고급전달공정 Advanced Transport Phenomena (ch. 24)

고급전달공정

Advanced Transport Phenomena (ch. 24)

Major: Interdisciplinary program of the integrated biotechnology

Graduate school of bio- & information technology Young-il Lim (N110), Lab. FACS

Young-il Lim (N110), Lab. FACS

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Ch. 24 Other mechanisms for mass transport

- The equation of change for entropy.

- The flux expression for heat and mass - Concentration diffusion and driving forces.

- Application of the generalized Maxwell-Stefan equations - Mass transfer across selectively permeable membranes - Diffusion in porous media

24.5 Mass transfer across selectively permeable membranes

Steel & Koros, 2005

- Slit pores and cylindrical pores.

- Hollow fiber membrane - Spiral-wound membrane

Vu & Koros, 2002

Ch. 24.5 Mass transfer across selectively permeable membranes

 0



 _ t c

- Negligible curvature: <<R

 the membrane surface radius of curvature is much bigger than the membrane thickness (20m). Mass transport id unidi rectional and perpendicular to the membrane surface

- Immobility of membrane: v

=0 (membrane velocity) - Pseudo-steady behavior:

Maxwell-Stefan equations for the multi-component system (Ch. 17)

N ..., , , , ),

N x N

x cD ( )

v v D (

x

x

x

1 1 2 3

1 1





















 



 

 







 

) v v D (

x x RT

z F x RT p

x V )

a ln (

x







 







 







         

1

Maxwell-Stefan equations of membrane for the mobile species

M M

M

D D

x





 1 

The interaction of each species with the membrane M is significant

 0

















( )

RT z F ) p p

RT ( x V

a

ln a

e M

BC: the total potential of each species is continuous dp

p V

V p ^M

e

p e p M

avg

,



  1

upstream downstream

cAM

cAe

cAe

0

z z

Ch. 24.5 Mass transfer across selectively permeable membranes

) v v D (

x x RT

z F x RT p

x V )

a ln (

x _T,p ^N _{ }







 







 





     



1

- Binary system: A (solute), B (solvent)

- Solute A diffuses through a membrane M under the influence of concentration gradient alone.

- dialysis, blood oxygenation, and gas-separations - N

is already known.

Ex. 24.5-1 Concentration diffusion between preexisting bulk phases

Fick’s law

) v v D (

x ) x

v v D (

x x dz

) x ln(

x d _{A B}

AB B A M

A AM

M A p

, T A A

A     



 



 



) v v D (

x x D

v x dz dx x

ln d

ln d

B A AB

B A AM

A A A

p , A T

A  

 



 



  

 1

) N N

( dz x

cD dx

N

 





A A

A

c v

N 

AB AM

AM B

A A A p , A T A AB

AM

AB AM

A D D

) D N N

( dz x

dx x

ln d

ln d D

D

D c D

N  

 

 



 



  



 





 

 

 1

) N N

( dz x

dx x

ln d

ln D d

c

N ^A _{A A B}

p , A T A AB

A   

 



  



 1 for D_AM  D_AB

upstream downstream

cAM

cAe

cAe

0

z z

Ch. 24.5 Mass transfer across selectively permeable membranes

- Binary system: A (solute), B (solvent)

- Solute A diffuses through a membrane M under the influence of concentration gradient alone.

- dialysis, blood oxygenation, and gas-separations - N

is already known.

Ex. 24.5-1 Concentration diffusion between preexisting bulk phases

Membrane permeability

AB AM

AM B

A A A p , A T A AB

AM AB AM

A D D

) D N N

( dz x

dx x

ln d

ln d D

D

D c D

N  

 

 



 



  



 





 

 

 1

) c

c ( P

N





A B

A

A( N N ) N

x

for  



 



 



 

 



 





 

 

x ln d

ln d D

D

D

N D ^{AM AM}

p , A T A AB

AM

AB

A AM

1

c

 K

c

 K

D 

P

p , A T A AB

AM

AB AM eff

,

A dln x

ln d D

D

D

D D

 

 



  

 

 





 

  1

Solute distribution (equilibrium eq.)

