Fundamentals of Mass Transfer

When a system contains two or more components whose concentrations vary from point to point, there is natural tendency for mass to be transferred, minimizing the concentration differences within the system.

Mechanisms of mass transfer

Molecular Transfer Convective Transfer

Mass transfer

~ random molecular motion in quiescent fluid

Molecular transfer Convective transfer

~ transferred from surface into a moving fluid Analogy between heat and mass transfer

Heat transfer Heat conduction Heat convection

Fundamentals of Mass Transfer

Fick’s law

dz j

_Az

= −  D

_AB

d 

^A

In vector form

Mass diffusivity

A A AB

j = −  D  

D

AB

Prandtl number



=  Pr

Schmidt number Lewis number

AB

AB

D

D

= 



=  Sc

AB AB

Cp

e k

D D

= 

=  L

dz C dx

J

_Az

= − D

_AB ^A

A

A

C

AB

x

J = − D 

Molecular Mass Transfer(Molecular Diffusion)

"Transport Phenomena" 2nd ed., R.B. Bird, W.E. Stewart, E.N. Lightfoot

Mass Concentration

the mass of A per unit volume of mixture

( ) 

_A

a) mass concentration





=



i i

Molecule of species A

dV in multicomponent muxture

- Total mass concentration(density)



= 



= 





A

i i A A

- mass fraction



 =

i i

1

Molar Concentration

~ the number of moles of A present per unit volume of mixture

( ) C

_A

b) molar concentration

=



i

C

i

- Total molar concentration

C

A A

A

M

C = 

For ideal gas



=

i

x

i

1 C

x

_A

= C

^A

- molecular weight of A

M

A

RT n V

P

_A

=

_A

RT

P V

C

_A

= n

^A

=

^A

P

_A - partial pressure of A

- Mole fraction

for ideal gas mixture

for gas for liquid

for ideal gas

RT P V

C = n

^total

=

C

y

_A

= C

^A 

=

i

y

i

1

P P RT

P RT P

C

y

_A

= C

^A

=

^A

=

^A Dalton’s law of gas mixture

Mass Average and Molar Average Velocity

a) mass average velocity









= 



= 



= 

i i i i i i

i i i i i

v v

v

v

- the absolute velocity of species i

relative to stationary coordinate axes vi

b) molar average velocity









= =

=

i i i i i i

i i i i i

v C x

v C C

v C V

c) diffusion velocity - the velocity of a particular species relative to the mass average or molar average velocity

v

v

_i

−

and

v

_i

− V

A species can have a diffusion velocity only if gradients in the concentration exist

v v

_B

− v

v

_A

−

v

B

v v

^A

Mass Diffusivity

t L L

L M t

L M dz

dC J

A A AB

2 3

2

1

1  =



 





 



 



= 

= − D

( Temp ., Pr essure , Compositio n )

AB

= f

D

Gases 5x10^-6 ~ 10^-5 m2/s Liquids 10^-10 ~ 10^-9

Solids 10^-14 ~ 10^-10

Molecular Mass and Molar Fluxes

D

AB

(

_A

)

_{AB A}

A

A

C v V C x

J = − = − D 

J

A

Molar flux

Mass flux

mass diffusivity(diffusion coefficient) for component A diffusing through component B

the molar flux relative to the molar average velocity

(

_A

)

_{AB A}

A A

v v

j =  − = −  D  

Convective Mass and Molar Fluxes

Molar flux

Mass flux relative to a fixed spatial coordinate system



 ^{= + }

 +







−

=

 +







−

=

=

i i A A

i i A

A AB

A A

AB A

A

v v n j n

n D D

(

_A

)

_{AB A}

A

A

C v V C x

J = − = − D 

V C x

C v

C

_{A A}

= − D

_AB



_A

+

_A  





= =

=

i i i i i i

i i i i i

v C x

v C C

v C V



+



−

=

i

i i A

A AB A

A

v C x x C v

C D

define

N =

_A

C

_A

v

_A the molar flux relative to a set of stationary axes



 ^{= +}

+



−

=

i i A

A i

i A

A AB

A

C x x N J x N

N D

concentration gradient contribution

bulk motion contribution

Related Types of Molecular Mass Transfer

Nernst-Einstein Relation

u

A Mobility of component A, or the resultant velocity of the molecule under a unit driving force

According to 2^nd law of thermodynamics, system not equilibrium will tend to move toward equilibrium with time

Driving force = chemical potential

c AB c

A A

u RT V

v − =   = − D  

(

_A

)

