Mass Transfer

transport of one constituent from a region of higher concentration to that of a lower concentration

mass average velocity1 1

1

v v

v

n n

i i i i

i in

i

i

molar average velocity1 1

1

v v

V

n n

i i i i

i in

i

i

c c

cc

absolute velocity of species i relative to stationary coordinate axs

molar flux relative to the molar average velocity

mass flux relative to the molar average velocity

molar flux relative to a set of stationary axes

mass flux relative to a set of stationary axes

,A

A z AB

dcJ D

dz

,A

A z AB

dj D

dz

, , ,N N NAA z AB A A z B zdy

cD ydz

n n nA AB A A A BD

concentration gradient contribution

bulk motion contribution

1

N Nn

A AM A A i

i

cD y y

for multicomponent mixture

convective mass transfer A c AN k c

molar mass transfer relative to fixed spacial coordinates

convective mass transfer coeff.

concentration difference between the boundary surface conc and the average conc of the fluid stream

Differential equation for mass transfer

equation of continuity for component A n 0AA Art

equation of continuity for the mixture v 0t

rate of mass production

in terms of molar units

equation of continuity for component A

equation of continuity for the mixture

N 0AA Ac

Rt

V 0A Bc

c R Rt

rate of molar production

depends on stoichiometry

if density is constant, 2v AA AB A Ac

c D c Rt

2 2 2

2 2 2A A A A

AB

c c c cD

t x y z

2 2 2

2 2 2 2

1 1A A A A AAB

c c c c cD

t r rr r z

Boundary conditions

1. Concentration at a boundary surface is specified

- A pure component in one phase and a mixture in the second phase, the concentration is at thermodynamic saturation conditions- For a gas mixture in contact with a pure volatile liquid or solid A, the partial pressure of A in the gas at the surface is saturation vapor pressure- For a liquid mixture in contact with a pure solid A, the concentration of A in the liquid at the surface is the solubility limit of A in the liquid- For a contacting gas and liquid, if both species in the liquid phase are volatile, the boundary condition at the gas-liquid surface is defined by Raoult’s law- For solutions where species A is only weakly soluble in the liquid, Henry’s law may be used

As A Ap x P

A Ap H x

Boundary conditions

2. A reacting surface boundary is specified

- The flux of aone species may be related to the flux of another species by chemical reaction stoichiometry- A finite rate of chemical reaction might exist at the surface- The reaction may be so rapid that CAs=0

2 3 ; 2 , 3B A C AA B C N N N N

A c AszN k c

3. The flux is zero at a boundary or at a centerline of symmetry

00 0

0 or 0A AA ABzz z

c cN D

z z

4. The convective mass transfer flux at the boundary surface is specified

0A c As Az

N k c c

Fabrication of silicon wafer by CVD

A1; rxn occurs only at the suface of growing Si thin film -> no homogeneous rxn

A2; gas phase is not externally mixed -> molecular diffusion dominates

A3; feed gas provides silane in high excess -> silance conc at boundary is constant

A4; flux is 1-dimensionalA5; thickness of Si film is very thin

-> diffusion path length (δ) is constantA6; mass transfer process within diffusion zone is at steady state

, , ,AA z AB A A z B zdy

N cD y N Ndz

0AyAx Az A

A

NN N cR

x y z t

,0

A zdN

dz

, 4

, 2

1 1

2 2

A z

B z

N mol SiH reacted

N mol H formed

, , ,21

A AB AA z AB A A z A z

A

dy cD dyN cD y N N

dz y dz

0,

0 1

As

A

yAB

A z Ay

A

cDN dz dy

y

0

,

1ln

1

AABA z

As

ycDN

y

formation of a tungsten thin film on a silicon wafer by CVD

2 63 ( ) ( ) ( ) 6 ( )H g WF g W s HF g

, , , ,AA z A mix A A z B z C zdy

N cD y N N Ndz

, 6

, 2

1 1

3 3

A z

B z

N molWF reacted

N mol H reacted

,1 2

A mix AA z

A

cD dyN

y dz

, 6

,

1 1

6 6

A z

C z

N molWF reacted

N mol HF formed

, , , ,3 6AA z A mix A A z A z A zdy

N cD y N N Ndz

steady state molecular diffusion (1-D, no chemical rxn)

N 0AA Ac

Rt

, , ,AA z AB A A z B zdy

N cD y N Ndz

unimolecular diffusion

0AyAx Az A

A

NN N cR

x y z t

,0

A zdN

dz

A vaporizes and diffuses into the gas phaseGas B has a negligible solubility in liquid A, and is chemically inert to A

