Cheat sheet
Two modes of mass transfer
Two types of boundary conditions
Diffusion and Bulk Motion
type1 : c A ( x 0, t ) c A,0
Diffusion : Fick ' slaw N A " DAB
dC A
, flux dx N A N A "* Area N A " N A
D AB (C A,1 C A,2 )
L D AB A(C A,1 C A,2 ) L
BulkMotion Q hA(C A,S C A , )
x
1 x A0 L x A 1 (1 xA0 )( ) 1 x Al N A "
cD AB L
T
L
r 2
P T = temperature, P = pressure
ci
P i RT
mt
N A
P i ci
ln( ) r 1
J A DAB V C A DAB V (CX A ) N A " cDAB x A x A ( N A " N B ") Same formula for mass and molar rewr itten N A "
c1
c2
)c
r 1
dx p A
2
M gen
d 2 pA dy
2
dM s
dc A
dt
dr
d 2 pA
p * mA , c A
dz
2
n Ag D AB
DA B dt
c A 2 c A1 DAB ( ) r r ln 2 r 1
1-d, spherical
d 2c A dx
2
D AB
1 dc A DAB dt dt
assume steady state d c A dx 2
d
( r 2
N A D AB
0
dc A
)0 dr c1
r 2 c
1
c A (r ) c1
r2 c2
dx 2
1 dc A
c A1
dx 2
dc A dr
2
d c A
c A (r )
D AB dt
assume no chemical reaction and steady state
r c A1 c A2 1 1
assume no chemical reaction
1 x A dx
c2 cD AB ln(1 x A0 )
c A2 c A1 1 ( ) r r ln 2 r 1
that is given
2
1 x Al
c A2 c A1 c c A1 ) ln r c A1 ln r1 A 2 r2 r ln ln 2 r1 r 1
Steady-state, Steady-state, no chemical reactions,
d 2c A
ln
c A2 c A1 r ln 2 r 1
Formula can be converted for the i nformation
cD AB dx A
L
)0
c A1 ln r 1
Cx A
dx A
1 x A 0
dr c1
c A 2 c A1 r ln 2 r 1
N A "
NA
cD AB
1 dp A
1 x A dx
c1 x c2
dc A
Mass diffusion equation
dr
cD AB dx A
c A 2 c A1
c A (r ) (
Only 1-D problems
( ) dx 1 x A dx
1 x A
r2
mostly x = 0 and x=L are used
d
cD AB
r 2
c1
cD AB dx A
seperation
c1
1
Two boundary conditions are needed
dx
L c A1
( r
dr dc A
solve for
0
1
(
4 D AB A(C A2 C A1 ) 1 1
d 2 p A
c A 2 c A1
dr r c A ( x ) c1 ln r c 2
4 D AB r1
Fick's Law = diffusive molar flux
dr dc A
c
1
M in M out
c2
d
2 LD AB
xi RT P i c RT i 1
1-d, cylindrical
spherical R mt
c1
0,
dc A
c1 dx dx c A ( x 0) c1x c 2
2
x c A1 L Steady-state, Steady-state, no chemical reactions,
1
d 2c A
c A ( x )
D AB A
(C A1 C A2 ) R mt cylindrical
R
, If D AB can not be be fou found
1-d, planer
Resistance of mass transfer,planer
3
D AB
1 x A0
Can be rewritten to molar of mass concentrations
N A
dc A
1 x Al
but sometimes sometimes it needs to be included
R mt
constant
0 constant flux dx x 0 Steady-state, Steady-state, no chemical reactions,
watch for advection, most times it can be i gnored
h convective mass transfer coefficient 2
ln
type2 :
N A "
r 1 1 c A1 c A2 r 1 1 1 r2 r 1
c A1 c A2 1 1 c A1 c A2 c A1 1 1 r r 1 1 1 r2 r1 r2 r1
0
c2
cA1 c A2 1 1 1 r 2 r2 r 1
