고급전달공정 Advanced Transport Phenomena (ch. 20)

고급전달공정

Advanced Transport Phenomena (ch. 20)

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. 20 Concentration distributions with more than one independent variable

- Ordinary differential equation (ODE) and partial differential equation (PDE).

- steady-state and unsteady-state

- 1D, 2D, 3D and time-space PDE.

- Laplace transformation (applicable to linear ODE or PDE)

20.1 Time-dependent diffusion (unsteady-state model)

(Four examples with reactions and without reaction)

Ex. 20.1-1 Unsteady-state evaporation of a liquid: Arnold problem

- A volatile liquid A evaporates into pure B in a tube of infinite length.

- T, P, and diffusivity are constant.

- A and B form an ideal gas mixture.

- B is insoluble in liquid A, and no radial difference in molar average velocity - The molar density (total density), c, is constant in the gas phase.

Ch. 20.1 Time-dependent diffusion (unsteady-state model)

Solution procedure:

1. draw a picture to summary the problem

2. define the symbols or use the symbols given in the problem 3. develop the model with assumptions

4. solve the model with BC (and/or IC) 5. analyze the solution and plot the results

Ex. 20.1-1 Unsteady-state evaporation of a liquid

- A volatile liquid A evaporates into pure B in a tube of infinite length.

- T, P, and diffusivity are constant.

- A and B form an ideal gas mixture.

- B is insoluble in liquid A, and no radial difference in molar average velocity - The molar density (total density), c, is constant in the gas phase.

Ch. 20.1 Time-dependent diffusion (unsteady-state model)

Ex. 20.1-1 Unsteady-state evaporation of a liquid

- A volatile liquid A evaporates into pure B in a tube of infinite length.

- T, P, and diffusivity are constant.

- A and B form an ideal gas mixture.

- B is insoluble in liquid A, and no radial difference in molar average velocity - The molar density (total density), c, is constant in the gas phase.

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

 0 dz dN_A

Ch. 20.1 Time-dependent diffusion (unsteady-state model)

Ex. 20.1-1 Unsteady-state evaporation of a liquid

N dz dN_Az

0



water air

Water evaporation into air

B A A B A AB A

A cv* x cD x x R x R

t

c x      



 ₂

2 2

z D x

z v x

t

x _A

AB

* A z A



 







 



 0



 z v

continuity of

equation

*

z v v (t )

continuity of

equation

* z

*

z  0

0 0

0 0

1  _



 

 

z A A

AB Bz

* Az

z z

x x

D c

N v N

0 2

0 1

0 0

0















) z

( x : BC

x ) z

( x : BC

) t

( x : IC

A

A A

A

Ch. 20.1 Time-dependent diffusion (unsteady-state model)

Ex. 20.1-1 Unsteady-state evaporation of a liquid

N dz dN_Az

0



water air

Water evaporation into air

B A A B A AB A

A cv* x cD x x R x R

t

c x      



 ₂

2 2

z D x

z v x

t

x _A

AB

* A z A



 







 



Ch. 20.1 Time-dependent diffusion (unsteady-state model)

Ex. 20.1-2 Gas absorption with rapid reaction

B A A B A AB A

A cv* x cD x x R x R

t

c x      



 ₂

) t ( z z z for

D c t

c

R A

AS

A  



 

 

 ₂ 0

2













 



 



















B

B BS A

AS R

R R A

A A

B B

c ) z

( c : BC

z D c

b z

D c ), a

t ( z z : BC

)) t ( z z ( c : BC

)) t ( z z ( c : BC

c ) z

( c : BC

c ) t

( c : IC

B B

5

1 4 1

0 3

0 2

0 1

0

0

- Gas A is absorbed by a stationary liquid solvent (S), containing solute B. Instantaneous reaction.

- Fick’s second law is adequate for this diffusion problem, because ?





 

 

 

 z (t ) z

z D c

t c

R B BS

B

2 2

Gas-liquid interface

z z_R

B+S

A A

C bB

aA  

Ch. 20.1 Time-dependent diffusion (unsteady-state model)

Ex. 20.1-2 Gas absorption with rapid reaction

B A A B A AB A

A cv* x cD x x R x R

t

c x      



 ₂

) t ( z z z for

D c t

c

R A

AS

A  



 

 

 ₂ 0

2

- Gas A is absorbed by a stationary liquid solvent (S), containing solute B - Fick’s second law is used.





 

 

 

 z (t ) z

z D c

t c

R B BS

B

2 2

Gas-liquid interface

z z_R

B+S

A A

C bB

aA  

Ch. 20.1 Time-dependent diffusion (unsteady-state model)

B A A B A AB

A

A cv* x cD x x R x R

t

c x       



 ₂

Ex. 20.1-3 Unsteady diffusion with first-order homogeneous reaction

- A diffuses in a liquid medium B and reacts with it irreversibly according to a

first-order reaction.