, 0B zd

Ndz

NB,z at z=z1 is zero -> NBz (net flux of B) is zero

,1

AB AA z

A

cD dyN

y dz

-> B is a stagnant gas

,1

AB AA z

A

cD dyN

y dz

11 A Az z y y 22 A Az z y y

2 2

1 1

,1

A

A

z yA

A z ABz y

A

dyN dz cD

y

2

1

,

2 1

1ln

1

AABA z

A

ycDN

z z y

2 1

2 1

,ln /

B B

B lm

B B

y yy

y y

2 1 1 2

2 1 2 1

,

1 1

ln 1 / 1 ln 1 / 1

A A A A

B lm

A A A A

y y y yy

y y y y

1 2

,

2 1 ,

A AABA z

B lm

y ycDN

z z y

For an ideal gas,n P

cV RT

AAp

yP

1 2

,

2 1 ,

A AABA z

B lm

p pD PN

RT z z p

Steady state diffusion of one gas through a second stagnant gas;Absorption, humidification

,0

A zdN

dz ,

1

AB AA z

A

cD dyN

y dz

0

1

AB A

A

d cD dy

dz y dz

10

1

A

A

d dy

dz y dz

1 2ln 1 Ay c z c 11 A Az z y y 22 A Az z y y

1 2 1

2

1 1

/11

1 1

z z z z

AA

A A

yy

y y

1 2 1

2

1 1

/z z z z

BB

B B

yy

y y

2

1

2

1

z

Bz

B z

z

y dzy

dz

average concentration of one of the species along the diffusion path

1 2 1

22

11

1

2 1 2 1

2 1 2 1

/

2 1

2 1

2 1

,

ln / ln /

z z z z

z B

zB

B B

B B B B

B B B B

B lm

ydz

yy y

z z

y y z z y y

y y z z y y

y

Vapor degreaser; cleaning metal parts

regulation; greaser cannot emit more than 1.0 kg TCE per day

Mw=131.4g/molvapor pressure=115.5mmHgDAB=0.088cm

2/s

2

1

,

2 1

1ln

1

AABA z

A

ycDN

z z y

3

10.0396

(0.082)(273 35)

P kg molc

RT m

1

115.5 10.152

1 760

AA

P mmHg atmy

P atm mmHg

48

,

1 0(0.0396)(0.088 10 )ln 1.197 10

5.0 0.2 1 0.152A zN

2

, 0.4234

A A z

D kgTCEW N

day

1 2

,

2 1 ,

A AABA z

B lm

p pD PN

RT z z p

film theory

1 2,

,

ABA z A A

B lm

D PN p p

RTp

1 2 1 2,c

A z c A A A A

kN k c c p p

RT

,

ABc

B lm

D Pk

p

kc is a function of the diffusion coefficient raised to an exponent varying from 0.5 to 1.0

pseudo-steady-state diffusion

when the length of the diffusion path changes a small amount over a long period of time

1 2

,

2 1 ,

A AABA z

B lm

y ycDN

z z y

1 2

,

,

AB A A

A z

B lm

cD y yN

zy

,,

A LA z

A

dzN

M dt

molar density of A in the liquid phase

1 2,

,

AB A AA L

A B lm

cD y ydz

M dt zy

01 2

, ,

0

/

t

t

t zA L B lm A

t zAB A A

y Mdt z dt

cD y y

0

1 2

2 2

, , /

2

t tA L B lm A

AB A A

z zy Mt

cD y y

0

1 2

2 2

, , /

2

t tA L B lm AAB

A A

z zy MD

c y y t

Formation of SiO2 thin film on a Si wafer - fabrication of solid state microelectronic devices

A1; oxidation of Si to SiO2 occurs only at Si/SiO2 interface-> unreacted Si serves as the sink for molecular mass transfer of O2 through the film

A2; O2 in the gas phase represents an infinite source for O2 transferA3; rate of SiO2 formation is controlled by the rate of molecular diffusion of O2 through the solid SiO2 layerA4; rxn is very rapid -> concentration of O2 at interface is zeroA5; the flux of O2 (A) through SiO2 (B) layer is 1-dimensionalA6; the rate of SiO2 film formation is slow

-> no accumulation of reactants or products within the SiO2 film

, 0A zd

Ndz

, , ,( )A A

A z AB A z B z

dc cN D N N

dz c

,A

A z AB

dcN D

dz as conc of O2 in SiO2 layer is dilute

0

,

0 As

A z AB A

c

N dz D dc

, AB AsA zD c

N

δ increases slowly with time -> pseudo-steady-state assumption

(molar rate of SiO2 formation) = (molar rate of accumulation of SiO2)

