joseph d. seymour - porelabporelab.no/wp-content/uploads/2017/09/porelab_oslo_dispersion_0917... ·...

NMR measurement of hydrodynamic dispersion in porous media: structure, dynamics and statistical

mechanics

Joseph D. SeymourCollege of Engineering Magnetic Resonance Lab

Chemical & Biological EngineeringCenter for Biofilm Engineering

Montana State University, Bozeman MT USA

Statistical Mechanics Playground

Granular Materials/Colloids- Statistical Mechanics

Sandbox[Jaeger, Nagel, and Behringer, RMP 68 1259 (1996); Seymour et al. PRL 84 266 (2000);

Brown et al. PRL 99 240602 (2007)]

Porous Media StatisticalMechanics Jungle Gym[Scheidegger, J. App. Phys. 25(8): 994

(1954); Saffman, JFM 6 321 (1959); Seymour and Codd, EPJ-App. Phys (2012)]

Pulsed Gradient Spin Echo (PGSE) NMR

Fourier Transform Relationship

Slope = D

Stejskal-Tanner Plot

Ln(E(q))

42q2(-/3)

[1/m]

FT

q = (2)-1g

Gradient (g) direction

R

r r + R

Mean Displacement - VelocityPhase shift ( = gv)

180o90o

Echo

RF

g

g

geff

Spin Echo

HydrodynamicDispersion

( ) .== dRRqiRPSqSqE S 2exp),()0(

)()(

Structure Function Relations

Order Disorder

Transport

vmax

v = 0

H2O

beads

241 m monodisperse beads 5 mm ID cylinder Low Re Range of Pe

Flow in a Model Bead Pack[Seymour and Callaghan, AIChE J. 1997]

Flow Diffraction in Monodisperse Bead Pack[Seymour and Callaghan, JMR 1996; AIChE J. (1997); Seymour et al. PRL (2004)]

0.001

0.01

0.1

10 2000 4000 6000 8000 10000

20 ms

100 ms

300 ms

E(q,)

q(m-1)

q = d-1 = (bead diam.)-1

-0.001 0.001 0.003 0.005 0.007

20 ms100 ms300 ms

Propagator Time Evolution

Z (mm)

Displacement Observation Time

()

P(Z, )

Z = d

2

2

zPD

zPv

tP

+

=

Hydrodynamic Dispersion Coefficient Peclet Number Scaling[Seymour and Callaghan, AIChE J. 1997]

= 2 periodic structure (Taylor dispersion)=1 mechanical dispersion~1.2 boundary layer dispersion (Pe lnPe)

DL ~ Pe

Pe1.4

o

p

o Ddv

DlvPe

> cD

*/c

Nonequilibrium Statistical Mechanics[R. Zwanzig Oxford Univ. Press 2001]

2nd order truncation K(t) = v2 exp(-t/)[Boon & Yip Molecular Hydrodynamics Dover (1997)]

Infinite continuedfraction[Mori Prog. Theor. Phys. (1965)]

1 ~s + s + s +

v2 (v4/v2) - v2 (s) =

v2 = /

Memory

Brownian Harmonic Oscillator

D(t)

v2 = o2

Transverse Dispersion in Model Bead Packs of Spheres

[Callaghan and Codd Phys. Fluids 13 421 (2001); Maieret al. Phys. Fluids 12 2065 (2000)]

Liquid Argon - Lennard-Jones potential [Rahman PRL (1964)]

0 0.5 1.0 1.5

TIME IN 10-12 SEC

0

1Molecular hydrodynamics:

D(t)

v2 = o2

Transverse Dispersion in a Polymer Foam [T.R Brosten et al..PRL (2009)]

d Dd = 0

l =

Direct measurement of transport phenomena length scale l

PGSE NMR Measured Transverse Backscattering in Cellular Foam[T.R Brosten et al. PRL (2009)]

Universality of Dynamics[T.R. Brosten et al. PRL 103 218001 (2009)]

Conclusions PGSE NMR & LB simulations determine a

transport length scale of cellular (foam) solids Introduced the memory function formalism for

the short time correlations of pre-asymptotic dispersion

Time dependent dispersion directly relates to structure and permeability

PGSE NMR and Statistical Mechanics ModelsCorrelate Structure and Transport

Acknowledgements Funding

US NSF Career Award CTS-0348076 (JDS) US NSF Career Award CBET-0642328 (SLC) DOE OS BER ERSP DE-FG02-07 ER64416 Equipment Funding NSF MRI and the M. J. Murdock

Charitable Trust

The late Prof. Sir Paul T. Callaghan New Zealand and H. Ted Davis Univ. Minnesota

Model Homogeneous Porous Media

dp=100 m

Spatial resolution 27.3 m/pixel over a 8 mm thick slice

Slice is 80 bead diameters thick!!