A  B  C

A A

AB A

A v w D w kw

t

w      



 ₂

) z , y , x ( w w

: boundary at

BC

) z , y , x ( w w

, t

at IC

A A

AI A

0

0







 ^{ } _ ^ _ ^







A

f ( x , y , z t, ) d t

) t t k exp(

) kt exp(

g )

z , y , x , t (

w

Here f is the solution of PDE with k=0, and w_AI(t=0)=0, whereas g is the solution with k=0 and wA0=0.

Ch. 20.1 Time-dependent diffusion (unsteady-state model)

B A A B A AB

A

A cv* x cD x x R x R

t

c x       



 ₂

Ex. 20.1-4 Influence of changing interfacial area on mass transfer at an interface

- One dimensional Fick’s second law:

2 2

z D c

t

c _A

A AB



 





t D erf z

c c

A AB A

1 4

0





Ch. 20.2 Steady-state transport in binary boundary layers

c t v

c z

v c y v c x v c t c Dt

Dc

z y

x  



 



 



 



 



 

2D steady equation of change - , , k, c_p and D_AB are constan

t

- Viscous dissipation is neglect ed

- External velocity at the bound ary layer, v_e

- Buoyant force leading to free convection

- Equation of continuity 0

0 



 



 





 y

v x

; v

v ^{x y}

- Equation of motion

) w w

( g ) T T ( y g

v dx

v dv y

v v x

v_x v^x _y ^x _e ^{e x}  _x  _  _x _A  _A









 



 







 



  ² ₂

- Equation of energy

- Equation of continuity of A

g p Dt v

Dv    

 ²

Dt T

DT ₂





 _A

B B A

A y

x

p )r

M H M

( H y k T y v T x v T

C  



 



 







 



  ²₂

A AB

A D

Dt

D  ₂

A A AB A

y A

x r

y D w

y v w x

v w 



 





 







 



  ² ₂

Ex. 20.2-1 Diffusion and chemical reaction in isothermal Laminar flow along a soluble flat plate

1 6

1 6

1

2 0 0 0

2



 



 



 

 





 



  

 





 

n v kc

dx c d c

D dx v

d v

c n A A

c A AB

Boundary layer balance

Ch. 20.2 Steady-state transport in binary boundary layers

x y

Ex. 20.2-2 Forced convection from a flat plate at high mass-transfer rates





 

 ^y _x

y v dy

) x x ( v

v 0 0

Ch. 20.2 Steady-state transport in binary boundary layers

D Sc w ,

w w w

Pr T ,

T T T v , v

AB w

A A

A w A

T T

v x v

 



 

 



 

 

 

 



 

 













0 0 0 0

1

2 2 0

0

y v

dy y x

v ) x ( v x

y v

v 







 









 



 



 

 



 







Ex. 20.2-2 Forced convection from a flat plate at high mass-transfer rates

Ch. 20.2 Steady-state transport in binary boundary layers

D Sc w ,

w w w

Pr T ,

T T T v , v

AB w A A

A A w

T T

v x v

 



 

 



 

 

 

 



 















0 0 0 0

1

 

 

 

 



















 







0 0

0 0

d ) d ) K , ( f exp(

d ) d ) K , ( f ) exp(

K , , (

2 2 0

0

y v

dy y x

v ) x ( v x

y v

v 







 









 



 



 

 



 



 

 

 









x v v

) x ( K v

x y v

where

2 2 1

0

Ch. 20.3 Steady-state boundary layer theory for flow around objects

Ch. 20.4 Boundary layer mass transport with complex interfacial motion

Ch. 20.5 Taylor dispersion in Laminar tube flow



 











 











 















 



 





 



2 2 2

1 1

z w r

r w r D r

z w R

v r t

w _{A A}

AB A

max , z A

0 2

1

0

 





 andr R r

A

r : w BC and BC

Cross-sectional average mass fraction by Taylor



 ^R _A

A w rdr

w R

2 0

2

Moving axial coordinate z  z v_z t

 

AB z A z

A D

v K R

Kt ), t v exp( z

Kt R

m

48 2 4

2 2 2

2  

 

 



This convective dispersion method is the best one for reasonably quick measurements of liquid

diffusivities

How to calculate diffusivity from experimental data?

20.6 Questions for discussion

1. What experimental difficulties might be encountered in using the system in Ex. 20.1-1 to measure gas-phase diffusivity?

2. What problems do you foresee in using Taylor dispersion technique for measuring liquid phase diffusivities?

3. The following equation is the solution of time-space PDE with convection and diffusion.

4. Laplace transformation is useful to solve the PDE?

0

2 2

2   

dZ )dX Z

dZ ( X d

 





 

 D t

Sx V

c t D

c _AB

A Fick

A A

A AB 4

0 2

0 0

2 4

A AB

A A

A AB

* A A

x t Sx D

V c

z D v c t

c  

 





 

 





Correction factor caused by Non-zero molar average velocity For high volatility liquids,   . How can we do?

 

AB z A z

A D

v K R

Kt ), t v exp( z

Kt R

m

48 2 4

2 2 2

2  

 

 



AB max

, z

p

D .

R v

L

2 2

8

 3

 

0 0

0

0 21

1 1 4

1

 







 







 

Z A

A A AB

A

dZ dX x and x

t D Z z

) , ( erf

) Z ( Z erf

x x