_A ^AB _c

A A

C RT V

v C

J = − = − D  

in homogeneous ideal solution at constant T & P

A c

=  + RT ln C



⁰

A

A AB

C

J = D − 

Fick’s Equation

Mass Transfer by Other Physical Conditions

Thermal diffusion(Soret diffusion)

~ produce a chemical potential gradient

Mass transfer by applying a temperature gradient to a multicomponent system Small relative to other diffusion effects in the separation of isotopes

Pressure diffusion

Mass fluxes being induced in a mixture subjected to an external force field Separation by sedimentation under gravity

Electrolytic precipitation due to an electrostatic force field Magnetic separation of mineral mixtures by magnetic force Component separation in a liquid mixtures by centrifugal force Knudsen diffusion

If the density of the gas is low, or if the pores through which the gas traveling are quite small

the molecules will collide with the wall more frequently than each other (wall collision effect increases)

the molecules are momentarily absorbed and than given off in the random directions

gas flux is reduced by the wall collisions







Differential Equations of Mass Transfer

Conservation of Mass(overall)

( )  = ⁰

 + 



 _



_{A V}

dV

dA t n v

Equation of Continuity for Component A Conservation of Mass of Component A

= 0

 −

 + 





_A ^A

r

_A

n t

y y , y A

y y , x A

x , x A

x x ,

A

y z n y z n z x n z x

n   −   +   −  



 + +

= 0







−





 

 + 





−





+

_+

x y z r x y z

y t x n

y x

n

^A _A

z z , z A

z z , A

= 0

 − + 





_A ^A

R

_A

t

N C

Differential Equations of Mass

Transfer-1

Equation of Continuity for the Mixture In binary mixture

= 0

 −

 + 





_A ^A

r

_A

n t

= 0

 −

 + 





_B ^B

r

_B

n t

( ) ( ) ( ^{− +} ) ^{= 0}



 +

 +  +





_{A B} ^{A B}

r

_A

r

_B

n t

n

v v

v n

n

_A

+

_B

= 

_{A A}

+ 

_{B B}

= 



=

 +



_{A B}

B

A

r

r = −

= 0



 + 





 v t

Differential Equations of Mass

Transfer-2

= 0

 −

 + 





_A ^A

r

_A

n t

v j

n

_A

A

=

A

+  

( v )

j

n

_A

A

=  

A

+    





( ^ ) ^{− = 0}





 +



+ 





^A _{A A}

A

v r

j t



v t v

t

^{A A A}

A

+     +    





 

 +



 

=0

= 0

−





 +



_A

A

A

j r

Dt

D

Differential Equations of Mass

Transfer-3

in terms of molar unit

= 0

 − + 





_A ^A

R

_A

t

N C

= 0

 − + 





_B ^B

R

_B

t

N C

But R_Ais not always –R_B

for only stoichiometry reaction ( A B)

( ) ( ) ( − + ) = ⁰

 + + 

+





_{A B} ^{A B}

R

_A

R

_B

t C N C

N

V C v

C v

C N

N

_A

+

_B

=

_{A A}

+

_{B B}

= C

C C

_A

+

_B

=

B

A

R

R = −

( ⁺ ) ^{= 0}

 − + 



 R

_A

R

_B

t V C

C

Differential Equations of Mass Transfer-4

A A

AB A

A

v r

t  =   +



 



 +   







₂

D v C

n

_A

= − D

_AB

 

_A

+ 

_A

V C y

C

N

_A

= − D

_AB



_A

+

_A

(

A B

)

A A

A

C

AB

y y N N

N = − D  + +

(

A B

)

A A

A AB

n n

n = −  D   +  +

or

If and are constant

 D

_AB

A A AB

A A

A

A

v v r

t

t +   +   =  +

 + 



         

 _D

²

A A

AB A

A

V C C R

t

C +   =  +





₂

D

or

= 0

−





 +



_A

A

A

j r

Dt D

A A AB

A

A

v r

t  =  +



 



 +  



    

 D

²

A A

A n r

t

= −  +



 

 

^g

t



v

= −    + 





(

^{v g}

)

e v

t U

 = −  + 



 

  +



 

^ ₂^{1 2}



A A

A

j v

n = +  

v

 v



 = +

  



 

  + +

 +

=

q v v U^ ₂¹v²

e

 

( p g )   g ( T T ) g (

A ^A

)

Dt v

D        

 = −  + −   − − − −

( ) ( ) ( )

( ) (

A ^A

)

A A

, A T ,

T A

T T

T T T

, T

A A

















 



