**Fick's law of diffusion** a relation between the flux of the diffusing substance and the concentration gradient responsible for molecular diffusion-mass transfer was first proposed by FICK in 1855 and is therefore referred to as the **Ficks law of diffusion. Film theory mass transfer.**

## Fick's Law

The flux of diffusion component A (diffusion flux of A) in the z-direction in a binary mixture of A and B is proportional to the molar concentration gradient.

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So Fick's law diffusion for a binary mixture for steady-state diffusion can be expressed as

**Fick's law of diffusion formula JA = - DAB. dCA/dz**

**Where, ****JA **is the molar flux of A in the z-direction relative to the molar average velocity [moles per unit area per unit time, kmol/m2.s]

**CA **is the concentration of A [moles of A per unit volume]

**dCA/dz** is the concentration gradient in the z-directions.

**D**AB is the proportionally constant, known as the molecular diffusivity or diffusion coefficient for component A diffusing through B, m2/s, and distance in the direction of diffusion m. The negative sign indicates/implies that diffusion occurs in the direction of a drop in concentration.

## Mass Transfer Coefficient

In most mass transfer operations, the turbulent flow is desired to increase the rate of transfer per unit area. In such cases, the mass transfer rate is expressed in trams of mass transfer coefficients. In turbulent flow, there are three regions of mass transfer but as it is very difficult to know the value of distance in turbulent regions through which diffusion occurs, it is considered that the entire resistance to mass transfer lies in a laminar sublayer of thickness z. The resistance offered by the effective laminar film thickness is approximately the same as the combined resistance offered by the three regions in turbulent motion. The flux equations for such situations are of the types.

#### Mass transfer coefficients formula:

**Flux** = (Mass transfer coefficient) X (concentration difference)

As the concentration can be expressed in some ways, we have a variety of types of concentration coefficients.

## Film Theory Mass Transfer

### The salient feature of Whitman's two-film theory is

1. Resistance to transfer in each phase is regarded as lying in a thin film close to the interface.

2. The transfer in these films is by a steady-state process of molecular diffusion.

3. The concentration gradient is assumed to be linear in these films and it is zero outside the films, i.e., zero in the bulk fluid.

4. The theory assumes that the turbulence in the bulk fluid dies out at the interface of the films.

5. The film's capacity is negligible i.e., the time taken for the concentration gradient to establish is small compared to the time of transfer.

### Film-penetration theory mass transfer

The penetration theory was proposed by Higbie. The main feature of this theory is:

1. As the time of exposure of fluid for mass transfer generally being short development/establishment of the concentration gradient of film theory is not possible.

2. The transfer is large because of fresh material brought to the interface by the eddies.

3. A process of unsteady-state transfer occurs for a fixed period at the freshly expressed surface.

4. Each fluid element resides for the same length of the period at the surface according to this theory, the mass transfer coefficient is proportional to the square root of the diffusivity.

### Surface Renewal Theory Definition

According to this theory proposed by Danckwerts

1. The eddies of the fluid at the surface are exposed to varying lengths of time.

2. Based on exposure-time histories, an age distribution for the surface elements is calculated.

3. The mass transfer coefficient is proportional to the square root of the diffusivity regardless of the nature of the surface renewal rate.

Take these Notes is, Orginal Sources: Unit Operations-II, KA Gavhane