Session 4 ic2011 wang
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Transcript of Session 4 ic2011 wang
Jinwu Wang, Post Doctoral AssociateSheldon Q. Shi, Assistant Professor
Forced Fluid Imbibitionin a Powder-Packed Column
Department of Forest ProductsMississippi State University
ObjectivesDevelop a tool to measure contact angles and surface energies for both– Spontaneous and – non-spontaneous imbibing liquids in powders
Current Problem– Spontaneous inbibition is not achieved in many
cases when the wetting angle is larger than 900
ExplanationWhen a rigid container is inserted into a fluid, the fluid will rise in the container to a height higher than the surrounding liquid
Capillary Tube Wedge Sponge
Professor John Pelesko and Anson Carter, Department of Mathematics, University of Delaware
Phys. Rev. Lett. (2007), Capillary Rise in Nanopores: Molecular Dynamics Evidence for the Lucas-Washburn Equation
Velocity Field around the Moving Meniscus
Liquid Behaviors in Powders
Assume that a powder-packed column consists of numerous capillary tubes: a wicking-equivalent effective capillary radius
The same governing equations as those applied to a capillary tube
Liquid
air
A powder-packed column with radius R
Capillary action
List of Variables:volume = πr2z g = gravityr = radius of capillary tubez = rising height, measured to the bottom of the meniscus, at time t ≥ 0ρ = density of the surface of the liquid γ = surface tensionθ = contact angle between the surface of the liquid and the wall of the tube
Free Body Diagram r
Z(t)
Poiseuille Viscous ForceGravitation Force Inertial Force
Surface TensionExternal vacuum }Driving Forces
Dragging Forces
Explanation of the Forces
Gravitational Force
Poiseuille Viscous Force
Vacuum Force
)cos(2 θγπr
zgrmgFw2ρπ==
dtdzzFdrag πη8=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+=⎟
⎠⎞
⎜⎝⎛==∑
2
2
222)(
dtdz
dtzdzr
dtdzzr
dtd
dtmvdF ρπρπ
Newton's Second Law of Motion
Surface Tension Force
πr2ΔP
Explanation of Differential Equation
Newton's Second Law of Motion:Net Force = Surface Tension Force +Vacuum
- Poiseuitte Viscous Force - Gravitational Force
Dividing by πr2, the differential equation becomes:
Zo = Z(0) = 0
gzdtdzz
rP
rdtdz
dtzdz ρηθγρ −−Δ+=⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+ 2
2
2
2 8)cos(2
Boundary Conditions:z(0) = 0 and z’(z∞) = 0
zrgdtdzzPrr
dtdz
dtzdzr 22
2
2
22 8)cos(2 πρπηπθγπρπ −−Δ+=⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
The Effective Zone of Forces
The size of each zone depends on the probe liquid properties and capillary structures
z
z0Inertial Force
Washburn Zone
Gravity Effective Zone
08cos22 =−−Δ+ gz
dtdzz
rP
rρηθγ
gP
grze ρρ
θγ Δ+=
cos2⎟⎟⎠
⎞⎜⎜⎝
⎛−
−= )(
)(ln8
2 tztzz
zzgr
te
eeρ
η
The Effect of Capillary Radius on Wicking
Lucas-Washburn equation:
( ) trtz2/1
2
2cos
⎟⎟⎠
⎞⎜⎜⎝
⎛=
ηθγ
Is valid whenCapillary diameter is smallAt initial rising periodViscous drag >> gravity forceDensity is low, inertia is small
Column Wicking Diagram
Non-spontaneous inbibitionwhen the contact angle is larger than 900
by applying vacuum spontaneous inbibition
Experimental Setup
