고급전달공정 Advanced Transport Phenomena (ch. 18)

고급전달공정

Advanced Transport Phenomena (ch. 18)

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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Announcement

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^th

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2010. 04. 21-23 (Wed. - Fri.)

2010. 04. 22 (Thu)  4. 29. PM 6-9 (communication

& discussion)

- FOA (fundamentals of Adsorption) conference, Japan 2010. 05. 22 – 28 (Sun. – Fri.)

2010. 05. 27 (Thu)  5. 20. PM 6-9

Ch. 18 Concentration distributions in steady-state diffusion problems

- Shell mass balance: first-order differential equation (convection=mass flux) + second-order differential equation (diffusion=relation between mass flux and concentration gradient) .

- Total molar flux (N

A

)

dz cD dx

j

^*_A

 

_AB ^A

















 j  c v  c ( v  v )  c v  c v

N

^{* * * *}

 



_{Az Bz}



A B

A A Az B

A

B A A A B

A A B

B A

A A

* A

N N

x v

c c N c

x

v x c

v x c x v

x v

x c v

c



 



 



 



 

 



 







) N N

( dz x

cD dx

N

_A

 

_AB ^A



_{A Az}



_Bz

- Homogeneous and heterogeneous reaction

- Homogeneous system: Source term (reaction) + convection term (bulk mass flux) + diffusion term (molecular mass flux)

18.0 diffusion problems in non-reacting and reacting systems

n A n

A

k c

R 

surface n

A surface n

Az

k c

N 

Ch. 18 Concentration distributions in steady-state diffusion problems

- Heterogeneous system: boundary condition at the surface (reaction) + convection term (bulk mass flux) + diffusion term (molecular mass flux)

Ch. 18 Concentration distributions in steady-state diffusion problems

- unsteady-state/steady-state mass balance on an infinitesimally small thickness

18.1 Shell mass balances: boundary conditions

source outlet

inlet

source outlet

inlet on

accumulati











 0

- IC (initial condition):

a. unsteady-state mass balance b. steady-state mass balance - BC (boundary condition):

a. unsteady-state mass balance b. steady-state mass balance - Heterogeneous reaction:

a. bulk mass transfer at the surface (between two phases) b. chemical reaction rate on the surface

0

0 c A, A,bulk A

c k N

) c

c ( k N







18.2 Diffusion through a stagnant gas film

) reaction no

( outlet inlet

source outlet

inlet









 0 0

water air

Water evaporation into air

dz dx x

N cD

N dz x

cD dx N

N , stationary is

air if

) N N

( dz x

cD dx N

A A

AB A

Az A A

AB A

Bz

Bz Az

A A

AB A

 



















1

0

1

1

1

0 dz C dx x

dz dN

A A

Az

 





18.2 Diffusion through a stagnant gas film

water air

Water evaporation into air

2 1

1

1 1

1

C z

C )

x ln(

dz C dx x

A A A









 

01 0 2

1 1

2 1

2 1

. x

x : BC

x x

: BC

z A A z

z A A z













Experimentally measured

 

^{2 1}

1 1

2 1

1 2 1 2 1

1 1 1

1 1 1 1

1

z z

z z A A

A

z z

z z

A A A

A

x x x

) z ( x

x x x

x









 

 







 





 

 







 





1 1

1

1 1 ^B

AB z

z B B

AB z

z A A

z AB ,

A lnx

z cD dz

dx x

cD dz

dx x

N cD 

 

 









18.3 Diffusion with a heterogeneous chemical reaction

Solid-catalyzed dimerization

B A  2

- Each catalyst particle is surrounded by a stagnant gas f ilm through which A has to diffuse to reach the catal yst surface

- Effective gas-film thickness () and main stream conc entrations (xA0, xB0) are known.

- Isothermal and no heat release from reaction - NBz=-0.5NAz (stoichiometric reaction)

dz dx x

. N cD

N x dz .

cD dx N

) N N

( dz x

cD dx N

A A

Az AB

Az A A

AB Az

Bz Az

A A AB

Az

5 0 1

5 0

 

















5 0 0 1

0

 



 









dz dx x

. cD dz

d dz dN

A A

AB Az

18.3 Diffusion with a heterogeneous chemical reaction

Solid-catalyzed dimerization

B A  2

0 2

1

_{0 0}









 A z

z A A

x : BC

x x

: BC

Experimentally measured

5 0 0

1  



 





 dz

dx x

. cD dz

d

_A

A AB









 . x

_A

) ( . x

_A

)

^z/

( 1 0 5 1 0 5

₀ ¹

dz y ln cD d

dz dy y

N cD

x . y

dz , dx x

. N cD

AB AB

Az

A A A

Az AB

2 2

5 0 5 1

0 1







 







 









 



0 1

5 0 1

5 0 1 2 2

A A AB

AB

Az . x

x ln .

cD dz

y ln cD d

N

18.4 Diffusion with a homogeneous chemical reaction

Absorption of CO₂ by a NaOH solution

with a irriversible homogeneous 1^st-order reaction

AB B

A  

dz D dc

dz cD dx

N

) N N

( dz x

cD dx N

AB A AB A

Az

Bz Az

A A AB

Az

















source outlet

inlet  

 0

z S c k S N

S

N

_{Az z}



_{Az z z}



_A





_{ 1}

0

A Az

k c dz

dN

0  

1

What assumptions are

imposed?