Note this image 512 pts!

Radial Distribution Function

0.50

0.75

1.00

1.25

1.50

-10 -8 -6 -4 -2 0(r-R)/d

Norm

aliz

ed V

eloc

ity a

0.50

0.75

1.00

1.25

1.50

1.75

2.00

-10 -8 -6 -4 -2 0

(r-R)/d

Norm

aliz

ed V

eloc

ity a

0.50

0.75

1.00

1.25

1.50

-10 -8 -6 -4 -2 0

(r-R)/d

Norm

aliz

ed V

eloc

ity a

0.50

0.75

1.00

1.25

1.50

-10 -8 -6 -4 -2 0

(r-R)/d

Norm

aliz

ed V

eloc

ity a

i.d. = 10 dp

i.d. = 50 dpi.d. = 25 dp

i.d. = 13 dp

Pe = 63Pe = 254

Velocity Radial Distribution Function

Chart11

0.86102585050.7854612635

0.88664294220.8217330431

0.93283117050.8860626852

0.99187247890.9181374602

1.0480095440.928540057

1.07724831420.9194266577

1.0761518490.9111234016

1.04290093750.8870860676

0.99928824370.8621923071

0.97219392460.8622044245

0.96613588090.9019471209

0.96691458170.9345227063

1.0214230881.0378805844

1.05424645591.0921331992

1.0563387761.1035049502

1.021045551.0771316205

0.9680906041.0370812801

0.93056956541.0145107708

0.90184197061.006812173

0.91924747981.0286106111

0.97380202081.0616978083

1.04146624371.0962215482

1.11334658091.1359908099

1.14134093551.1442020477

1.13108074071.1404995315

1.08391948551.1250968658

1.02196396311.0961247522

0.94947035581.0520966198

0.91433356611.0450065404

0.93525690061.0869610919

1.00825786551.152586052

1.09739527511.2418589043

1.17242061421.2943637071

1.19796532881.2995277173

1.16993348461.2736700985

1.08801282081.2086023576

1.00112040251.1437621772

0.92901841891.0966785117

0.89908545051.0752084508

0.93064188171.1117852371

1.0246670581.2118556784

1.14200402371.3516695347

1.24551641191.4555988456

1.29574558571.4949572456

1.27045350831.4451236828

1.17919613471.3361210851

1.05784921391.1860028271

0.93726592341.0573593669

0.82396859640.9696366155

0.76903669270.953636489

0.73847572031.0449190954

0.70499180581.1642164654

0.60244012441.2334292989

0.4648185651.1831689485

0.30298377380.9596866586

0.17545925710.6203071643

0.08202814040.3185425883

0.02793827940.1497089553

0.00578120140.0495892834

0.00023529710.0084350661

00.0002747712

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

0-0.0000832713

0-0.0012258905

0-0.0042056035

0-0.0062112299

0-0.004981864

0-0.0024957703

0-0.0010959574

0-0.0000758363

00

00

00

00

00

00

00

00

00

00

00

00

Pe=64 Velocity

Pe=253 Velocity

(r-R)/d

Normalized Velocity a

Velocity dist

Velocity radial distribution functionVelocity radial distribution functionVelocity radial distribution functionVelocity radial distribution function

Taken from exp 32, gradient values 1,4 = 720 mT/m difference, exp 20, grad values 1,3 = 120 mT/mTaken from exp 152, gradient values 1,4 = 720 mT/m difference, exp 156, grad values 1,3 = 120 mT/mTaken from exp 211, gradient values 1,4 = 720 mT/m difference, exp 215, grad values 1,3 = 120 mT/mTaken from exp 49, gradient values 1,4 = 720 mT/m difference, exp 62, grad values 1,3 = 120 mT/m

10 mm thick10 mm thick10 mm thick10 mm thick

# r incr =100# r incr =100# r incr =100# r incr =100

FOV (mm) =22mm/s8mm/sFOV (mm) =22mm/s8mm/sFOV (mm) =32mm/s8mm/sFOV (mm) =6

offset for chartvel offsetvel offsetoffset for chartvel offsetvel offsetoffset for chartvel offsetvel offsetoffset for chartvel offsetvel offset

particle diam (mm) =0.1-0.02-0.08-0.05particle diam (mm) =0.1-0.2-0.2-0.2particle diam (mm) =0.1-0.05-0.1-0.1particle diam (mm) =0.10.20.10

