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# Session 4 ic2011 wang

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### Session 4 ic2011 wang

1. 1. Forced Fluid Imbibitionin a Powder-Packed Column Jinwu Wang, Post Doctoral Associate Sheldon Q. Shi, Assistant Professor Department of Forest Products Mississippi State University
2. 2. ObjectivesDevelop a tool to measure contact anglesand surface energies for both– Spontaneous and– non-spontaneous imbibing liquids in powdersCurrent Problem– Spontaneous inbibition is not achieved in many cases when the wetting angle is larger than 900
3. 3. ExplanationWhen a rigid container is inserted into a fluid, the fluid willrise in the container to a height higher than the surroundingliquid Capillary Tube Wedge Sponge Professor John Pelesko and Anson Carter, Department of Mathematics, University of Delaware
4. 4. Velocity Field around the Moving MeniscusPhys. Rev. Lett. (2007), Capillary Rise in Nanopores: Molecular Dynamics Evidence for the Lucas-Washburn Equation
5. 5. Liquid Behaviors in Powders A powder-packed column with radius R air Assume that a powder-packed Liquid column consists of numerous capillary tubes: a wicking-equivalent effective capillary radius The same governing equations as those applied to a capillary tubeCapillary action
6. 6. Free Body Diagram r Surface Tension External vacuum }Driving Forces Poiseuille Viscous Force Gravitation Force Inertial Force Z(t) Dragging ForcesList of Variables:volume = πr2zg = gravityr = radius of capillary tubez = rising height, measured to the bottom of the meniscus, at time t ≥ 0ρ = density of the surface of the liquidγ = surface tensionθ = contact angle between the surface of the liquid and the wall of the tube
7. 7. Explanation of the Forces Surface Tension Force 2π r γ cos( θ ) Gravitational Force Fw = mg = ρπr 2 zg Poiseuille Viscous Force Fdrag = 8πη z dz dt Vacuum Force πr2ΔP Newtons Second Law of Motion d (mv) d ⎛ 2 dz ⎞ ⎛ d 2 z ⎛ dz ⎞2 ⎞∑ F= dt dt ⎝ dt ⎠ 2 ⎜ = ⎜πr zρ ⎟ = πr ρ z 2 + ⎜ ⎟ ⎟ ⎜ dt ⎝ dt ⎠ ⎟ ⎝ ⎠
8. 8. Explanation of Differential Equation Newtons Second Law of Motion:Net Force = Surface Tension Force +Vacuum - Poiseuitte Viscous Force - Gravitational Force ⎛ d 2 z ⎛ dz ⎞ 2 ⎞ dz ρπr 2 ⎜ z 2 + ⎜ ⎟ ⎟ = 2πrγ cos(θ ) + πr 2 ΔP − 8πηz − ρgπr 2 z ⎜ dt ⎝ dt ⎠ ⎠ ⎟ dt ⎝ Dividing by πr2, the differential equation becomes: ⎛Zo = Z(0) =⎛0dz ⎞ 2 ⎞ 2 d 2z ρ ⎜z +⎜ ⎟ ⎟ = γ cos(θ ) + ΔP − 8 ηz dz − ρgz ⎜ dt ⎝ dt ⎠ ⎟ r 2 ⎝ ⎠ r 2 dt Boundary Conditions: z(0) = 0 and z’(z∞) = 0
9. 9. The Effective Zone of Forces The size of each zone depends on the probe Gravity Effective liquid properties and Zone capillary structures z Washburn Zone z0 Inertial Force 2 8 dz γ cosθ + ΔP − 2 ηz − ρgz = 0 r r dt 8η ⎛ ze ⎞ 2γ cos θ ΔPt = 2 ⎜ z e ln ⎜ − z (t ) ⎟ ⎟ ze = + r ρg ⎝ z e − z (t ) ⎠ ρgr ρg
10. 10. The Effect of Capillary Radius on Wicking Lucas- Washburn equation: 1/ 2 ⎛ γ r cos θ ⎞ z (t ) =⎜ ⎟ 2 ⎜ 2η ⎟ t ⎝ ⎠Is valid when Capillary diameter is small At initial rising period Viscous drag >> gravity force Density is low, inertia is small
11. 11. Column Wicking Diagram Non-spontaneous inbibitionwhen the contact angle is larger than 900 by applying vacuum spontaneous inbibition