A A

AB

k c

dz c

D d

_{2 1}

2

0  

0 0

2

1 _{0 0}











dz / dc or , N

: BC

c c

: BC

L A Az z

z A A

AB AB

A A

D / L k cosh

L ) ( z

D / L k cosh

c c

2 1 2 1

0

1 



 



 



2

x

x

e

cosh e









18.5 Diffusion into a falling liquid film (gas absorption)

Forced-convection mass transfer

Absorption of gas A (O₂) by liquid B (water)











 



 





 



2

1 x

v ) x (

v_{z max}

dx D dc

) N N

( dx x

cD dx N

) x ( v c ) N N

( dz x

cD dx N

A AB Bx

Ax A

A AB Ax

z A Bz

Az A

A AB Az























outlet inlet 

 0

dx dN dz

dN

_{Az Ax}



 0

What assumptions are

imposed?

2 2

x D c

z

v

_z

c

^A _AB ^A



 





0 3

2

0 1

0 0 0















A x x A A A z

dx / dc : BC

c c

: BC

c : BC

A A

v / z D erf x

c c

1 4

0



2



2 2

1 x

D c z

c

v

_max

x

^A _AB ^A



 





 





 



 



 





 

18.7 Diffusion and chemical reaction inside a porous catalyst

Effective diffusion in a porous catalyst pellet

dr D dc N

) N N

( dr x

cD dx N

A A

Ar

Br Ar

A A

A Ar













source outlet

inlet  

 0

R AR A r

r A A

c c

: BC

c c

: BC









2

1

₀

What assumptions are

imposed?

A Ar

) r k ac N

r dr (

d

1 2

2

 

 

 ^{k a / D R} 

sinh

r D

/ a k sinh r

R c

c

A A AR

A



 

1 1

r r ac k ) r r ( N

r

N_{Ar r}   _{Ar r r}    _A  

 4 ² _ 4 ² ₁ 4 ²

0

A A

A

) k ac

dr r dc dr (

d

D r 1

₂ ²



₁











 



















D R a coth k

D R a ( k

c RD

dr D dc R N

R W

A A

AR A

R r A A Ar

AR

1 1

2 2

1 4

4 4

water air

Water evaporation into air (N₂ and O₂)

18.8 Diffusion in a three-component gas system

1=water, 2=Nitrogen, 3=Oxygen

- tertiary system

- Maxwell-Stefan equation for the mixture - Mass conservation of each component - Air is stationary

15 0 75

0 1

0 2

45 0 0

1

3 2

1 0 1

. x

, . x

, . x

, L z at : BC

) pressure vapor

( . x

, z

at : BC

L , L

, L

, ,













3 2

1

0, , , and dz

dN _z









3 2

1 0

3 1

2 1

and , , dz ,

dN

x x

x

z   









2 21 1 1

2 2 1 21 2

3 2

3 2 2 3 23 1

2 2 1 21 2

2 2 3

1 2

2

1

0 1 1

1

cD x ) N N x N x cD ( dz

dx

N N

, stationary is

air if

) N x N x cD ( ) N x N x cD ( dz

dx

) N x N x cD ( dz

: dx equation MS

z z z

z z

z z

z z

z z



























 _{ }



 



water air

Water evaporation into air (N₂ and O₂)

18.8 Diffusion in a three-component gas system

1=water, 2=Nitrogen, 3=Oxygen

3 13 1 3

2 12 1 2

3 2

1 1

cD x N dz

dx

cD x N dz

dx

x x

x

z z













 



 





 



 





13 1

3 12

1 2

1 1

cD

) z L ( exp N

cD x

) z L ( exp N

x

x _L ^z _L ^z

The boundary condition at z=0



 



 



 



 



13 1 3

12 1 2

10 1

cD L exp N

cD x L exp N

x

x _L ^z _L ^z

Transcendental equation for N

_1z

18.9 Questions for discussion

1. What arguments are used for N

_B

=0?

2. How to measure the diffusivity by means of the examples?

3. Distinguish between homogeneous and heterogeneous reaction

4. Diffusion controlled reaction (18.3)?

5. Local rate of chemical reaction in Eq. 18.3-9.