,AB As

A z

D cN S S

B

B

Sd

M

dt

0 0

t

B AB As

B

M D cd dt

2 B AB As

B

M D ct

equimolar counterdiffusion , ,A z B zN N

, , ,AA z AB A A z B zdc

N D y N Ndz

,A

A z AB

dcN D

dz

2 2

1 1

,

A

A

z c

A z AB Az c

N dz D dc

1 2,

2 1

ABA z A A

DN c c

z z

A AA

n pc

V RT for ideal gas,

1 2,

2 1

ABA z A A

DN p p

RT z z

, 0A zd

Ndz

2

20A

d c

dz 1 2Ac C z C 1

1 2

1

1 2

A A

A A

c c z z

c c z z

One dimensional systems with chemical reaction

homogeneous rxn; occurs uniformly throughout a given phaseheterogeneous rxn; takes place in a restricted region within or at a

boundary of the phase

N 0AA Ac

Rt

only for homogenous rxn

diffusion controlled; when the rxn rate is instantaneous relative to the rate of diffusion

reaction controlled; when the rxn rate at the surface limits the mass transfer rate

diffusion with heterogeneous 1st order chemical reaction

diffusion controlled

2 23 ( ) 2.5 ( ) 2 ( ) ( )C s O g CO g CO g

no homogeneous chemical rxn occurs along the diffusion path -> RO2=0

As the coal particle is oxidized, the particle shrinks with time. It is desired to predict the size of the particle with time

22

sin1 1 10

sin sin

Ar AAAA

r N NNcR

t r r rr

22

10

Ard r N

drr

2 2

2 24 4 0O r O rr r r

N r r N r

2

2

0O rd r N

dr 2 2

2 2O r O r

r Rr N R N

1

N Nn

A AM A A i

i

cD y y

2 2 22.5 and 1.25O r COr O r CO rN N N N

2 23 ( ) 2.5 ( ) 2 ( ) ( )C s O g CO g CO g

22 2 2 2 2 2mix

O

O r O O O r COr CO r N

dyN cD y N N N N

dr

2

2 2 2 2 2 2mix

1 10

2.5 1.25

O

O r O O O r O r O r

dyN cD y N N N

dr

2

2 2 2 2mix0.2

O

O r O O O r

dyN cD y N

dr

2 2

2

2

mix

1 0.2

O O

O r

O

cD dyN

y dr

2

0Or R y

2, 0.21Or y

2

2

0O rd r N

dr

2 22

2

0.21-mix2

2 0

0.2

0.2 1 0.2

O O

O rR

O

cD dydrr N

yr

2

2

-mix2 1 1ln0.2 1.042

O

O r

cDr N

R

instantaneous rxn

the moles of oxygen transferred per time is the product of the oxygen flux and the cross sectional area

22 2

-mix24 4 ln 1.0420.2

O

O O r

cDW r N R

negative because the direction of oxygen flux from the bulk gas to the surface is opposite to the increasing r direction from r=R to infinity

the material balance for carbon

22 2

-mix3 3 34 ln 1.042

2 2.5 2.5 0.2

O

C CO O

cDW W W R

(input carbon rate) – (output carbon rate) = rate of carbon accumulation

output rate of carbon

2 23 ( ) 2.5 ( ) 2 ( ) ( )C s O g CO g CO g

carbon accumulation rate24C C

C C

dV dRR

M dt M dt

2-mix 230 4 ln 1.042 4

2.5 0.2

O C

C

cD dRR R

M dt

2

2 2

-mix12 ln 1.042

Ci f

C

O

R RM

cD

2 23 ( ) 2.5 ( ) 2 ( ) ( )C s O g CO g CO g

for alternative rxn with instantaneous rxn at the surface

2 2C s +O g CO g

22 2 2 2 2 2mix

O

O r O O O r COr CO r N

dyN cD y N N N N

dr

2

2 2 -mix

O

O r O

dyN cD

dr

2 2 2-mix4O O OW RcD y

2

2 2

-mix24 4 ln 1.0420.2

O

O O r

cDW r N R

if the rxn is not instantaneous

2 2 2 2-mix

4O O O O sW RcD y y

As s AsRN k c

2 2

2

O s O R

O s

s

c Ny

c k c 2

2 2 2-mix4

O R

O O O

s

NW RcD y

k c

2 2 2

2 24 4O O R O rW R N r N 2

2 2 2

-mix2-mix1

O

O R O O

s

DR N RcD y

k R

2 2

2

2

-mix

-mix

4

1

O O

OO

s

RcD yW

D

k R

diffusion with homogeneous 1st order chemical reaction

one of the constituents of a gas mixture is preferentially dissolved in a contacting liquid(absorption of A into B)