Ch. 24.5 Mass transfer across selectively permeable membranes

- Binary system: S (solute) and W (water solvent) at isothermal condition.

- The solute do not pass through membrane (solute rejecting membrane).

- Solvent flow rate?

- Filtrate composition?

- Composite membrane = thin selective layer + thick non-selective backing for mechanical strength

- Consider just the thin layer and intra-membrane behavior

Ex. 24.5-2 Ultra-filtration (macromolecule reject) and reverse osmosis (small molecule reject)

M-S equation

M WM

W SM

S p

cRT D

c N D

c

N

   



 





 



1

upstream downstream

N ..., , , , ),

N x N x cD ( )

v v D (

x

x ^N x ^N 1 123

1 1

















 _{   }



 







 







 

) v v D (

x x RT

z F x RT p

x V )

a ln (

x _T,p ^N _{ }







 







 





      



1

cRT p V ) c

a ln RT RT (

x D

v ) x

v v D (

x x

cRT p V ) c

a ln RT RT (

x D

v ) x

v v D (

x x

W W p , T W W

WM W W S

W SW

W S

S S p , T S S

SM S S W

S SW

W S









 













 



 M-S membrane equation for solute

M-S membrane equation for water M-S membrane equation









 c V , and dG RTd ln a , G dx 0

note

0

z z 

Ch. 24.5 Mass transfer across selectively permeable membranes

Ex. 24.5-2 Ultra-filtration (macromolecule reject) and reverse osmosis (small molecule reject)

M WM

W SM

S p

cRT D

c N D

c

N

    

 





 



upstream downstream

1

0











 _ ^ _



 ( )

RT z F ) p p RT ( x V a

lna ^,avg _{M e M e}

e M

Se SM S

e

M

a

ln a V p RT p  

) (

) D (

c N D

c N

p cRT _{e e M M}

WM W SM

S

ext

    

   



 





 

 





0

z z  Osmotic pressure is seldom

known but it is small

cRT p V ) c

a ln RT RT (

x D

v ) x

v v

D ( x

x

p , T S S

SM S S W

S SW

W

S

    

 



dz ) dx x ln ( ln

D c D

c N x D

c D

c

N x

S S WM

S SW

S W

SM W SW

W

S





 



 



 







 

 



 







  1

- High flow rate

M-S membrane equation for solute

Ch. 24.6 Mass transport in porous media

- Applications: catalysis, adsorption, particle technology, membrane, …

- Mass transport mechanism: 1) ordinary diffusion by MS equation, 2) Knudsen diffusion, 3) viscous flow intr a-membrane, 4) surface diffusion (creeping of adsorbed molecules along the surfaces of pores0, 5) therma l transpiration (thermal analog of viscous slip, 6) thermal diffusion

- Limiting situation in this discussion:

- free-molecule flow of gases: no significant interaction between the intrapore species

- continuum flow of gases or liquids: Pore size is big compared to molecules, the intrapore fluid can be de scribed by the generalized hydrodynamic theory, the M-S equation can be used for multicomponent syste m

- : porosity of membrane (V_pore/Vmembrane)

- : tortuosity, about 10 in carbon molecular sieve

Free molecule transport: Knudsen diffusion

dz dc M

a RT k

N

A geo

A

   8

Molecule transport inside porous media by effective Knudsen diffusion dz

dc M

RT

N a

A

A

 

 

 3 2

8

Bae et al., 2009

Ch. 24.6 Mass transport in porous media

- Membrane cross-sectional area (S) and length (L) - Constant temperature (isothermal condition) - The ideal gas law holds throughout the system

- Develop an expression for the total pressure in each reservoir as a function of time.