 





 

−

−

−

−



+

 − + 

 − + 

=



Boundary Conditions

Initial Conditions

at t = 0 , C

_A

= C

_A0

or 

_A

= 

_A0

Case 1) The concentration of the surface may be specified Boundary Conditions

,

,

,

,

_{1 1 0 1}

1 A A A A A A A A

A

A

C y y x x ,

C = = =  =   = 

Case 2) The mass flux at the surface may be specified

0

1

,

1

, 0 ,

=

 

−

=

=

=

=

z A surface AB

z , A A A

A A

A

dz

D d j

) e impermeabl (

j N

N j

j

Case 3) The rate of heterogeneous chemical reaction may be specified

Case 4) The species may be lost from the phase of interest by convective mass transfer

surface n

A surface n

z ,

A

k C

N =

( ⁻

_

)

=

_A,

surface ,

A surface c

z ,

A

k C C

N

Steady State Diffusion of A through Stagnant B

= 0



−

+ z A,z z z z

,

A

SN

SN

At z=z₁, gas B is insoluble in liquid A Mass Balance Equation

(

_{A,z B,z}

)

A A

AB z

,

A

y N N

dz C dy

N = − D + +

 z

Liquid A

A z

N

z A z

N

_+

S

Flowing of gas B

z

1

z = z

2

z =

= 0 dz dN

_A,z

= 0 dz dN

_B,z

= 0

 N

_B,z

~ Component B is stagnant

= 0

z ,

N

B

dz dy y

N C

^A

A AB z

,

A

= − −

1 D

2 2

1 1

A A

,

A A

,

y y

z z at

y y

z z at

=

=

=

=

Steady State Diffusion of A through Stagnant B-1





=

²

− −

1 2

1

1

A A

y y

A A AB

z z z ,

A

y

C dy dz

N D

( ) (

_{1 2}

)

ln , 1 2 ln

, 2 1

1 2 1

2 1

2

,

1

ln 1

_{A A}

B AB B

A A

AB A

A AB

z

A

y y

y z z

C y

y y

z z C y

y z

z

N C −

= −

−

= −

−

−

= D − D D

( ) ( )

(

² ₂

) (

¹₁

)

1 2

1 2

1 1

1 1

A A

A A

B B

B B

ln ,

B

ln y y

y y

y y ln

y y y

−

−

−

−

= −

= −

Steady state diffusion of one gas through a second gas

~ absorption, humidification

P y P

RT and P V

C = n =

_A

=

^A

(

_{2 1}

)

¹ _,ln ²

,

B A A

AB z

A

P

P P

z z RT

N P −

= D −

For ideal gas

Film Theory

~ a model in which the entire resistance to diffusion from the liquid surface to the main gas stream is assumed to occur in a stagnant or laminar film of constant thickness d.

z

N

A_,

Flowing of gas B

= 0 z

d

= z

Liquid A

Slowly moving gas film

ln ,

2 1

,

B A A

AB z

A

P

P P

RT

N P −

= D d

(

_{1 2}

) (

_{1 2}

)

, A A

c A

A c z

A

P P

RT C k

C k

N = − = −

= d

ln , B

AB

c

P

k D P

Convective Mass Transfer

1.0

~ 0.5 AB

k

c

 D

actual

Main fluid stream in turbulent flow

Concentration Profile

= 0 dz dN

_A,z

dz dy y

N C

^A

A AB z

,

A

= − −

1 D

2 2

1 1

A A

,

A A

,

y y

z z at

y y

z z at

=

=

=

= 1  = 0



 





− dz dy y C dz

d

_A

A

D

AB

If C and are constant under isothermal and isobaric conditions

D

_AB

1 0

1  =



 





− dz dy y dz

d

_A

A

( 1 )

_{1 2}

ln − y

_A

= C z + C

−

( 1) ( 2 1)

1 2

1

1

1 1

1

^{z z z z}

A A A

A

y y y

y

^{− −}

 

 





