고급전달공정 Advanced Transport Phenomena (ch. 19)

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고급전달공정

Advanced Transport Phenomena (ch. 19)

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. 19 Equation of change for multicomponent systems

- Mass balance over an arbitrary differential fluid element  Equation of continuity in a multicomponent mixture.

- momentum/conduction/mass flux  diffusion equations (²v, ²T, ²cA) - Equation of change = equation of motion, equation of energy and equation of

continuity (=conservation laws)



 















 





j v

n

r ) n t (

- The law of conservation of mass in a finite volume of x, y, and z

 The equation of continuity for species.

19.1 the equations of continuity for a multicomponent mixture

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

 















 





j v

n

r ) n t (

- The equation of continuity for each species.

19.1 the equations of continuity for a multicomponent mixture

- equation of continuity for the mixture = equation of continuity.



   



   





 v j r

t

^

v



   





 0    v

What assumption is used for this equation

of continuity?

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- The equation of continuity for each species in mass.

19.1 the equations of continuity for a multicomponent mixture



   



   





 v j r

t v

t    







- The equation of continuity for each species in molar quantity.



  



   



 c v j R

t

c

_* _*



 







 



_* ^N

R t cv

c

1

 v





 0



 







 cv

^* ^N

R

1

0



     



   





  v j r

t



   



   



 c x v j R

t

c x

^* ^*



     



   





  v j r

t



 



   



    



_* _* ^N

R x

R j

x t cv

c x

1

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- Binary systems with constant mass diffusivity (DAB)

19.1 the equations of continuity for a multicomponent mixture



     



   





  v j r

t



 



   



    



_* _* ^N

R x

R j

x t cv

c x

1

A A

AB

A

v

A

D r

t           





 

²

- Binary systems with constant mole diffusivity (cD_AB)

B A A

B A

AB A

A

cv

*

x cD x x R x R

t

c x        



₂

- Binary systems with zero velocity and without reaction (v* = 0, RA=0, RB=0)

A

cD

AB

x

t

c x   

²



A

D

AB

c

t

c   

₂



Fick’s second law of di ffusion

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- Three equations of change = three conservation laws

19.2 Summary of the multicomponent equations of change

e flux

energy

flux momentum

n flux

mass







 _

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- Three equations of change = three conservation laws

19.3 Summary of the multicomponent fluxes





 





















































M j T H

k q

flux molecular energy

) v )(

( ] ) v ( v [ flux

molecular momentum

D j

flux molecular

mass

N

t A

AB A

1

3 2

- Diffusion flux - Viscous flux

- Conduction heat flux - Diffusion thermo effect

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Ex. 19.4.1: simultaneous heat and mass transport

19.4 Use of the equations of change for mixtures

dy e de

: balance energy

dy j dN

: balance mass

y y

Ay A









 0 0

(a) Mole fraction profile, xA(y)?

(b) Temperature profile, T(y)?

Assumption: steady-state, no reaction, no convectio n, ideal gas of A, constant P, no radial heat transf er, constant physical properties.



















 



 









 



 







 



 



 



k N C

k y N C

A , p A

Ay y

A A Ay AB

/ y

A A A

A A

A Ay AB

A , Ay P

e e T

T T )), T

T T ( C H

( dy N

k dT e

x ln x

N cD x ,

x x

, x dy dx x

N cD

1 1

1 1 1

1 1

0 0 0

0

0 0

0

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Ex. 19.4.2: Concentration profile in a tubular reactor

19.4 Use of the equations of change for mixtures

dr r dv dr

d : r

balance momentum

r r c r D r

z v c , n :

balance mass

z r

A AS

A z A

0 1 0 1













 



 



(a) Mole concentration profile, cA(y)?

Assumption: steady-state, isothermal, catalytic r eaction, parabolic velocity, diffusion of A, c onstant P, no radial heat transfer, ignoring pr oduct A & B.



 

 



 



 



 



 







 







0 0 0

2

3 3

1 1

1

d e

d e c

, c r r c r D r

z ) c R ( r

v

R ) ( r

v ) r ( v

A A AS A

max A , z

max , z z

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Ex. 19.4.3: Catalytic oxidation of CO

19.4 Use of the equations of change for mixtures

z z

z

A iz

N N

N

dz , dN j :

balance mass

3 2

1 2

1

0 0









(a) Mole concentration profile, cA(y)?

Assumption: steady-state, isothermal, catalytic r eaction, parabolic velocity, diffusion of A, c onstant P, no radial heat transfer, ignoring pr oduct A & B.

) x x cD (

) N x x cD (

N dz

dx

) x cD (

N dz

dx

z z

z

3 1 13

3 3 1 12

3 1

3 13

3 3

2 2 3

2 1

2 1 1















 







 

 ^

2 2 2

30 3 3 13

x ln x N _z cD

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19.5 Dimensional analysis of the equations of change for binary mixtures

- Equation of continuity

 0





 v

- Equation of motion

g p

Dt v

Dv    

 ²

- Equation of energy Dt T

DT  ₂

- Equation of continuity of A

A A DAB

Dt

D  ₂

- Dimensional analysis: dimensionless quantity, dimensionless group

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19.5 Questions for discussion

1. Equation of change for reacting mixtures?

2. Flux equations for reacting mixtures?

3. Under what conditions is divergence of v (v) zero?

4. Mass and molar based equations of continuity (mass balance) are physically equivalent.

For what kinds of problems is there a preference for one form over the other?

5. Interpret physically each term in the equations in Table 19.2.3?

z v y

v x

v v ^x ^y ^z

i i

i



 



 



 



 







 3 1

z v v y v v x v v t v Dt

Dv _x

x z x y

x x x



 



 



 



 

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19.5 Questions for discussion

1. Gradient p = p 2. Divergence v = v

3. Substantial time derivative (p 83) of c = Dc/Dt

c t v

c z

v c y

v c x

v c t

c Dt

Dc

z y

x

  



 



 



 



 



 

z v y

v x

v v

^x ^y ^z



 



 



 



