Evaporation • Freezing -

Melting


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Evaporating Water into Air

Liquid water experiences dynamic departures of water molecules from its surface, called evaporation, together with arrivals of molecules from adjacent vapor, called condensation. When air is saturated, evaporation and condensation are in equilibrium.


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The partial pressure of water vapor, i.e., that portion of total atmospheric pressure that is due to the presence of H2Ov


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Relative humidity (RH)= partial water pressure

maximum water pressure


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Relative humidity (RH)= partial water pressure

maximum water pressure

maxmaxmax ρρϕ ===

pTR

TRp

pp

Relative humidity (RH)= actual water vapour content

maximum water vapour content


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Daily Humidity

Patterns

Figure 7.10


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P=611exp[0.0829 T-0.2881 10-3 T2+4.403 10-6T3

The temperature to which the air must be cooled (at constant pressure and without changing the moisture) for it to become saturated

Dew point temperature


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Chilled Mirror Dew Point

Mirror is chilled until dew is formed. The temperature at which saturation is achieved is determined by observing condensation on a chilled surface (mirror).

Mirror

Optical Sensor

Advantages • Very high accuracy • High reliability

Disadvantages • Need clean mirror • Expensive

Cooler


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70’s energy crisis -> isolation

roofs started to collapse after few years >why??

Flat -roof

20 oC

5 oC

wood

Water+

vapour barrier

gypsum board


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Pictures by Henk Schellen


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Flat -roof

20 oC, 60 %

5 oC

wood

Water+

vapour barrier

gypsum board


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Flat -roof

20 oC, 60 %

5oC

wood

Water+

vapour barrier

gypsum board


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20oC > max 2337 Pa

60% = 1402 Pa

Cool down 5oC

1402 Pa > 12oC

CONDENSATION!!!


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Flat -roof

20 oC, 60 %

5oC

wood

Water+

vapour barrier

gypsum board

120C max

condensation


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Condensation

- isolation

- bathroom fungi growth

- musea (wall paintings)

- wood


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Capillary condensation If the vapour pressure of water within a porous eventually filling the pores. This process is known as capillary condensation. For capillary condensation to occur, the water vapour pressure must exceed its saturation vapour pressure.

BUT: in porous materials the saturation vapour pressure varies!!!

This is due to the pressure drop across a curved liquid surface, and is described by the Kelvin equation


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Capillary

r

p<0

−==

RTrpph l

vs

v

ργ2exp

h = relative humidity (0-100%)

Kelvin equation

Negative pressure

See proof dictaat

Capillary pressure


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William Thomson, 1st Baron Kelvin of Largs (1824–1907)

Born 26 June 1824(1824-06-26) Belfast

Died 17 December 1907 (aged 83)[1] Largs

Residence Cambridge, Glasgow, Belfast

Nationality British Institutions University of Glasgow Known for Joule–Thomson effect

Thomson effect (thermoelectric) Mirror galvanometer Siphon recorder Kelvin material Kelvin water dropper Kelvin wave Kelvin–Helmholtz instability Kelvin–Helmholtz mechanism Kelvin–Helmholtz luminosity Kelvin transform Kelvin's circulation theorem Kelvin bridge Kelvin sensing Kelvin equation Magnetoresistance Four-terminal sensing Coining the term 'kinetic energy'

http://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin


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0.001 µm ~30% water

100 %

0.01 µm 90%

0.001 µm 30%


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Condensation

• Dehumidifiers


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• Warning against humidity (electronics)

Moist

Silica impregnated with CoCl2


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Raindrop

positive pressure

==

rRTh

lv

v 2exp0, ρ

γρρ

r

> 1

It is very difficult to form clouds with pure water vapor (nucleation problem)


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Applying Kelvin equation Drop in its vapor. The vapor pressure of a drop is higher than that of

a liquid with a planar surface. One consequence is that an aerosol of drops (fog) should be unstable

To see this let us assume that we have a box filled with many drops in a gaseous environment, some drops are larger than others

The small drops have higher vapor pressure than the large drops, hence more liquid evaporates from their surface