−

= −

−

−

ln , 1

2 1 2

ln

_{B B} ^B

B B

B

B

y

y y

y y

dz dz

y = y = − =





Pseudo Steady State Diffusion through a Stagnant Gas Film

dt dz N M

A z

A

L A, ,

= 

z

t

z t t at

z z t

at

=

=

=

= ,

,

0

₀

the length of the diffusion path changes a small amount over a long period of time ~ pseudo steady state model

ln , B

A A

AB A

A AB

z ,

A

y

y y

z z C y

ln y z z

N C

^{1 2}

1 2 1

2 1

2

1

1 −

= −

−

−

= D − D

Molar density of A in the liquid phase

ln ,

2 , 1

B A A

AB A

L A

y y y

z C dt

dz M

= −

 D

( − ) ^ _ ^{ − } _ ^

= 

2

2 0 2

2 1

ln ,

,

z z

y y

C

M

t y

^t

A A

AB

A B

L A

D

( − ) ^ _ ^{ − } _ ^

= 

2

2 0 2

2 1

ln ,

,

z z

t y y

C

M

y

_t

A A

A B

L A

D

AB z

N

A_,

S

Flowing of gas B

z

0

z = z

t

z =

Liquid A

Diffusion through a Isothermal Spherical Film

= 0

r ,

N

B

( ^r

²

^N

_A,r

) ^{= 0}

dr d

1 0

2

 =



 





− dr dx x r C

dr

d

_A

A

D

AB

( ) ( ) ( ) ( )1 ₁¹ 1 ₂

1 1

1 2

1

1

1 1

1

^{r r}

r r

A A A

A

x x x

x

⁻

−

 

 





−

= −

−

−

for constant temperature is constant

C D

_AB

Temperature T₁

r₁

r₂ Gas film

Temperature T =² T¹

( ) ( ) ^ _ ^ − ^ _ ^

−

−

= 



=

₌

1 2 2

1 ,

2

1

1

ln 1 1

1 4 4

1 A

A AB

r r r A

A

x

x r

r N C

r

W D

(

_{A,r B,r}

)

A A

AB r

,

A

x N N

dr C dx

N = − D + +

Since B is insoluble in liquid A

Diffusion through a Nonisothermal Spherical Film

2 3

1 n

AB,1 AB

r r  

 



=  D

D

1 0

2 3

1

2 1

  =





 





 

 





− dr

dx r

r x

RT r P

dr

d

ⁿ _A

A

D

AB,1

( )  ( ) 

( )

^{( )}

( )

^{( )}

  ^ 

 





−

−

−

+

= 



=

_{= + +}

1 2 2

1 2 1 2 2

1 1 ,

2

1

1

ln 1 1

1

2 1

4 4

1 A

A n n

n AB,1 r

r r A

A

x

x r

r r

n RT

N P r

W D

1 0

2

 =



 





− dr dx x r C

dr

d

_A

A

D

AB

2 n  − if

Temperature T₁

r₁

r₂ Gas film

Temperature

n

r T r

T  

 



= 

1 2 1 2

Diffusion with Heterogeneous Chemical Reaction

1 0 1

2

1

  =

 





− dz

dx x dz

d

_A

A

 

 





−

= d

2 0 , 1

1 ln 1 2

A AB

z

A

x

N C D

Gas A Gas A & B

Sphere with coating of catalytic material

B A → 2

Edge of hypothetical Stagnant gas film

Catalytic surface A

B Z=0

Z=d

Z x_A

x_B

x_A0

z A z

B

N

N

_,

= −

₂¹ _,

dz dx x

N C

^A

A AB z

A

2 , 1

− 1−

= D

( 1

¹₂

)

_{1 2}

( 2 ln

₁

) ( 2 ln

₂

)

ln

2 − x

_A

= C z + C = − K z − K

−

0

, ,

0

₀

= d

=

=

=

A

A A

x z

at

x x

z at

( 1 −

₂¹

x

_A

) ( = 1 −

₂¹

x

_A0

)

^{1−( )zd}

Irreversible and instantaneous rxn Solid catalyzed dimerization of CH₃CH=CH₂

Diffusion with Slow Heterogeneous Chemical Reaction

1 0 1

2

1

  =

 





− dz

dx x dz

d

_A

A

( large )

1 ln 1 1

2

1 2 0

1 1

,

k

x k

N C

A AB

AB z

A

 

 





− d

+

= d

D D

B A → 2

z A z

B

N

N

_,

= −

₂¹ _,

dz dx x

N C

^A

A AB z

A

2 , 1

− 1−

= D

( ¹

₂¹

)

_{1 2}

ln

2 − x

_A

= C z + C

−

(

_{A z A}

)

z A A

A A

C k C N

k x N

z at

x x

z at

1 ,

1 , 0

,

, 0

=

= d

=

=

=

assume pseudo 1^storder rxn

( ) 

^d

( − )

^{−( )d}



 



 −

=

−

_A ^z

z Az

A

x

C k

x N

₂¹ ₀ ¹

1 2

1

1

2 1 1 1

k

AB

Da =

₁

d D

Damkohler No.: the effect of the surface reaction on the overall diffusion reaction process