Ex. 24.6-1 Knudsen diffusion

2. Macroscopic mass balance

) p p

( dt K

dp RT

V

dt W dp RT

V dt

V dc

A A

A A

A A

A

2 1

2

2 2









H₂ p₀

N₂ p₀

dz dc M

RT

N a

A

A

 

 

 3 2

8

) p p

( K

L

) p p

( RT M

RT S a

W

A A

A

A A

A A

2 1

2

1

2 3 8





 

 



 

1. Molecule transport rate by effective Knudsen diffusion (from reservoir 1 to 2)

) e

p ( p

) e

p ( p

V t RTK . H

V t RTK N

N N

48 7

0 2

2

0 2

2 1 1 2 1 1













1

0

0 1

0 2

1









N H

A A

p , p p

: IC

p p

p

Final Exam: Report by June 10, 2010

A porous carbon membrane (L=20 m) is used for the separation of the methane and carbon dioxide mixture. Since the molec ule size methane is bigger than that of carbon dioxide, methane is rejected from the membrane. The upstream (natural g as) is methane-rich but the down stream is carbon dioxide-rich.

The carbon membrane is composed of slit pores with the mean pore width (H) of 0.68 nm and with accessible pore width (Ha) of 0.34nm (H = Ha + CH4)

A student wants to calculate the flux (J) of methane in this membrane for the methane pure system. See the following figure. T he upstream is kept at 20 bar and the downstream is vacuumed to 0.01 bar. Temperature is constant (T=25 ^oC).

1) Calculate the Knudsen diffusion-based flux (J) using Eq. (24.6-1).

2) Since the pore width is very small (0.68nm), the solute is strongly influenced by the membrane wall. Thus, this transport is called the confined molecular transport. At this case, the flux is often obtained by

va cu um

CH₄

p₂ = 0.01 bar CH₄

p₁ = 20 bar

L=20m

 = 0.3

 = 10

a = 0.34nm



 ^{  }





 





1 2

1

p p

ads p A

p ads A

ads

A dp

p L

p D ln L d

D D RT

J

76 11 08

1 017

1 , k 0. ,k . , k .

p k

p k p k

k _{l b m}

b b m l

ads   

 





2) continued: Calculate the molar flux based on the confined diffusion. DA is obtained from EMD (equilibrium

molecular dynamics) in the confined slit pore, using the provided LJ-based EMD code (Nicholson, 2010).

3) Compare experimental value carried out in a CMS at T=25^oC and P=20 bar. Analyze the simulation results and

experimental data. Find reasonable answer for the

difference between the theoretical and experimental values.

Lenard-Jones (LJ) parameters for CH4

CH4 = 0.381 nm

CH4/kB = 148.1 K

References

1. Lim et al., J. Membr. Sci., 2010.

2. Allen & Tildesley, Computer simulation of liquids, 1989.

3. D. Nicholson, SIMAPOS v7.7 user manual, 2009.

10^-2 10^-1 10⁰ 10¹ 0

2 4 6 8 10 12 14

(a) H=0.65nm

Fugacity (f , bar) Adsorption density (ads, mmol/cm3)

_ads^GCMC at T=298K

_ads^GCMC at T=308K

_ads^GCMC at T=318K

_ads^model at T=298K

_ads^model at T=308K

_ads^model at T=318K

10^-2 10^-1 10⁰ 10¹

0 2 4 6 8 10 12 14

(b) H=0.68nm

Fugacity (f , bar) Adsorption density (ads, mmol/cm3)

10^-2 10^-1 10⁰ 10¹

0 2 4 6 8 10 12 14

(c) H=0.72nm

Fugacity (f , bar) Adsorption density (ads, mmol/cm3)

10^-2 10^-1 10⁰ 10¹

0 2 4 6 8 10 12 14

(d) H=0.75nm

Fugacity (f , bar) Adsorption density (ads, mmol/cm3)

Final Exam: Report by June 10, 2010

76 11 08

1 017

1 , k 0. , k . , k .

p k

p k p k

k _{l b m}

b b m l

ads   

 





Lenard-Jones (LJ) parameters for CH4

CH4 = 0.381 nm

CH4/kB = 148.1 K