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 means 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 DµT^1.5/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 when the units are cm/s and if mole fraction is chosen 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, k_g and k_y are related according to the relation k_G/P = k_Y/P². According to the film theory, the mass transfer coefficient, k_l, and diffusivity are related as k_l µ D as boundary layer theory predicts that k_l α D ^0.67. For mass transfer of a solute, A present in a dilute mixture of A and B, the term P_B,M tends to total pressure P.

The relation between diffusivity and mass transfer coefficient: 

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

Theories which explain about mass transfer coefficient calculation are:

1. Film Theory: considered as a 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/ρD_v).

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 (k_cD/D_v) 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 S_h / (R_e.S_c). Reynolds analogy   gives St = f / 2.

According to Danckwerts surface renewal theory, the mass transfer coefficient, k_l’ is given by (D_AB .S)^0.5."S" in Danckwerts surface renewal theory is a fraction of the surface renewed per unit time.

According to the penetration theory, the average mass transfer coefficient k_L,av is given by

2 (D_AB / p t)^0.5.

For example let a certain mass transfer process, k_l = 1 x 10 ^{- 3} cm/s and D_AB = 1 x 10^-5 cm²/s the film thickness in cm is 0.01cm. 

The Knudsen diffusivity is dependent on the molecular velocity and a 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 the 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