1.0 mm i.d. column1.3 mm i.d. column2.5 mm i.d. column5.0 mm i.d. column

include imaging factor of 1.1 (image dimensions are 1.1 times actual dimensions measured with calipers)include imaging factor of 1.1 (image dimensions are 1.1 times actual dimensions measured with calipers)include imaging factor of 1.1 (image dimensions are 1.1 times actual dimensions measured with calipers)include imaging factor of 1.1 (image dimensions are 1.1 times actual dimensions measured with calipers)

include2mms/s8mm/sexp32exp 28exp20include2mms/s8mm/sexp246exp252exp248include2mms/s8mm/sexp 211exp 221exp 215includevel includevel includeexp 49exp61exp 62

r (mm)(r-R)/doffset for figureoffset for figureoffset for figure10 mm vel10 mm10 mm velr (mm)(r-R)/doffset for figureoffset for figureoffset for figure10 mm vel10 mm10 mm velr (mm)(r-R)/doffset for figureoffset for figureoffset for figure10 mm vel10 mm10 mm velr (mm)(r-R)/doffset for figureoffset for figureoffset for figure10 mm vel10 mm10 mm vel

0-5-5.02-5.08-5.057.22E-010.8615.45E+085.94E-013.04E+000.7850-6.5-6.7-6.7-6.73.66E-010.6159.59E-018.51E-012.20E+000.8870-12.5-12.55-12.6-12.67.72E-010.9979.83E-019.02E-013.72E+000.8960-25-24.8-24.9-259.42E-011.1721.10E+091.05E+003.94E+000.971

0.0090909091-4.9090909091-4.9290909091-4.9890909091-4.95909090917.44E-010.8876.04E+086.57E-013.18E+000.8220.0090909091-6.4090909091-6.6090909091-6.6090909091-6.60909090913.66E-010.6151.03E+009.14E-012.06E+000.8300.0136363636-12.3636363636-12.4136363636-12.4636363636-12.46363636367.62E-010.9841.01E+009.31E-013.78E+000.9120.0272727273-24.7272727273-24.5272727273-24.6272727273-24.72727272738.86E-011.1031.02E+099.79E-013.89E+000.957

0.0181818182-4.8181818182-4.8381818182-4.8981818182-4.86818181827.83E-010.9337.52E+088.18E-013.43E+000.8860.0181818182-6.3181818182-6.5181818182-6.5181818182-6.51818181824.03E-010.6761.10E+009.76E-011.73E+000.6970.0272727273-12.2272727273-12.2772727273-12.3272727273-12.32727272737.71E-010.9951.05E+009.62E-013.86E+000.9320.0545454545-24.4545454545-24.2545454545-24.3545454545-24.45454545458.06E-011.0031.01E+099.68E-013.96E+000.974

0.0272727273-4.7272727273-4.7472727273-4.8072727273-4.77727272738.32E-010.9928.71E+089.48E-013.56E+000.9180.0272727273-6.2272727273-6.4272727273-6.4272727273-6.42727272734.66E-010.7831.16E+001.03E+001.61E+000.6480.0409090909-12.0909090909-12.1409090909-12.1909090909-12.19090909097.90E-011.0201.07E+009.78E-014.01E+000.9680.0818181818-24.1818181818-23.9818181818-24.0818181818-24.18181818187.90E-010.9841.02E+099.77E-013.96E+000.976

0.0363636364-4.6363636364-4.6563636364-4.7163636364-4.68636363648.79E-011.0481.00E+091.09E+003.60E+000.9290.0363636364-6.1363636364-6.3363636364-6.3363636364-6.33636363645.24E-010.8801.18E+001.05E+001.63E+000.6570.0545454545-11.9545454545-12.0045454545-12.0545454545-12.05454545457.92E-011.0221.09E+001.00E+004.13E+000.9960.1090909091-23.9090909091-23.7090909091-23.8090909091-23.90909090918.16E-011.0161.04E+091.00E+004.17E+001.028

0.0454545455-4.5454545455-4.5654545455-4.6254545455-4.59545454559.04E-011.0771.08E+091.17E+003.56E+000.9190.0454545455-6.0454545455-6.2454545455-6.2454545455-6.24545454555.78E-010.9711.19E+001.06E+001.75E+000.7050.0681818182-11.8181818182-11.8681818182-11.9181818182-11.91818181827.79E-011.0061.12E+001.03E+004.16E+001.0030.1363636364-23.6363636364-23.4363636364-23.5363636364-23.63636363647.91E-010.9851.01E+099.72E-013.80E+000.935



0.0727272727-4.2727272727-4.2927272727-4.3527272727-4.32272727278.38E-010.9998.36E+089.10E-013.34E+000.8620.0727272727-5.7727272727-5.9727272727-5.9727272727-5.97272727276.38E-011.0711.15E+001.02E+002.30E+000.9240.1090909091-11.4090909091-11.4590909091-11.5090909091-11.50909090917.77E-011.0041.11E+001.02E+004.05E+000.9770.2181818182-22.8181818182-22.6181818182-22.7181818182-22.81818181828.12E-011.0111.01E+099.71E-014.07E+001.001

