What is Diffusivity and how it is related to Mass Transfer and its Coefficient, How they are related



Diffusivity ‘D’ is defined as the ratio of the molar flux to the corresponding concentration gradient and its units are m2/s. The diffusivity of a component mean it tells about the mobility characteristic of the component and it is a function of temperature, pressure, nature, and concentration of the other components. Diffusivity of gases at atmospheric pressure in cm2/s is in the range less than 1 and for liquids is of the order 1 x 10-5. Diffusivity of a gas generally varies with temperature and pressure according to the relation DµT1.5/P and for liquids it varies by D µT. Diffusivity of liquids can be estimated by Wilke–Chang equation.

 Mass transfer coefficient ‘k’ is defined as molar flux = (mass transfer coefficient) X (concentration difference). We consider concentration difference not concentration gradient and the units will change according to the choice of the concentration selection, which we take into consideration.  If mole/ volume is used then the units are cm/s and if mole fraction is choosed then the unit will be the units of flux, (mole/ cm2 s) due to the reason that mole fractions are dimensionless. The ratio of mass flux for diffusion of A to the mass flux through non-diffusing B for equimolar counter-diffusion is greater than one. The mass transfer coefficients, kg and ky are related according to the relation kG/P = kY/P2. According to the film theory, the mass transfer coefficient, kl, and diffusivity are related as kl µ D as boundary layer theory predicts that kl α D 0.67. For mass transfer of a solute A present in a dilute mixture of A and B, the term PB,M tends to total pressure P.

Relation between diffusivity and mass transfer coefficient

Mass transfer coefficient is the ratio of molecular diffusivity to the thickness of stagnant layer (given by film theory)

Theories which explain about mass transfer coefficient calculation are:

1. Film Theory: considered as steady state model

2. Boundary Layer Theory

3. Penetration Theory

4. Surface Renewal Theory

5. Surface Stretch Theory

6. Combination of Film and Surface Renewal Theory

Some dimensionless groups

Corresponding to Prandtl number in heat transfer, the dimensionless group in mass transfer is Schmidt number (μ/ρDv).
Schmidt number is a ratio of momentum diffusivity to mass diffusivity. Schmidt number for gases is of the order of 1 and for liquids is 0.1.
Sherwood number (kcD/Dv) is a ratio of flow velocity to diffusion velocity or Convective flux to diffusive flux. For evaporation from a spherical naphthalene ball in a stagnant medium, Sherwood number is equal to 2. 
Corresponding to Nusselt number in heat transfer, the dimensionless group in mass transfer is Sherwood number.  
Stanton number for mass transfer is defined as Sh / (Re.Sc). Reynolds analogy   gives St = f / 2.


According to Danckwerts surface renewal theory, the mass transfer coefficient, kl’ is given by (DAB .S)0.5."S" in Danckwerts surface renewal theory is fraction of the surface renewed per unit time.
According to the penetration theory, the average mass transfer coefficient kL,av is given by
2 (DAB / p t)0.5.
For example let a certain mass transfer process, kl = 1 x 10 - 3 cm/s and DAB = 1 x 10-5 cm2/s the film thickness in cm is 0.01cm. 
The Knudsen diffusivity is dependent on the molecular velocity and pore radius of the catalyst.  A gaseous solute having mass diffusivity equal to 0.5cm2/s diffuses into a porous solid having a porosity of 0.5 and a porosity of 2 then the effective diffusivity in the porous solid is 0.125 cm2/s. Knudsen diffusion occurs when the ratio of mean free path to the pore diameter is much greater than one. In Knudsen diffusion molecule – pore wall collision is important. Knudsen diffusivity is independent of total pressure it increases with the square root of temperature and inversely with the square root of molecular weight, it falls in the range of 10 –1 to 10 – 4 cm2/s.
The term permeability is defined as permeability=solubility X diffusivity

 



Approximate Diffusivities of gases at standard atmospheric pressure, 101.325 KPa:
 
s.no
System
Temperature, 0C
Diffusivity,m2/s  X 10-5
Reference
1
H2 - CH4
0
6.25
Chapman,s.&T.G. Cowling
2
O2 - N2
0
1.81
,,
3
CO – O2
0
1.85
,,
4
CO2 – O2
0
1.39
,,
5
Air – NH3
0
1.98
Wintergeist
6
Air – H2O
25.9
59.0
2.58
3.05
Gilliland
7
Air – C2H5OH
25.9
1.02
Gilliland
8
Air – n-Butanol
25.9
59.0
0.87
1.04
International critical table
9
Air – Ethyl Acetate
25.9
59.0
0.87
1.06
Gilliland
10
Air – Aniline
25.9
59.0
0.74
0.90
Gilliland
11
Air – Chlorobenzene
25.9
59.0
0.74
0.90
Gilliland
12
Air - Toluene
25.9
59.0
0.86
0.92
Gilliland

Approximate Diffusivities of Liquids at 1 atm, pressure:




s.no
System
Temperature,
 0C
Solute con:
Kmole/m3
Diffusivity,
m2/s  X 10-9
solute
solvent
1
Cl2
Water
16
0.12
1.26
2
HCl
Water
0
,,
10
,,
16
9
2
9
2.5
0.5
2.7
1.8
3.3
2.5
2.44
3
NH3
Water
5
15
3.5
1.0
1.24
1.77
4
CO2
Water
10
20
0
0
1.26
1.21
5
NaCl
Water
18
,,
,,
,,
,,
0.05
0.2
1.0
3.0
5.4
1.24
1.36
1.54
1.28
0.82
6
Methanol
Water
15

0
0.91
0.96
7
Acetic acid
Water
12.5
,,
18
1.0
0.01
1.0
0.50
0.83
0.90
8
Ethanol
Water
10
,,
16
3.75
0.05
2.0
0.50
0.83
0.90
9
n-butanol
Water
15
0
0.77
10
Co2
Ethanol
17
0
3.2
11
chloroform
ethanol
20
2.0
1.25