Vacuum Regulator
Vacuum Pump
Vacuum Gauge
Sample Liquid
Rising Rate by Image Analysis
0 s 150 s 2 s 410 s 614 s 700 s65 s Imbibing was recorded by camera videoScale was referenced with a caliperAdvancing front line vs. time processed by ImageJ image analysis
γmJ/m2
ηmPa.s
ρg/cm3
Hexane 18.4 0.326 0.65Water 72.8 1 1
Methanol 22.5 0.54 0.79
Assuming full wetting, i.e. contact angle is zero. Rising rates: Water > Hexane > Methanol
Experimental: Hexane > Methanol > WaterSome energy is not used for rising in water and
methanol imbibitions
Energy loss due to
Contact angle, partial wetting (water)
Polar liquid swelling (methanol)
Heat of wetting, (water & methanol)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 100 200 300
Ris
ing
Hei
ght (
m)
Time (s)
Hexane Replicate 1
Hexane Replicate 2
Methanol, Experimental
Water, Experimental
Hexane, theta = 0
Methanol, theta = 0
Water, theta = 0
Observations
Reproducibility & Vacuum: Hexane
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 20 40 60 80 100
Ris
ing
heig
ht (m
)
Time (s)
Replicate 1
Replicate 2
Replicate 3
Replicate 4
replicate 5
Vacuum 453 Pa
Vacuum 1050 Pa
Vacuum 4700 Pa
Vacuum 5800 Pa
Reproducibility is good for hexane imbibitionsRising rates increase with the vacuum
Reproducibility & Vacuum: Water
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 100 200 300 400 500 600
Ris
ing
Hei
ght (
m)
Time (s)
Replicate 1Replicate 2Replicate 3Replicate 4Replicate 5Replicate 6Vacuum 2237 PaVacuum 2362 PaVacuum 2658 PaVacuum 2856 Pa
Reproducibility for water is not as good as hexane imbibitionsRising rates increase with the vacuum
Experimental Data: EG & Glycerol
00.010.020.030.040.050.060.070.080.09
0.1
0 100 200 300 400
Ris
ing
Hei
ght (
m)
Time (s)
Vacuum 2353 PaVacuum 2106 PaVacuum 2053 PaVacuum 2160 PaVacuum 2266 PaVacuum 2160 Pa
0
0.01
0.02
0.03
0.04
0.05
0.06
0 500 1000 1500R
isin
g H
eigh
t (m
)Time (s)
Vacuum 2,914 PaVacuum 26,319 PaVacuum 26,553 Pavacuum 23,496 PaVacuum 22,668 Pa
γmJ/m2
ηmPa.s
ρg/cm3
Hexane 18.4 0.326 0.65Ethylene glycol 48 16.1 1.113
Glycerol 64 1420 1.261
Ethylene glycol imbibes very slowly without external vacuum
Glycerol cannot imbibe spontaneously
Results and DiscussionDefine the effective capillary radius with hexane The effect of polar liquidsEnergy loss constantContact angle with waterVacuum induced slip
Effective Capillary Radius from Hexane
08cos22 =−−Δ+ gz
dtdzz
rP
rρηθγ
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−= )(
)(ln8
2 tztzz
zzgr
te
eeρ
η
gP
grze ρρ
θγ Δ+=
cos2
Quasi state ma=0 No external vacuum, ΔP = 0 Full wetting, cos(θ) = 1 No swelling & release of heat of wetting
Effective Capillary Radius (r) R2
Replicate 1 1.41E-06 1.00Replicate 2 1.41E-06 1.00Replicate 3 1.56E-06 0.98Replicate 4 1.20E-06 1.00Replicate 5 1.10E-06 0.99
Average 1.34E-06COV (%) 13.80
Average effective Capillary Radius
mr μ 1034.1 6−×=
Effect of Polar Liquid
rGR
GRrmm
mvms ⋅
−+−
= )1(2
2
πρδρπ
S.Q. Shi and D.J. Gardner, A new model to determine contact angles on swelling polymer particles by the column wicking method, Journal of Adhesion Science and Technology, 14 (2000) 301-314.