0.0818181818-4.1818181818-4.2018181818-4.2618181818-4.23181818188.16E-010.9727.94E+088.64E-013.34E+000.8620.0818181818-5.6818181818-5.8818181818-5.8818181818-5.88181818186.17E-011.0361.14E+001.01E+002.35E+000.9470.1227272727-11.2727272727-11.3227272727-11.3727272727-11.37272727277.74E-011.0001.11E+001.01E+004.10E+000.9890.2454545455-22.5454545455-22.3454545455-22.4454545455-22.54545454558.19E-011.0201.03E+099.89E-014.24E+001.044

0.0909090909-4.0909090909-4.1109090909-4.1709090909-4.14090909098.11E-010.9667.67E+088.35E-013.50E+000.9020.0909090909-5.5909090909-5.7909090909-5.7909090909-5.79090909096.08E-011.0211.15E+001.02E+002.36E+000.9490.1363636364-11.1363636364-11.1863636364-11.2363636364-11.23636363647.84E-011.0131.11E+001.02E+004.14E+000.9970.2727272727-22.2727272727-22.0727272727-22.1727272727-22.27272727277.97E-010.9921.04E+091.00E+004.07E+001.002

0.1-4-4.02-4.08-4.058.11E-010.9678.12E+088.84E-013.62E+000.9350.1-5.5-5.7-5.7-5.75.92E-010.9941.17E+001.03E+002.42E+000.9740.15-11-11.05-11.1-11.17.94E-011.0261.10E+001.01E+004.17E+001.0050.3-22-21.8-21.9-227.84E-010.9761.02E+099.81E-013.92E+000.966




0.1363636364-3.6363636364-3.6563636364-3.7163636364-3.68636363648.57E-011.0211.08E+091.17E+004.17E+001.0770.1363636364-5.1363636364-5.3363636364-5.3363636364-5.33636363646.15E-011.0321.16E+001.03E+002.91E+001.1690.2045454545-10.4545454545-10.5045454545-10.5545454545-10.55454545457.73E-010.9981.08E+009.93E-014.44E+001.0710.4090909091-20.9090909091-20.7090909091-20.8090909091-20.90909090917.81E-010.9711.03E+099.87E-014.07E+001.002

0.1454545455-3.5454545455-3.5654545455-3.6254545455-3.59545454558.12E-010.9681.04E+091.13E+004.02E+001.0370.1454545455-5.0454545455-5.2454545455-5.2454545455-5.24545454556.02E-011.0111.13E+001.01E+002.97E+001.1960.2181818182-10.3181818182-10.3681818182-10.4181818182-10.41818181827.76E-011.0021.07E+009.81E-014.35E+001.0480.4363636364-20.6363636364-20.4363636364-20.5363636364-20.63636363647.96E-010.9901.04E+091.00E+004.20E+001.033

0.1545454545-3.4545454545-3.4745454545-3.5345454545-3.50454545457.81E-010.9319.27E+081.01E+003.93E+001.0150.1545454545-4.9545454545-5.1545454545-5.1545454545-5.15454545456.15E-011.0331.12E+009.89E-012.97E+001.1950.2318181818-10.1818181818-10.2318181818-10.2818181818-10.28181818187.71E-010.9951.07E+009.86E-014.17E+001.0070.4636363636-20.3636363636-20.1636363636-20.2636363636-20.36363636367.98E-010.9931.04E+091.00E+004.07E+001.002

0.1636363636-3.3636363636-3.3836363636-3.4436363636-3.41363636367.57E-010.9028.41E+089.15E-013.90E+001.0070.1636363636-4.8636363636-5.0636363636-5.0636363636-5.06363636366.33E-011.0641.10E+009.80E-012.96E+001.1890.2454545455-10.0454545455-10.0954545455-10.1454545455-10.14545454557.70E-010.9941.09E+001.00E+004.12E+000.9930.4909090909-20.0909090909-19.8909090909-19.9909090909-20.09090909098.02E-010.9981.02E+099.85E-013.98E+000.981



0.1909090909-3.0909090909-3.1109090909-3.1709090909-3.14090909098.74E-011.0419.69E+081.06E+004.25E+001.0960.1909090909-4.5909090909-4.7909090909-4.7909090909-4.79090909096.59E-011.1071.12E+009.91E-012.80E+001.1250.2863636364-9.6363636364-9.6863636364-9.7363636364-9.73636363647.49E-010.9681.08E+009.91E-014.03E+000.9710.5727272727-19.2727272727-19.0727272727-19.1727272727-19.27272727278.02E-010.9991.03E+099.91E-014.04E+000.995

0.2-3-3.02-3.08-3.059.34E-011.1131.07E+091.17E+004.40E+001.1360.2-4.5-4.7-4.7-4.76.66E-011.1191.13E+001.00E+002.76E+001.1100.3-9.5-9.55-9.6-9.67.45E-010.9621.07E+009.82E-014.05E+000.9780.6-19-18.8-18.9-197.99E-010.9941.03E+099.90E-014.04E+000.994



