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Molecular Diffusion in Gases
Diffusion plus Convection
 =

−

+
 +
 (
)

Molecular Diffusion in Gases
Equimolar Counterdiffusion
In terms of mole fraction,
A
B
 = 
∗

A
B

= −

Molecular Diffusion in Gases
Uni-component Diffusion


 = −
+ 


 
 = −
 
http://sst-web.tees.ac.uk/external/U0000504/Notes/ProcessPrinciples/Diffusion/Default.htm
Molecular Diffusion in Gases
Example
Water in the bottom of a narrow
metal tune is held a t a constant
temperature of 293 K. The total
pressure of air (assumed dry) is
1.01325  105 Pa and the temperature
is 293 K.
Water evaporates and diffuses through the air in the tube, and the
diffusion path z2-z1 is 0.1524m long. Calculate the rate of evaporation
of water vapor at 293 K and 1 atm pressure. The diffusivity of water
in air is 0.250 x 10-4 m2/s. Assume that the system is isothermal.
Introduction to
Mass Transfer II
Outline
3. Molecular Diffusion in Gases
Diffusion with Varying Cross-sectional Area
4. Molecular Diffusion in Liquids
5. Molecular Diffusion in Solids
6. Prediction of Diffusivities
Molecular Diffusion in Gases
Example:
Diffusion through a varying cross-sectional area
A sphere of naphthalene having a radius of 2.0 mm
is suspended in a large volume of still air at 318 K
and 1.01325x105 Pa. The surface temperature of
the naphthalene can be assumed to be at 318 K and
its vapor pressure at 318 K is 0.555 mm Hg. The DAB
of naphthalene in air at 318 K is 6.92x10-6 m2/s.
Calculate the rate of evaporation of naphthalene
from the surface.
Molecular Diffusion in Gases
Given:
DAB = 6.92x10-6 m2/s
pA1 = (0.555/760)*(101325) =
74.0 Pa
pA2 = 0
r1 = 0.002 m
  
 = −
 
* The radius of the sphere
decreases slowly with time
Molecular Diffusion in Gases
  
 = −
 
ṅ
 

=−

( −  ) 
Where A = 4 2
Substitution and rearranging,
ṅ
 
 = −

2
4
( −  )
Molecular Diffusion in Gases
∞
1
ṅ
 
 = −
2
4

2
1
1

( −  )
The left side of the equation will be
∞
1
ṅ
ṅ
 =
2
4
4
∞
1
1
ṅ 1
1
 =
[ − ]
2

4 1 ∞
Molecular Diffusion in Gases
∞
1
ṅ
 
 = −
2
4

2
1
1

( −  )
The right side of the equation will be
 
−

2
1
1
   − 1
 = −

 − 

 − 2
Molecular Diffusion in Gases
∞
1
ṅ
 
 = −
2
4

2
1
1

( −  )
Solving for the rate of evaporation,
   − 1
ṅ = −41


 − 2
ANS: 4.9 x 10-9 mol/s
Outline
3. Molecular Diffusion in Gases
Diffusion with Varying Cross-sectional Area
4. Molecular Diffusion in Liquids
5. Molecular Diffusion in Solids
6. Prediction of Diffusivities
Molecular Diffusion in Liquids
Gas Model
Gases are made of
continuous free space
throughout which are
distributed moving
molecules.
For gases,
Kinetic theory is well developed
 ≫ 
http://www.bbc.co.uk/bitesize/ks3/science/chemical_material_behaviour/behaviour_of_matter/revision/4/
Molecular Diffusion in Liquids
Liquid Model
A continuous phase of
arranged molecules close to
each other but held together
by strong intermolecular forces
Dispersed throughout the
phase are “holes” of free space
The structure is more complex.
Molecular Diffusion in Liquids
Rate of Diffusion
 > 
BUT only about 100 times faster….
Molecular Diffusion in Liquids
Equations for Diffusion
 =

−

+
 +
 (
)

1. For equimolarcounterdiffusion,  = −

 = −

where



 = = ( ) + ( )



Molecular Diffusion in Liquids
Equations for Diffusion
 =

−

+
 +
 (
)

2. For unicomponent diffusion,  = 0
 
 = − (1 + )
 
NOTE:


 +  = ( )

( )  average value for the

molar density of the mixture
Molecular Diffusion in Liquids
Example
An ethanol (A) – water (B) solution in the form of a
stagnant film 2.0 mm thick at 293 K is in contact at one
surface with an organic solvent in which ethanol is
soluble and water is insoluble. Hence NB = 0. At point 1
the concentration of ethanol is 16.8 wt% and the
solution density ρ1 = 972.8 kg/m3. At point 2 ethanol
concentration is 6.8 wt% and ρ2 = 988.1 kg/m3. The
diffusivity of ethanol is 0.740x10-9 m2/s. Calculate the
steady-state flux NA.
Outline
3. Molecular Diffusion in Gases
Diffusion with Varying Cross-sectional Area
4. Molecular Diffusion in Liquids
5. Molecular Diffusion in Solids
6. Prediction of Diffusivities
Molecular Diffusion in Solids
What do we expect?
Rate of Diffusion
 >  > 
Outline
3. Molecular Diffusion in Gases
Diffusion with Varying Cross-sectional Area
4. Molecular Diffusion in Liquids
5. Molecular Diffusion in Solids
6. Prediction of Diffusivities
Predicting Diffusivities
For gases at low density
- almost independent of concentration
- increase with temperature
- vary inversely with pressure
For liquids and solids,
- strongly concentration-dependent
- generally increase with temperature
Predicting Diffusivities
Empirical Equations
For gases,
1. See Table 2-324 Perry’s
2. Chapman and Enskog Equation
DAB = diffusivity in m2/s
T = temperature in K
MA = molecular weight of A in kg/kmol
MB = molecular weight of B in kg/kmol
σAB = average collision diameter
ΩD,AB= collision integral based on
Lennard-Jones potential
Predicting Diffusivities
Empirical Equations
For gases,
3. Gilliland Equation
 =
1
1
1.38  10−7  3 ( +  )


1
(3
+
1
3 )2
DAB = diffusivity
T = temperature
MA = molecular weight
V = molar volume
P= pressure
Predicting Diffusivities
Empirical Equations
For liquids,
4. See Table 2-325 Perry’s
5. Stokes Einstein Equation
4. Wilke and Chang Equation
Predicting Diffusivities
All diffusivities have units m2/s
Therefore, their ratios are dimensionless groups
Dim. Group
Ratio
Equation
Prandtl, Pr
molecular diffusivity of momentum /
molecular diffusivity of heat
Schmidt, Sc
momentum diffusivity/ mass diffusivity
Lewis, Le
thermal diffusivity/ mass diffusivity
 

ν



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