-
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