0.2272727273-2.7272727273-2.7472727273-2.8072727273-2.77727272739.09E-011.0841.05E+091.14E+004.36E+001.1250.2272727273-4.2272727273-4.4272727273-4.4272727273-4.42727272736.47E-011.0871.11E+009.84E-012.65E+001.0680.3409090909-9.0909090909-9.1409090909-9.1909090909-9.19090909097.91E-011.0221.10E+001.01E+004.22E+001.0190.6818181818-18.1818181818-17.9818181818-18.0818181818-18.18181818187.93E-010.9871.05E+091.01E+004.01E+000.988

0.2363636364-2.6363636364-2.6563636364-2.7163636364-2.68636363648.57E-011.0229.52E+081.04E+004.25E+001.0960.2363636364-4.1363636364-4.3363636364-4.3363636364-4.33636363646.40E-011.0751.09E+009.71E-012.63E+001.0580.3545454545-8.9545454545-9.0045454545-9.0545454545-9.05454545458.04E-011.0381.11E+001.02E+004.33E+001.0440.7090909091-17.9090909091-17.7090909091-17.8090909091-17.90909090918.03E-010.9991.07E+091.03E+004.02E+000.990

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0.86363636363.63636363643.61636363643.55636363643.58636363640.00E+000.0002.33E+082.53E-010.00E+000.0000.86363636362.13636363641.93636363641.93636363641.93636363640.00E+000.0008.85E-027.85E-020.00E+000.0001.29545454550.45454545450.40454545450.35454545450.35454545456.78E-030.0091.15E-011.06E-012.51E-010.0612.59090909090.90909090911.10909090911.00909090910.90909090910.00E+000.0002.53E+072.43E-020.00E+000.000

0.87272727273.72727272733.70727272733.64727272733.67727272730.00E+000.0002.31E+082.51E-010.00E+000.0000.87272727272.22727272732.02727272732.02727272732.02727272730.00E+000.0008.79E-027.80E-020.00E+000.0001.30909090910.59090909090.54090909090.49090909090.49090909099.78E-060.0007.03E-026.46E-021.59E-020.0042.61818181821.18181818181.38181818181.28181818181.18181818180.00E+000.0002.43E+072.34E-020.00E+000.000

0.88181818183.81818181823.79818181823.73818181823.76818181820.00E+000.0002.24E+082.44E-010.00E+000.0000.88181818182.31818181822.11818181822.11818181822.11818181820.00E+000.0008.70E-027.72E-020.00E+000.0001.32272727270.72727272730.67727272730.62727272730.62727272730.00E+000.0005.55E-025.10E-020.00E+000.0002.64545454551.45454545451.65454545451.55454545451.45454545450.00E+000.0002.36E+072.27E-020.00E+000.000

0.89090909093.90909090913.88909090913.82909090913.85909090910.00E+000.0002.22E+082.42E-010.00E+000.0000.89090909092.40909090912.20909090912.20909090912.20909090910.00E+000.0008.57E-027.60E-020.00E+000.0001.33636363640.86363636360.81363636360.76363636360.76363636360.00E+000.0005.18E-024.76E-020.00E+000.0002.67272727271.72727272731.92727272731.82727272731.72727272730.00E+000.0002.32E+072.23E-020.00E+000.000

0.943.983.923.950.00E+000.0002.25E+082.45E-010.00E+000.0000.92.52.32.32.30.00E+000.0008.45E-027.50E-020.00E+000.0001.3510.950.90.90.00E+000.0004.94E-024.54E-020.00E+000.0002.722.22.120.00E+000.0002.36E+072.27E-020.00E+000.000

8.39E-019.19E+083.88E+005.95E-011.13E+002.49E+007.74E-011.09E+004.15E+008.04E-011.04E+094.06E+00

Velocity dist

-5-5-5

-4.9825-4.9825-4.9825

-4.965-4.965-4.965

-0.7010251288-0.4587654011-0.3349390613

-4.93-4.93-4.93

-0.5950996874-0.4587654011-0.3349390613

-4.895-4.895-4.895

-0.5975463525-0.3731282402-0.2721149576

-0.4891742528-0.3731282402-0.2721149576

-4.8425-4.8425-4.8425

-4.825-4.825-4.825

-0.4374348613-0.2974338665-0.2153129702

-0.3832488115-0.2874910858-0.2092907933

-4.7725-4.7725-4.7725

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-4.685-4.685-4.685

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-0.1128023153-0.016900066-0.025671581

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0.05838845190.03081393490.0131376327