12. 12. Experimental SetupVacuum Vacuum Vacuum Sample LiquidRegulator Pump Gauge
13. 13. Rising Rate by Image Analysis0s 2s 65 s 150 s 410 s 614 s 700 s Imbibing was recorded by camera video Scale was referenced with a caliper Advancing front line vs. time processed by ImageJ image analysis
14. 14. Observations 0.07 Hexane Replicate 1 0.06 Energy loss due to Hexane Replicate 2Rising Height (m) 0.05 Contact angle, Methanol, Experimental partial wetting 0.04 (water) Water, Experimental 0.03 Polar liquid Hexane, theta = 0 swelling (methanol) 0.02 Methanol, theta = 0 Heat of wetting, 0.01 (water & methanol) Water, theta = 0 0 0 100 200 300 Time (s) γ η ρ Assuming full wetting, i.e. contact angle is mJ/m2 mPa.s g/cm3 zero. Rising rates: Water > Hexane > Methanol Hexane 18.4 0.326 0.65 Experimental: Hexane > Methanol > Water Water 72.8 1 1 Some energy is not used for rising in water and Methanol 22.5 0.54 0.79 methanol imbibitions
15. 15. Reproducibility & Vacuum: Hexane 0.07 Replicate 1 0.06 Replicate 2 0.05 Replicate 3 Replicate 4 0.04 Rising height (m) replicate 5 0.03 Vacuum 453 Pa 0.02 Vacuum 1050 Pa 0.01 Vacuum 4700 Pa Vacuum 5800 Pa 0 0 20 40 60 80 100 Time (s) Reproducibility is good for hexane imbibitions Rising rates increase with the vacuum
16. 16. Reproducibility & Vacuum: Water 0.14 Replicate 1 0.12 Replicate 2 Replicate 3 0.1 Replicate 4 Rising Height (m) 0.08 Replicate 5 Replicate 6 0.06 Vacuum 2237 Pa Vacuum 2362 Pa 0.04 Vacuum 2658 Pa 0.02 Vacuum 2856 Pa 0 0 100 200 300 400 500 600 Time (s)Reproducibility for water is not as good as hexane imbibitionsRising rates increase with the vacuum
17. 17. Experimental Data: EG & Glycerol 0.1 0.06 0.09 Vacuum 2,914 Pa 0.08 0.05 Vacuum 26,319 Pa Rising Height (m) 0.07 Vacuum 26,553 PaRising Height (m) 0.04 0.06 vacuum 23,496 Pa Vacuum 2353 Pa 0.05 0.03 Vacuum 22,668 Pa Vacuum 2106 Pa 0.04 Vacuum 2053 Pa 0.02 0.03 0.02 Vacuum 2160 Pa 0.01 Vacuum 2266 Pa 0.01 0 Vacuum 2160 Pa 0 0 100 200 300 400 0 500 1000 1500 Time (s) Time (s) γ η ρ mJ/m2 mPa.s g/cm3 Ethylene glycol imbibes very slowly Hexane 18.4 0.326 0.65 without external vacuum Ethylene glycol 48 16.1 1.113 Glycerol cannot imbibe spontaneously Glycerol 64 1420 1.261
18. 18. Results and DiscussionDefine the effective capillary radius withhexaneThe effect of polar liquidsEnergy loss constantContact angle with waterVacuum induced slip
19. 19. Effective Capillary Radius from Hexane2 8 dz γ cosθ + ΔP − 2 ηz − ρgz = 0 Effective Capillary Radius (r) R2r r dt Replicate 1 1.41E-06 1.00 ⎛ ⎞ Replicate 2 1.41E-06 1.00 8η zet = ⎜ z e ln − z (t ) ⎟ Replicate 3 1.56E-06 0.98 r 2ρg ⎜ ⎝ z e − z (t ) ⎟ ⎠ Replicate 4 1.20E-06 1.00 Replicate 5 1.10E-06 0.99 2γ cos θ ΔP Average 1.34E-06 ze = + COV (%) 13.80 ρgr ρg Quasi state ma=0 Average effective No external vacuum, ΔP = 0 Capillary Radius Full wetting, cos(θ) = 1 No swelling & release of heat of r = 1.34 × 10 −6 μm wetting
20. 20. Effect of Polar Liquid 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 liquid π R 2 ρ m − (1 + δ v ) G m Gm, unit column mass ofrs = ⋅r the material (g/m) πρ m R − G m 2 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.
