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aggregated

porous

fibrousmembrane

catalyst

scale of

heterogeneity

scale of

the phase

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Species: 1 a small one (water)

2 a large one (protein)

The matrix M is solid phase that

can resist stresses, and

can transmit support forces

membrane support

Our Species

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Driving Forces

dz

dp

Vdz

d

dz

ad

RTF 222

2

ln

Fz

dz

dpV

dz

dFxFxF Fz2211

dz

dpV

dz

d

dz

adRTF

111

1

ln

Fz

on the components separately

on the mixture as a whole

internal forces cancel

N

mol of i

N

mol of mixture

2211 zzz xx with 2211 VxVxV

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ui

vi

ui

vi

w u vi i i

diffusive velocity

slipping past the matrix

viscous velocity

transport by convection

whole (overall) velocity

1

2

3

Three Velocities

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F vV

viscous friction coefficient

VcB

hydraulic permeability

B dpore

2

32

B

dsphere

2 2

2170 1

B

dfibre

2 2

280 1

Viscous Flow and Friction

void fraction

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complications in channels of colloidal dimensions:

surface adsorption

changes diameters

particles cause

increased friction

Complications

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Viscous Selectivity

viscous velocity of a centerline

species higher than average

v

v

vi

vi

i 1

viscous velocity of adsorbing

species lower than average i 1

v vi i

v v vi i i i i i 1volume balance

viscous selectivity

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Transport Equations

1,1212,121 uuuxF M

2,2122,112 uuuxF M

F vV v v1 1 v v2 2

Structuredmodel

w u v1 1 1

w u v2 2 2

diffusive

viscous

whole

Non-structuredmodel

1,1212,121

wwwxFM

2,2121,212 wwwxF M

whole velocities in one go

(ksi)

(zeta)

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Mason Equations

MMV

MMM

xx

x

,222,111

21,22,12,2,11

2,12,1

MMV

MMM

xxx

,222,111

12,12,11,2,122,11,2

MMV

MMMV

MMxx

xxxx

.222,111

21,112,121,2,12,1

2

1

,2,2

MMV

MMMV

MMxx

xxxx

,222,111

12,222,121,1,22,1

2

2

,1,1

Approaches identical when:

Non-structured Structured

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w1

dp

v1

u1 Structured Non-structured

Vdp

dzuM1 1,

Vdpdz

vV 1

Vdp

dzwM1 1,

w u v1 1 1 11

1

1 1

, ,M V M

Single Liquid Permeant

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Permeant - Matrix Coefficient

a rough estimate

d1 dp

u1

u d

at wall

11

2

d dp

dz

pore

4

Vdp

dz

V

d du

pore

at wall4 2

1

1

pore

wallat

pore

wallat

Md

d

dd

V 21

1

,1 48

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Friction Coefficients vs Pore Diameter

permeant-permeant

permeant-matrix

viscous V pored 2

1

,1

poreM d0

2,1 pored 2,1

M,1

Vln

lndp

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Wide Pores

write N- equations per mol of mixture:

vxvvxxFx M,112,12111

vxvvxxFx M,221,22122

summing the equations:

vvxxF VMM ,22,11

identical to viscous flow through a tube

Wide pores: w vw v

1

2

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Two similar Permeants

1 2 1

MMM ,2,1

MV

M

2

2,12,1

MV

VMM

,1

MV

VMM

,2

2,12,1

viscous flow makes the N-coefficientlarger than the S-coefficient

MM ,1,1

Permeants similar, with slipping

MV

M

2

1,21,2

viscous flow makes the N-coefficientsmaller than the S-coefficient

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Dilute Solute, no Slipping

M,1M,22,12,1

M,221,21,2

1,M V

22,1,2,2 1 MM

Protein (2) dilute, no slipping of water (1)

w v v1 1