0.03309109480.06217428910.023873334

0.10760183630.07453576410.0351316962

0.14092042050.12416182070.067655039

0.16348489240.08689730370.0463901191

0.19852985120.15920617170.084608915

0.23139985680.14617019220.0766196686

0.27062365470.13854991440.0756511943

0.20426298230.2179850360.1110191728

0.34788264220.16102721110.0938081481

0.3792616940.12051450220.0803135072

0.28331019410.16844533810.0865768329

-4.405-4.405-4.405

0.30255427750.13768930170.0965161186

0.2073233860.15175537720.0844735225

0.24929177580.08969151340.0561177707

0.1849653050.09001146160.0432948195

0.13892079790.09260242070.0737161245

0.11617470210.03517424750.040358912

0.06724200920.03586582980.0199510131

0.0529727567-0.0037852961-0.0030955844

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-4.195-4.195-4.195

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0.0190892030.0322936790.0396212837

0.06735467830.08748402630.0748929602

0.11318244110.11364868360.0679651568

0.10857850470.09641184810.0913032663

0.14145669830.14552007510.1280312905

0.21230802890.15941576170.0993716633

0.22162193930.16326516640.1153713678

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0.25306182560.19434095690.1486519751

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0.30420222360.22656153570.1737897619

0.33985708430.21047666880.1461797596

0.35900054370.22491198490.1682278225

0.31362684070.218013240.1351279405

0.36669215450.20263528650.130622232

0.32738918380.22561825140.1754417439

0.30108795530.22190145770.1709117326

0.28946145260.18483508080.1214974829

0.25365981770.18177390270.1335313518

0.28376059980.17226893220.1384770556

0.25643086820.16391605980.1330423269

0.1406859470.11105694820.0697505313

0.19399317790.11306260790.0973632003

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0.13089693170.10332734720.1018570907

0.07129545730.00575208320.0213670583

0.06642300980.02187226980.0418710774

0.013809252-0.00279854130.034417704

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0.02668634770.04075872090.0424828889

0.09439722210.06090626970.0487198513

0.11678710760.12060416010.0928491282

0.15588355220.14816983590.0987055493

0.24397250910.11107868350.0827925105

0.19554862040.17405717460.1292560045

0.2545410050.26003525560.1553052359

0.27521913240.22211648150.1466533302

0.30066806840.21366489470.1310882269

0.32873532560.23017204670.1544693056

0.31018342970.29296158340.1800367242

0.33735001090.26541214440.1628207298

0.34621865850.25527374640.1555705664

0.36382555510.23854296710.1593366466

0.34603703450.30691587760.1828462693

0.33841918310.27456682790.1558571087

0.34244644320.23538934940.1509934041

0.3073727250.22952354680.1455573421

0.35807090530.23302680460.1340144981

0.30704825160.20629289050.1199933175

0.27177030790.16260088260.1075203018

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-0.4239160932-0.2990769485-0.3436635117

-0.458781871-0.361623542-0.386216927

-0.5674503614-0.4011915952-0.4129821265

-0.5789817077-0.4433405551-0.4397242352

-0.630495416-0.4928331598-0.4822125361

-0.6694500802-0.541953503-0.5139537238

-0.6917582553-0.5677175976-0.524273272

-0.7433016905-0.6150329441-0.5639736457

8 mm slice 53

8 mm slice 55

16 mm slice 56

(r-R)/dp

1.0 mm i.d., 1000 rinc

1.0mm column

Radial distribution functionsRepack column

Duplicate Data from 9/12/05

# r incr =1000# r incr =100# r incr =100

FOV (mm) =3.5FOV (mm) =3.5FOV (mm) =3.5

offset for chartoffset for chart

particle diam (mm) =0.1particle diam (mm) =0.10.07particle diam (mm) =0.10.07

include imaging factor of 1.1 (image dimensions are 1.1 times actual dimensions measured with calipers)include imaging factor of 1.1 (image dimensions are 1.1 times actual dimensions measured with calipers)

exp53exp 55exp 56includeexp53exp 55exp 56includeexp15

r (mm)(r-R)/d8 mm thick slice8mm16mmr (mm)(r-R)/doffset for figure8 mm8mm16mmr (mm)(r-R)/doffset for figure8 mm

0-50-5-4.935.93E+03-0.609.16E+03-0.41105834461.16E+04-0.30602015770-5-4.935.87E+08-0.4355837862

0.00175-4.98250.0159090909-4.8409090909-4.77090909091.08E+04-0.281.27E+04-0.18609992821.43E+04-0.14348238290.0159090909-4.8409090909-4.7709098.21E+08-0.210387784

0.0035-4.9650.0318181818-4.6818181818-4.61181818181.68E+040.131.68E+040.08298422691.72E+040.03102840290.0318181818-4.6818181818-4.6118181.07E+090.0306821307