This tends to condense into larger drops Within a population a drops of different sizes, the bigger drops will

grow at the expense of the smaller one, these drops will sink down and at the end bulk liquid fills the bottom of the box


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Porous media

wetting propeties

θ

r

vapor 0cos2

<−r

θγ

10,

<v

v

ρρ

Hydrophilic surfaces

liquid

θ

r vapor

liquid

0cos2>−

rθγ

10,

>v

v

ρρ

Hydrophobic surfaces


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R

p<0

rp wn

cγ2

=

−==

RTrpph wn

vs

v

ργ2exp

coupled

capillary

Porous media

ρθψ

gpc )(

=

== ψ

RTMg

pph

vs

v exp

macro


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Relative

Humidity


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Hygroscopic curve

Hysteresis


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Very,very slow (6 months) + temperature


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Dynamic Vapour sorption


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Relation capillary pressure <-> RH Pressure pore relative humidity

bar %

0 ∞ 100

0.1 15 µm 99.993

1 1.5 µm 99.93

15 100 nm 98.9

100 15 nm 93

500 3 nm 70

1000 1.5 nm 48

5000 0.3 nm 2.6

So never in one measurement

vapour

liquid


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Pore size classification

Micropores r<1 nm p/po < 0.1 >1000 bar

Mesopores 1 < r < 25 nm

Macro pores r>25 nm p/po >0.96 <15 bar

IN SMALL PORES (first filled)


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Drying cracks concrete especially: high performance concrete (HPC)

• Early age pavement cracking is a persistent problem – Runway at Willard Airport (7/21/98) – Early cracking within 18 hrs and

additional cracking at 3-8 days


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Autogenous Shrinkage

-250

-200

-150

-100

-50

0

50

0 20 40 60 80 100Age (d)

Aut

ogen

ous

Shr

inka

ge (1

0-6 m

/m)

OPC1, w/c = 0.40SCC1, w/c = 0.39SCC2, w/c = 0.33SCC3, w/c = 0.41SCC4, w/c = 0.32HPC1, w/c = 0.25SCC2-2SCC2-slag


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Autogenous shrinkage: why only low w/c?

0.50 0.50 w/cw/c

0.30 0.30 w/cw/c

Cement grains initially separated by

water

Initial set locks in paste structure

Chemical shrinkage ensures some porosity remains even at α=1

“Extra” water remains in small pores even at α=1

Pores to 50 nm emptied

Internal RH and pore fluid pressure reduced as smaller

pores are emptied

Autogenous Autogenous shrinkageshrinkage

Increasing degree of hydration

0.50 0.50 w/cw/c

0.30 0.30 w/cw/c

Cement grains initially separated by

water

Initial set locks in paste structure

Chemical shrinkage ensures some porosity remains even at α=1

“Extra” water remains in small pores even at α=1

Pores to 50 nm emptied

Internal RH and pore fluid pressure reduced as smaller

pores are emptied

Autogenous Autogenous shrinkageshrinkage

Increasing degree of hydration

HPC : concrete made with low moisture content


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Mechanism of shrinkage

• Both autogenous and drying shrinkage dominated by capillary surface tension mechanism

• As water leaves pore system, curved menisci develop, creating reduction in RH and underpressure within the pore fluid

Hydratio product

Hydration product


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Visualize scale of mechanism Capillary stresses present in pores with radius between 2-50 nm

Note the dimensions

•C-S-H makes up ~70% of hydration product •Majority of capillary stresses likely present within C-S-H network

*Micrograph take from Taylor “Cement Chemistry” (originally taken by S. Diamond 1976)


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BE AWARE

LOW MOISTURE CONTENT

REV


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0102030405060708090

100

0 5 10 15 20 25 30

Sqrt Area

n (%

)

Representative Elementary Volume (area) REV

Choice error


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Same moisture content


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RH

θ


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Porous media

ρθψ

gpc )(

= )(exp θψ fRTMgh =

=

θ

hysteresis

How/What to measure in porous material


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Question ?

Liquid ‘fast’ Vapour ‘slow’


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Membrane method (hard materials)

sample

P

semi-permeable membrane

water drainage/wetting

Slow measurement (order weeks)

)(θcc pp =

Comination liquid/vapour

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