Diffusion with Homogeneous Chemical Reaction

AB B

A + →

1

0

,

,

S − N

_+

S − k C S  z =

N

_A

z z z z A

z A

dz N

_A,z

= − D

_AB

dC

^A

( 0 )

0

, , 0

,

0

=

=

=

=

=

dz dC

or N

L z at

C C

z at

A z

A

A A

First order reaction

1

0

,^z

+

_A

=

A

k C

dz dN

1

0

2

2 ^A

−

_A

=

AB

k C

dz C D d

2

0

2

2

−   =



 d d

0 A A

C

= C

 ζ = z L  = k

₁

L

²

D

_AB Thiele modulus

0

, 1

1

, 0

=





=



=



=



d d at

at

Diffusion with Homogeneous Chemical Reaction-1



 



 



= 

−

=

= = ⁰

tanh

0 0

,

L

C dz

N dC

^{A AB}

z A z AB

z A

D D

ζ sinh cosh

₂

1

 + 

=

 C C

( )

 





−

= 







−



= 

 cosh

1 cosh cosh

ζ sinh sinh

cosh cosh

( )

( )

 

AB AB

A A

L k

L z L

k C

C

D D

2 1 2 1

0

cosh

1

cosh −

=

( )



= 

= 

 tanh

dz

dz

L 0 L

0 0

0

, A A

A avg

A

C C

C

C

Gas Absorption with Chemical Reaction in an Agitated Tank

effect of chemical reaction rate on the rate of gas absorption in an agitated tank

Example) The absorption of SO₂ or H₂S in aqueous NaOH solutions

Semiquantative understanding can be obtained by the analysis of a relatively simple model

Gas A in

Surface area of the bubble is S

Volume of liquid phase is V

Liquid B

Gas Absorption with Chemical Reaction in an Agitated Tank-1

Assumptions:

1) Each gas bubble is surrounded by a stagnant liquid film of thickness d, which is small relative to the bubble diameter

2) A Quasi-steady state concentration profile quickly established in the liquid film after the bubble is formed

3) The gas A is only sparingly soluble gas in the liquid, so that we can neglect the convection term.

4) The liquid outside the stagnant film is at a concentration C_Ad, which changes so slowly with respect to time that it can be considered constant.

Z=0 Z=d

C_Ad Liquid film

Liquid-gas interface

Gas in bubble

Main body of liquid

Without reaction

With reaction

Gas Absorption with Chemical Reaction in an Agitated Tank-2

1

0

2

2 ^A

−

_A

=

AB

k C

dz C D d

= d

d

=

=

=

A A

A A

C C , z at

C C , z at

0 ₀

( )







− +



= 

sinh

sinh cosh

B cosh

sinh C

C

A

A

ζ

0

0 A A

C

= C

 ζ = z d  = k

₁

d

²

D

_AB

d d

=

=

_A

z A

AB

Vk C

dz dC S D

1

-

( d ) 

+

= 

sinh S

V

B cosh 1

( ) ^ _ ^

 





 d +

− 

 

= 

=

⁼

d

sinh S

V cosh cosh

sinh C

N N

AB A

z z ,

A

1

0 0

D



The total rate of absorption with chemical reaction

ζ sinh cosh

₂

1

 + 

=

 C C

Assumption (d)

Gas Absorption with Chemical Reaction in an Agitated Tank-3

for large value of , dimensionless surface mass flux increases rapidly with  and becomes nearly independent of V/Sd.

for very slow reactions, the liquid is nearly

saturated with dissolved gas

Chemical rxn is fast enough to keep the bulk of the solution almost solute free, but slow enough to have little effect on solute transport in the film

"Transport Phenomena" 2nd ed., R.B. Bird, W.E. Stewart, E.N. Lightfoot

Diffusion and Chemical Reaction inside a Porous Catalyst

We make no attempt to describe the diffusion inside the

tortuous void passages in the pellet. Instead, we describe the

“averaged diffusion” of the reactant ~ effective diffusivity

( ) 4 0

4

4

² _, ^{2 2}

,

 r − N

_+

 r +  r + R  r  r =

N

_A

r r r r A

r A

( ) ( )

A r r

, r A

r r , A r

R r r

N r N

lim r

²

2 2

0

− =

+

→ ^





( r N

Ar

) r R

A

dr

d

₂

,

2

=

"Transport Phenomena" 2nd ed., R.B. Bird, W.E. Stewart, E.N. Lightfoot