0.00525-4.94754.42E+03-0.70102512888.37E+03-0.45876540111.10E+04-0.33493906130.0477272727-4.5227272727-4.45272727271.86E+040.251.75E+040.12758178791.78E+040.06730365440.0477272727-4.5227272727-4.4527271.17E+090.1223101496

0.007-4.930.0636363636-4.3636363636-4.29363636361.39E+04-0.071.47E+04-0.05698496671.64E+04-0.02065301440.0636363636-4.3636363636-4.2936361.01E+09-0.0302850166

0.00875-4.91255.98E+03-0.59509968748.37E+03-0.45876540111.10E+04-0.33493906130.0795454545-4.2045454545-4.13454545451.09E+04-0.271.26E+04-0.18773809181.51E+04-0.09733594330.0795454545-4.2045454545-4.1345458.61E+08-0.1727523492

0.0105-4.8950.0954545455-4.0454545455-3.97545454551.31E+04-0.121.46E+04-0.06056357181.64E+04-0.0180335610.0954545455-4.0454545455-3.9754558.95E+08-0.1398370489

0.01225-4.87755.95E+03-0.59754635259.69E+03-0.37312824021.20E+04-0.27211495760.1113636364-3.8863636364-3.81636363641.80E+040.211.79E+040.15104316831.84E+040.1012533690.1113636364-3.8863636364-3.8163641.07E+090.0310333624

0.014-4.867.55E+03-0.48917425289.69E+03-0.37312824021.20E+04-0.27211495760.1272727273-3.7272727273-3.65727272731.95E+040.311.85E+040.19080531151.88E+040.12717176120.1272727273-3.7272727273-3.6572731.17E+090.1247007165

0.01575-4.84250.1431818182-3.5681818182-3.49818181821.59E+040.071.58E+040.01410421091.71E+040.02554040630.1431818182-3.5681818182-3.4981821.10E+090.0571909903

0.0175-4.8250.1590909091-3.4090909091-3.33909090911.16E+04-0.221.28E+04-0.17900726221.53E+04-0.08404901630.1590909091-3.4090909091-3.3390919.62E+08-0.0754020512

0.01925-4.80758.31E+03-0.43743486131.09E+04-0.29743386651.29E+04-0.21531297020.175-3.25-3.181.12E+04-0.251.34E+04-0.14164739891.56E+04-0.0675093780.175-3.25-3.1800009.56E+08-0.0810451061

0.021-4.799.11E+03-0.38324881151.10E+04-0.28749108581.30E+04-0.20929079330.1909090909-3.0909090909-3.02090909091.63E+040.101.69E+040.08734829171.77E+040.05932272440.1909090909-3.0909090909-3.0209091.11E+090.0710266729

0.02275-4.77250.2068181818-2.9318181818-2.86181818182.00E+040.341.94E+040.2481894461.91E+040.14473086050.2068181818-2.9318181818-2.8618181.22E+090.1769319612

0.0245-4.7551.08E+04-0.27243005351.17E+04-0.24162509931.37E+04-0.17055545760.2227272727-2.7727272727-2.70272727271.82E+040.221.76E+040.12999840391.79E+040.070768170.2227272727-2.7727272727-2.7027271.18E+090.1348827824

0.02625-4.73759.01E+03-0.39058881.31E+04-0.15219557591.46E+04-0.11748118710.2386363636-2.6136363636-2.54363636361.28E+04-0.141.37E+04-0.12229209921.55E+04-0.07385039660.2386363636-2.6136363636-2.5436361.01E+09-0.0276653896

0.028-4.721.12E+04-0.23875268761.29E+04-0.16671673331.45E+04-0.12226083210.2545454545-2.4545454545-2.38454545451.02E+04-0.311.21E+04-0.22399736021.45E+04-0.13292670640.2545454545-2.4545454545-2.3845459.14E+08-0.1211373748

0.02975-4.70251.24E+04-0.16405795381.37E+04-0.11622637671.51E+04-0.08741333280.2704545455-2.2954545455-2.22545454551.34E+04-0.101.48E+04-0.05129028781.61E+04-0.03514894280.2704545455-2.2954545455-2.2254551.02E+09-0.018650924

0.0315-4.6850.2863636364-2.1363636364-2.06636363641.90E+040.281.86E+040.19879896111.85E+040.10779924940.2863636364-2.1363636364-2.0663641.24E+090.1944328915

0.03325-4.66751.33E+04-0.09965386641.45E+04-0.05913814041.57E+04-0.04678753280.3022727273-1.9772727273-1.90727272732.02E+040.361.89E+040.21463074971.86E+040.1131370260.3022727273-1.9772727273-1.9072731.31E+090.2545799215

0.035-4.651.31E+04-0.11280231531.52E+04-0.0169000661.61E+04-0.0256715810.3181818182-1.8181818182-1.74818181821.56E+040.051.54E+04-0.01299909741.64E+04-0.01738653780.3181818182-1.8181818182-1.7481821.12E+090.0813358127

