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6. MASS DIFFUSION
1. Modes of Mass Transfer
- Diffusion
› Due to concentration difference (high to low)
› Analogous to heat conduction
› Other driving forces: temperature, pressure, electrical, etc
- Convection
› Due to bulk movement
- Mixture Composition
Parameter
Mass density
Molar concentration
Constituent
Total
Fraction
Relationship
i = WMi Ci
2. Fick’s Law of Diffusion
- Form
Mass density basis:
P = i Mass flux = ji
Molar concentration basis: P = Ci Mass flux = Ji*
- DAB
› specific to molecular species (e.g. A)
› also depends on other components in the mixture (e.g. B)
 binary diffusion coefficient
2
› For a binary mixture in 1D systems
d
dm A
j A , kg / m 2 / s   D AB A  D AB
dx
dx
dC A
dx
J *A , kmol / m 2 / s   D AB
 CD AB A
dx
dx
- Reference Frame

The fluxes jA and JA* are written wrt a reference frame that is
moving at a mass (or molar) average velocity of the mixture

Need to develop absolute fluxes n"A or N "A that are relative to
a fixed coordinate system
n"A   A v A
N "A  C A v A
› Relationships
v  mass average velocity  1 ( A v A   B v B )  m A v A  mB v B

v*  molar average velocity  1 (C A v A  C B v B )  x A v A  x B v B
C
Redefine the relative fluxes:
j A   A (v A  v )   A v A   A v  n"A   A v
J "A  C A (v A  v * )  C A v A  C A v *  N "A  C A v *
For a binary mixture:
J *A  J *B  0
j A  jB  0
For a stationary medium:
v  0  j A  n"A
*
 N "A
v*  0  J A
 the mass and heat transfer analogy is complete
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3. Mass Conservation Law
- Mass Conservation in a CV
- General Mass Diffusion Equation
For a homogeneous, stationary, binary medium:
› Rectangular coordinates:
  D  A     D  A     D  A   n   A






x  AB x  y  AB y  z  AB z  A
t
› Cylindrical coordinates:
› Spherical coordinates:
Some boundary conditions:
› constant concentration at the surface:
x A (0, t )  x A, s
› impermeable surface:
x
 CD AB A
0
x x  0
For convection-diffusion systems:
 Pv x Pv y Pv z    x  y  z   P
  
  G 
 





x

y

z

x

y

z
t

 

where: P = I or Ci
n"A or N "A include both diffusion and convection
 A
C A
 n"A  n A
 N "A  N A

t
t
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4. Mass Diffusion in Solids (without chemical reaction)
- Some Examples
› Semi-permeable membranes
» Packaging
» Hemodialysis
» Gas separation (e.g. He from natural gas)
› Movement of chemical through soil
› Drying solids
› Extraction of metals from ores
- 1D, Steady-State, Stationary, Homogeneous, Non-reacting
Governing equations:
Boundary conditions:
Solve for concentration profiles through the medium:
m A  (m A, s 2  m A, s1 ) x  m A, s1
L
The rate of mass diffusion:
m A, s 2  m A, s1 (m A, s1  m A, s 2 )
"
n A  j A ( why ?)  D AB

L
L / D AB
In similar way, we can get:
C ( x A, s1  x A, s 2 )
N "A  J *A 
L / D AB
See Table 14.1 for other solutions for mass diffusion
Note: for an ideal gas: Ci  pi / T , C  PT / T and xi  pi / PT
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› Sample 1: Hemodialysis in an artificial kidney.
Artificial kidney uses a membrane to separate blood on one
side and dialyzing fluid on other side as shown below.
Dialyzing fluid has solutes such as K+ and Cl- at noramal
physiological levels but initially no urea or uric acid. The
membrane is 0.025 mm thick and the total area is 2.0 m2.
The concentration of urea in the blood is 0.20 g urea/L of
blood.
Blood
Membrane
Dialyzing fluid
a)
b)
Estimate the total mass transfer coefficient, assuming
kblood = 1.25 x 10-5 m/s, kdial. fluid = 3.33 x 10-5 m/s, and
DAB = 2.20 x 10-5 m2/s
Estimate the rate of urea removal from the blood.
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5. Evaporation in a Column
› pure liquid A in a container is open to atmosphere at the top
› A is volatile, and evaporates into air (component B)
› air moving across top of the container
What are the concentration profiles and the rate of mass
transfer in the headspace?
N "  N "A  N B"  ?

N "A 
Rearrange it, realizing for ideal gases, x A  p A / PT :
CD AB dx A
CD AB dp A
N "A  

1  x A dx
PT  p A dx
Assuming steady-state conditions, N "A  constant .
dp A 
d 
1

0
dx  PT  p A dx 
with boundary conditions:
p A x  0  p A, o
p A x  L  p A, L
Note that p A, o  Hx A, o from Henry’s law
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Solving for it leads to:
x/L
PT  p A  PT  p A, L 
1  x A  1  x A, L 


or


PT  p A, o  PT  p A, o 
1  x A, o  1  x A, o 
Because x A  xB  1, we can get:
x/L
x/L
x B  x B, L 


x B, o  x B, o 
The rate of mass diffusion is:
CD AB dx A CD AB  1  x A, L  D AB PT  PT  PA, L 
 

N "A  

ln 
ln 
1  x A dx
L
1

x
RTL
P

P
A, o 
A, o 

 T
Sample 2. Estimate the evaporation rate of water into dry air
at 20 C and 1 atm. The liquid H2O is contained in a beaker
and the distance between the liquid surface and the top of the
beaker is 10 cm.

C
 2.5kPa
DH2O-air = 2.5 x 10-5 m2/s, p 20
A, H O
2
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6. Mass Diffusion with Chemical Reactions
› Consider the stationary media (Component B is fixed)
› Component A is diffusing through B and reacting
› 1D, steady-state condition
Gov. eq.:
with BC:
If the reaction rate is N A  k1C A , the general solution is:
C A ( x)  C1e  mx  C 2 e  mx , where m  k1 / D AB
Solve for the boundary conditions:
C A, o
Example 14.5
C A ( x)  2mx
e m( 2 L  x )  e mx
e
1
The rate of mass diffusion is:
dC A
N "A
  D AB
dx x  0
x 0


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7. Transient Diffusion
Taking an 1D, stationary medium as an example, the
governing equation is:
2
 2C A C A

T  T

D AB

t
x 2 t
x 2
 if defining the dimensionless groups for mass transfer similar
to those used in heat conduction:
D t
Fom  AB2
Fo  t2
Lc
Lc
h L
Bi  hL
Bim  m
k
D AB
we can use all the heat transfer solutions for mass transfer
problems (both analytical solutions and charts)!
Sample 3. A large sheet of material 40 mm thick contains
dissolved hydrogen having a uniform concentration of 3
kmole/m3. The sheet is exposed to a fluid stream that causes
the concentration of the dissolved hydrogen to be reduced
suddenly to zero at both surface, and maintained thereafter. If
DH2-sheet = 9 x 10-7 m2/s, how much time is required to bring the
density of dissolved hydrogen to a value of 1.2 kmole/m3 at the
center of the sheet?