r, average capillary radius (m)rs, average capillary
radius after material swelling (m)R, inner radius of the
column tube (m) ρm, material density
(g/cm3 )δv, volume shrinkage
after absorbing probe liquidGm, unit column mass of
the material (g/m)
Characteristics of Packing tubes
Name Water Methanol Ethylene glycol GlycerolVolume Shrinkage (%) 15.0 13.8 17.4 20.0
Inner d (mm) 3.77 3.84 3.78 3.83G0 (tube weight) (g) 5.87 4.06 4.05 4.07
G1 (g) 6.56 4.63 4.58 4.64G2 (g) wet weight 7.35 5.03 5.29 5.14
Packing Length, mm 161.4 126.9 127.5 129.3wetting Length,mm 91.4 69.0 72.7 40.0
density (g/cm3) 0.38 0.39 0.37 0.38Gm (g/m) 4.27 4.53 4.19 4.38
wet (g/g wood) 2.01 1.25 2.34 2.85Wet(g/cm) 0.09 0.06 0.10 0.12
r/rs 0.75 0.80 0.78 0.71
Derivation of Energy Loss ConstantQuasi-state ma = 0; External vacuum ΔP = 0 Deformable materials, r into rs
Energy loss is proportional to shrinkage and reverse proportional to r2 by CFitting with methanol imbibition data, i.e. cos(θ) =0
08cos2222 =−−−Δ+ gz
dtdzz
rrCP
rr
s
vs ρηπδθγ
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−= )(
)(ln8
2 tztzz
zzgr
te
ee
s ρη
grc
gP
grrz vs
e ρπδ
ρρθγ
22
cos2−
Δ+=
C (J/m) R2
Rep. 1 5.59E-07 1.00Rep. 2 4.88E-07 1.00Rep. 3 5.57E-07 0.99Rep. 4 5.62E-07 0.98Rep.5 4.45E-07 0.98
Average 5.52E-07Cov 9.6%
mJC / 1052.5 7−×=
Average energy loss constant
Contact Angle with Water
Quasi-state ma = 0; External vacuum ΔP = 0 Deformable materials, r into rs
Energy loss is proportional to shrinkage and reverse proportional to r2 by CFitting with water imbibition data to calculate cos(θ)
08cos2222 =−−−Δ+ gz
dtdzz
rrCP
rr
s
vs ρηπδθγ
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−= )(
)(ln8
2 tztzz
zzgr
te
ee
s ρη
grc
gP
grrz vs
e ρπδ
ρρθγ
22
cos2−
Δ+=
θ (°) R2
Rep. 1 63 0.99Rep. 2 57 0.99Rep. 3 65 0.97Rep. 4 48 0.93Rep. 5 53 0.84Rep. 6 64 0.94
Average 58COV (%) 12.6
mJC / 1052.5 7−×=
mr μ 1034.1 6−×=
75.0/ =rrs
The water contact angles calculated from the model (58°) is in agreement with the sessile drop results (60°) from the literature T. Nguyen and W. E. Johns, Wood Sci. Technol. 12, 63–74 (1978).
V. R. Gray, For. Prod. J. 452–461 (Sept. 1962).
Effect of Vacuum
Under vacuum, the rise of the liquid proceeds much faster than predicted even with con(θ) = 1, clearly
indicating a slip radius δ in the interface
Slip under Vacuum
capillary force: RF nsionSurfaceTes
θγ cos2=
viscous drag: ( ) ( )dt
tdztzR
Fviscous η2
8=
ESF-Exploratory Workshop Microfluidic: Rome, Sept. 28-30, 2007
Force without Slipr
Z(t)
Gravity: mgFGravity =
( ) ( )dt
tdztzR
Fviscous ηδ 2)(
8+
=
Effect of Slip under Vacuum
D.I. Dimitrov, A. Milchev, and K. Binder, Capillary rise in nanopores: Molecular dynamics evidence for the Lucas-Washburn equation, Physical Review Letters, 99 (2007).
Full Models
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−+= )(
)(ln
)(8
2 tztzz
zzgr
te
ee
s ρδη
grc
gP
grrz vs
e ρπδ
ρρθγ
22
cos2−
Δ+=
mJC / 1052.5 7−×=
mr μ 1034.1 6−×=
75.0/ =rrs
0)(
8cos2222 =−
+−−Δ+ gz
dtdzz
rrCP
rr
s
vs ρηδπ
δθγ
SurfaceTension
Vacuum
EnergyLoss
Viscous Drag
gravitySlip Radius
Swelling
Slip Radius under Vacuum
y = 5E-10x + 2E-06R² = 0.898
0.0E+00
2.0E-06
4.0E-06
6.0E-06
8.0E-06
1.0E-05
1.2E-05
1.4E-05
1.6E-05
1.8E-05
0 5,000 10,000 15,000 20,000 25,000 30,000
Slip
Rad
ius (
m)
Vacuum (Pa)
HexaneMethanolWaterEthylene GlycolGlycerol
Assuming forced wetting under vacuum, cos(θ)=1Slip radius is roughly proportional to vacuumContact angle and slip radius cannot be decoupled except for figuring out slip
radius with alternative methods
ConclusionsRising rates of imbibitions can be measured precisely with an image acquisition and analysis systemThe effect of swelling and heat of wetting can be calibrated by hexane and methanolContact angles for other polar and partial wetting liquids can thus be measured reasonablyVacuum induced slip; the slip and partial wetting were coupling together such that contact angle could not be measured separately in this investigation. Further investigation is needed to correlate the extent of slip and vacuum.
Thank you for your attentions
Questions or Comments
?