0.03675-4.63251.44E+04-0.02822802491.56E+040.00917234591.64E+04-0.00427126860.3340909091-1.6590909091-1.58909090911.03E+04-0.301.18E+04-0.23895625871.45E+04-0.13291820790.3340909091-1.6590909091-1.5890919.06E+08-0.1294606619

0.0385-4.6151.56E+040.05838845191.59E+040.03081393491.67E+040.01313763270.35-1.5-1.431.05E+04-0.291.30E+04-0.16238840681.57E+04-0.06271951530.35-1.5-1.4300009.15E+08-0.1209281353

0.04025-4.59751.53E+040.03309109481.64E+040.06217428911.69E+040.0238733340.3659090909-1.3409090909-1.27090909091.72E+040.161.84E+040.18574746051.92E+040.1499453490.3659090909-1.3409090909-1.2709091.20E+090.1514182581

0.042-4.581.64E+040.10760183631.66E+040.07453576411.71E+040.03513169620.3818181818-1.1818181818-1.11181818182.27E+040.522.25E+040.44567520552.17E+040.2974876990.3818181818-1.1818181818-1.1118181.44E+090.3826089602

0.04375-4.56251.69E+040.14092042051.74E+040.12416182071.76E+040.0676550390.3977272727-1.0227272727-0.95272727272.10E+040.412.06E+040.32706629732.05E+040.22531717470.3977272727-1.0227272727-0.9527271.37E+090.3131390075

0.0455-4.5451.72E+040.16348489241.68E+040.08689730371.73E+040.04639011910.4136363636-0.8636363636-0.79363636361.37E+04-0.081.48E+04-0.04708546211.66E+04-0.0046189770.4136363636-0.8636363636-0.7936361.04E+090.000860697

0.04725-4.52751.77E+040.19852985121.79E+040.15920617171.79E+040.0846089150.4295454545-0.7045454545-0.63454545459.46E+03-0.361.13E+04-0.27517631191.39E+04-0.16560332170.4295454545-0.7045454545-0.6345458.14E+08-0.2178942892

0.049-4.511.82E+040.23139985681.77E+040.14617019221.78E+040.07661966860.4454545455-0.5454545455-0.47545454551.24E+04-0.171.34E+04-0.1391866821.46E+04-0.12422902960.4454545455-0.5454545455-0.4754558.46E+08-0.1863312405

0.05075-4.49251.88E+040.27062365471.76E+040.13854991441.77E+040.07565119430.4613636364-0.3863636364-0.31636363641.71E+040.151.70E+040.0924980091.61E+04-0.0343207580.4613636364-0.3863636364-0.3163648.74E+08-0.1597731566

0.0525-4.4751.78E+040.20426298231.88E+040.2179850361.83E+040.11101917280.4772727273-0.2272727273-0.15727272731.47E+04-0.011.54E+04-0.00969994821.43E+04-0.14309606020.4772727273-0.2272727273-0.1572736.73E+08-0.3534365137

0.05425-4.45751.99E+040.34788264221.79E+040.16102721111.80E+040.09380814810.4931818182-0.06818181820.00181818186.65E+03-0.558.92E+03-0.42637620299.42E+03-0.43603790520.4931818182-0.06818181820.0018183.34E+08-0.6792764999

0.056-4.442.04E+040.3792616941.73E+040.12051450221.78E+040.08031350720.50909090910.09090909090.16090909092.22E+03-0.853.57E+03-0.77063863165.16E+03-0.6910670650.50909090910.09090909090.1609091.42E+08-0.8639151735

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0.06125-4.38751.92E+040.30255427751.76E+040.13768930171.81E+040.09651611860.55681818180.56818181820.63818181821.07E+03-0.937.91E+02-0.94916202956.18E+02-0.96298752930.55681818180.56818181820.6381827.96E+07-0.9235096269

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0.06475-4.35251.85E+040.24929177581.68E+040.08969151341.74E+040.05611777070.58863636360.88636363640.95636363641.07E+03-0.938.31E+02-0.94656402565.39E+02-0.96773506580.58863636360.88636363640.9563647.78E+07-0.9252426149

0.0665-4.3351.75E+040.1849653051.69E+040.09001146161.72E+040.04329481950.60454545451.04545454551.11545454551.02E+03-0.937.72E+02-0.95035630695.33E+02-0.96812381360.60454545451.04545454551.1154557.83E+07-0.9247553516

0.06825-4.31751.68E+040.13892079791.69E+040.09260242071.77E+040.07371612450.62045454551.20454545451.27454545451.00E+03-0.937.44E+02-0.95214031225.20E+02-0.96887792570.62045454551.20454545451.2745457.98E+07-0.9233405608

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