c A1 c A2 DAB 1 1 r 2 r2 r 1
Evaporation
Homogenous, zeroth, cylindrical
Raoult ' s Law
1 d
P A (0) PAsat xA (0) c A (0)
(r
r dr
P A (0)
1 x A (0) ( ) PA (0), H Henry ' s constant H S solubility coefficient, has to be given Chemical reactions Heterogeneous or Homogeneous Heterogeneous - different phases, requires catalyst
r
dc A
dr
dc A dr
k
c2
D AB
dc A
kc A ( x 0)
0
k
D AB k
D AB
x
kL D AB
)
1
cAL kL
DAB
kc AL kL
typeII : 2
d c A dx
2
0, rapid diffusion, small consumption 1, slow diffusion, rapid consumption
dc A dx
l
2 2
l
to
2 N Ag D AB
D AB
dc A dx
c A ( x )
N Ag
D AB
l
l ) cA1 , BC 2 : cA ( x ) cA1 2 2
x c1
N Ag
2 D AB
c x c2
c A1
N Ag 2 D AB
c A1e ml
c1 c A1
0 (
L2 4
mx
mx
c2e
c2e
ml
Initial Condition : c A ( x, t
0) c A,0
Type 2 dc A
c A ( x, t ) 2 c A,i
mx
e
ml
mx
c2e
ml
c A1e
ml
0
c A1e
ml
2 cosh( ml )
ml
2cosh( ml )
c A1 2 cosh( ml )
c A ( x) c A1
4 D ABt
x 2
c A ( x) c A1 (1
N A " N Ag ( x)
c A, i ) erf
2
x )
c A ( x , t ) c A,0
D AB
e
*(c A,0
x2
Time dependent
k 1
(c A1 c2 )e
1
D AB t
0
BC 2 : c1e
c2
D AB A
N A *" dx x 0 Boundary Condition 2:
c1 c A1 c2
)
Table B.2 in the appendix of the book
D AB
0
BC1: c1 c2 c A1
0
2 DAB t
Boundary Condition 1:
0
K1C A
c A ( x) c1e
x
x L
dx x L Second order
m2
erf (
dc A
BC 2 :
All data on erf can be found on
N A
BC1: c A ( x 0) c A1
D AB
BC1: c A ( x
c2
4 D AB
Differential Equation
d 2 c A
c1
c A1
4
Homogeneou s ,1st order
Homogeneous reaction, zeroth order, planer
dx
N Ag R 2
bottom boundary, impermeable
N Ag either consumed of produced x
2
the units provided
kc AL
1
N Ag r
C A can be easily converted to
D AB (1 N A "
4 D AB
N Ag K1C A
D AB
N Ag R
c A ( x, t ) c A, s
x c A ( x, t ) c A,0 (c A, i c A,0 ) erf 2 D AB t
2
, planer
c2
c A ( x )
c2
4
0) c A, i
Boundary Condition 2:c A ( x , t ) cA, i c A,i c A, s
N Ag r
Boundary Condition 1:c A ( x 0, t ) c A, s
r
c1 ln r c2
4
D AB dt
Type 1
c1
2
c A (r )
c AL kL
1
N Ag r
known
0
dx x c A ( x ) c1 x c 2 c1
1 dc A
Initial Condition : c A ( x, t
2
c A (r )
c2 c A1
BC1: c A ( x L) cAL
BC 2 : D AB
dx 2
c1
D AB 2
kcA ( x 0)
from, x 0toL
N Ag r 2
D AB 2
c A (r )
EX : Heterogeneous
2
d 2 c A
0
D AB
N Ag r
Heterogeneous - mathematically same as type II BC
dx
dr
c1 0
Homogeneous - same phase
d 2 c A
N Ag
)
BC1: c A (r R ) c A1
RT Henry ' s Law
N A "( x 0)
dc A
Time dependence,semi-infinte system
e
ml
2cosh( ml )
emx
cosh( m(1 x)) cosh( ml )
)e mx
D ABt N A *" 4 D AB t ( )e D AB
N *" x x ( A )(1 erf ) D AB 2 DABt