21. 21. 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 3.83 G0 (tube weight) (g) 5.87 4.06 4.05 4.07 G1 (g) 6.56 4.63 4.58 4.64 G2 (g) wet weight 7.35 5.03 5.29 5.14 Packing Length, mm 161.4 126.9 127.5 129.3 wetting Length,mm 91.4 69.0 72.7 40.0 density (g/cm3) 0.38 0.39 0.37 0.38 Gm (g/m) 4.27 4.53 4.19 4.38 wet (g/g wood) 2.01 1.25 2.34 2.85 Wet(g/cm) 0.09 0.06 0.10 0.12 r/rs 0.75 0.80 0.78 0.71
22. 22. Derivation of Energy Loss Constant Quasi-state ma = 0; External vacuum ΔP = 0 C (J/m) R2 Deformable materials, r into rs Rep. 1 5.59E-07 1.00 Energy loss is proportional to shrinkage and reverse Rep. 2 4.88E-07 1.00 proportional to r2 by C Rep. 3 5.57E-07 0.99 Fitting with methanol imbibition data, i.e. cos(θ) =0 Rep. 4 5.62E-07 0.98 2rs Cδ 8 dz Rep.5 4.45E-07 0.98 γ cos θ + ΔP − 2v − 2 ηz − ρgz = 0 Average 5.52E-07 r2 πr rs dt Cov 9.6% 8η ⎛ ze ⎞t = 2 ⎜ z e ln ⎜ − z (t ) ⎟ ⎟ rs ρ g ⎝ z e − z (t ) ⎠ Average energy loss 2rsγ cos θ ΔP cδ constant ze = + − 2v ρgr 2 ρg πr ρg −7 C = 5.52 × 10 J /m
23. 23. Contact Angle with Water2rs Cδ v 8 dz r = 1 . 34 × 10 − 6 μ m γ cos θ + ΔP − 2 − 2 ηz − ρgz = 0r 2 πr rs dt C = 5 . 52 × 10 − 7 J / m 8η⎛ ze ⎞t = 2 ⎜ z e ln − z (t ) ⎟ rs / r = 0 . 75 ⎜ rs ρ g ⎝ z e − z (t ) ⎟ ⎠ θ (°) R2 2rsγ cos θ ΔP cδ ze = + − 2v Rep. 1 63 0.99 ρgr 2 ρg πr ρg Rep. 2 57 0.99 Quasi-state ma = 0; External vacuum ΔP = 0 Rep. 3 65 0.97 Deformable materials, r into rs Rep. 4 48 0.93 Energy loss is proportional to shrinkage and reverse Rep. 5 53 0.84 proportional to r2 by C Rep. 6 64 0.94 Fitting with water imbibition data to calculate Average 58 cos(θ) COV (%) 12.6The water contact angles calculated from the model (58°) is in agreement with the sessile drop T. Nguyen and W. E. Johns, Wood Sci. Technol. 12, 63–74 (1978).results (60°) from the literature V. R. Gray, For. Prod. J. 452–461 (Sept. 1962).
24. 24. Effect of VacuumUnder vacuum, the rise of the liquid proceeds muchfaster than predicted even with con(θ) = 1, clearly indicating a slip radius δ in the interface
25. 25. Slip under Vacuum
26. 26. r Force without Slip 2γ cosθ capillary force: FSurfaceTesnsion = R Gravity: FGravity = mg dz (t ) Z(t) ηz (t ) 8 viscous drag: Fviscous = R2 dt ESF-Exploratory Workshop Microfluidic: Rome, Sept. 28-30, 2007
27. 27. Effect of Slip under Vacuum dz (t ) ηz (t ) 8 Fviscous = (R + δ ) 2 dtD.I. Dimitrov, A. Milchev, and K. Binder, Capillary rise in nanopores: Molecular dynamics evidence for the Lucas-Washburn equation, Physical Review Letters, 99 (2007).
28. 28. Full Models Vacuum Viscous Drag Swelling 2 rs Cδ v 8 dz γ cos θ + Δ P − − ηz − ρ gz = 0 r 2 πr 2 ( rs + δ ) 2 dt Surface Energy Slip gravity Tension Loss Radius 8η ⎛ ze ⎞t = ⎜ z e ln ⎜ − z (t ) ⎟ ⎟ r = 1.34 × 10 −6 μm ( rs + δ ) ρ g ⎝ 2 z e − z (t ) ⎠ C = 5.52×10−7 J / m 2rsγ cos θ ΔP cδ v ze = + − 2 rs / r = 0 . 75 ρgr 2 ρg πr ρg
29. 29. Slip Radius under Vacuum 1.8E-05 1.6E-05 y = 5E-10x + 2E-06 1.4E-05 R² = 0.898 1.2E-05 Slip Radius (m) 1.0E-05 8.0E-06 6.0E-06 Hexane Methanol 4.0E-06 Water 2.0E-06 Ethylene Glycol Glycerol 0.0E+00 0 5,000 10,000 15,000 20,000 25,000 30,000 Vacuum (Pa) Assuming forced wetting under vacuum, cos(θ)=1 Slip radius is roughly proportional to vacuum Contact angle and slip radius cannot be decoupled except for figuring out slipradius with alternative methods
30. 30. ConclusionsRising rates of imbibitions can be measured preciselywith an image acquisition and analysis systemThe effect of swelling and heat of wetting can becalibrated by hexane and methanolContact angles for other polar and partial wetting liquidscan thus be measured reasonablyVacuum induced slip; the slip and partial wetting werecoupling together such that contact angle could not bemeasured separately in this investigation. Furtherinvestigation is needed to correlate the extent of slip andvacuum.