0Ac

, , ,AA z AB A A z B zdc

N D y N Ndz

if there is little fluid motion and if the concentration of A is small

,A

A z AB

dcN D

dz

N 0AA Ac

Rt

1A AR k c , 0A z A

dN R

dz 1 0

AAB A

d dcD k c

dz dz

1 1 2 1cosh / sinh /A AB ABc c k D z c k D z

0at 0 A Az c c

at 0Az c

0

0

1

1

1

sinh /cosh /

tanh /

A AB

A A AB

AB

c k D zc c k D z

k D

0 1, 0

1

/

tanh /

AB A ABA z z

AB

D c k DN

k D

molar mass flux at the liquid surface

penetration theory

0 1, 0

1

/

tanh /

AB A ABA z z

AB

D c k DN

k D

as the rxn rate increases, 0, 10

0A z AB AzN D k c

1 2,A z c A AN k c c

,

ABc

B lm

D Pk

p film theory;

boundary layer theory; 1/2 1/3Sh 0.664Re Scc L LAB

k L

D Sc

ABD

2/3~c ABk D

1/ 2~c ABk D

~c ABk D

two- and three-dimensional systems

2 2

2 20A A

c c

x y

,Ac x y X x Y y

1

sin sinhA nn

n x n yc A

W W

1

sin sinhA A nn

n x n Lc c x A

W W

simultaneous heat and mass transfer

vapor condensation on a cold surface

N 0AA Ac

Rt

, 0A z

dN

dz

if A is diffusing through a stagnant gas

,1

AB AA z

A

cD dyN

y dz

if the temperature profile is of the form

1 1

nT z

T z

1 1

3/2 3 /2

1 1

n

AB AB ABT T

T zD D D

T z

1 1/n

P Pc

RT RT z z

1

/2

,

1 11

nAB T A

A z

A

PD z dyN

RT y z dz

1/4

4/99/16

0.670RaNu 0.68

1 0.492 / Pr

LL

over a small temperature range

avg

,1

AB AA z

A

cD dyN

y dz

1 2avg

,

2 1 ,

AB A A

A z

B lm

cD y yN

z z y

total energy flux

liquid 2 3 1 2 , 1 2z

c A z A

qh T T h T T N M H H

A

enthalpy of A per unit mass

assume T2 , ( )c AB avgh cD

2

2 or

A AA

p Py

P P

1 2avg

,

2 1 ,

AB A A

A z

B lm

cD y yN

z z y

check ; liquid 2 3 1 2 , 1 2z

c A z A

qh T T h T T N M H H

A

simultaneous momentum and mass transfer;

dissolution of one of the components of a gas mixture by a liquidtime of contact is short

0AyAx Az A

A

NN N cR

x y z t

,,0

A yA x NN

x y

, , ,AA x AB A A x B xc

N D x N Nx

, , ,AA y AB A A y B yc

N D x N Ny

, , ,A x A A x B x A xN x N N c v

,A

A y AB

cN D

y

2

20A Ax AB

c cv D

x y

2 2

max 2

12

2

A AAB

y y c cv D

x y

2 2

max 2

12

2

A AAB

y y c cv D

x y

at 0 0Ax c

at 0 0Ac

yy

0at A Ay c c

5.1213 39.318

0

105.64

204.75

0.7857 0.1001

0.03500

0.01811

A Ax L y n n

A Ax y

n

n

c ce e

c c

e

e

if solute A penetrates only a short distance into the liquid film

2

max 2A A

AB

c cv D

x y

0

max,

ABA y Ay

D vN c

x

0

max

, 1 erf4

A A

AB

c x cD x

v

exp

ABc

Dk

t

unsteady state diffusion

diffusion in a semi-infinite medium

2

2A A

AB

c cD

t z

0, ,0 for all A Aot c z c z

at 0, 0, for 0A Asz c t c t

at , , for all A Aoz c t c t

erf2

As A

As Ao AB

c c z

c c D t

, 0

ABA z As Aoz

DN c c

t

diffusion in a finite-dimensional medium

2

2A A

AB

c cD

t z

at 0 for 0A Aoc c t z L

at 0 for 0A Asc c z t

at for 0A Asc c t L t

2

/2

1

4sin , 1,3,5,...D

n XA As

nAo As

c c n ze n

c c L

2

/2

1

4cos , 1,3,5,...D

n XABAz As Ao

n

D n zN c c e n

L L

2

4 ABD

D tX

L