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 of less than 1 and for liquids is of the order 1 x 10-5. The 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. The 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 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 the 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.

The relation between diffusivity and mass transfer coefficient: 

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

Theories that explain 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 the Prandtl number in heat transfer, the dimensionless group in the 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’s number is equal to 2.

Corresponding to the Nusselt number in heat transfer, the dimensionless group in the mass transfer is the Sherwood number.

Stanton number for mass transfer is defined as S_h / (R_e.S_c). Reynolds analogy gives St = f / 2.

According to Danckwert’s surface renewal theory, the mass transfer coefficient, k_l’ is given by (D_AB .S)^0.5.”S” in Danckwert’s surface renewal theory is a fraction of the surface renewal 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 the 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:

Approximate Diffusivities of Liquids at 1 atm, pressure:

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