Various Separation Processes Have Certain Basic Principles That Can Be Classified into 3 Fundamental Transport Processes

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Where, Permeability (PM) is – [pic 22]

When there are several solids 1, 2, 3,…., in series and L1, L2,…, represents the thickness of each, then equation becomes –

[pic 23]

Where, PA1- PA2 is the overall partial pressure difference.

- Diffusion in porous solids that depends on structure

The solid sometimes may act as porous barrier or as porous catalyst pellets and is normally surrounded by a single body of fluid. The inward or outward movement of the solutes through the pores of the solid is mainly by diffusion. This movement may occur inside the pore or at the surface of the adsorbed solute.[pic 24]

In this case the diffusion can be understood in two parts –

- Diffusion Inside the pores

- Surface Diffusion[pic 25]

Diffusion inside pores

At low pressure the mean free path of the molecules may be larger than the diameter of the passage when the diffusion occurs inside the fine pores of the solid. The collision with wall becomes important compared to collision among molecules. The diffusion of this kind is known as “Knudsen diffusion”.

To quantify Knudsen diffusivity a simple Equation based on kinetic theory of gases was proposed as follows:[pic 26]

Where, rp radius of passage and v is the average velocity of the molecules due to their thermal energy which is defined as –

[pic 27]

Here, T is the temperature in K and M is the molecular weight.

The flux due to Knudsen diffusion is similar to Fick’s law:

[pic 28]

Surface diffusion

The diffusion of adsorbed molecules on the surface due to concentration gradient is kwon as surface diffusion. If the fractional coverage of the surface is less than unity then the some of the active sites remain empty. Adsorbed molecule having energy greater than the energy barrier tends to migrate to an adjacent vacant site. This migration is visualized to occur by “hoping” mechanism.

The flux due to surface diffusion may be written similar to Fick’s law:[pic 29]

Where, Ds is the surface diffusion coefficient (m2/s) and Cs is the surface concentration of the adsorbed molecules (kmol/m2). Js is the number of moles transported across unit distance on the surface normal to the direction of transport (kmol/m.s).

DIFFUSION IN LIQUIDS

In liquids, an increase in temperature increases the rate of diffusion. However, since liquids are incompressible, the rate of diffusion is not affected by the pressure. Diffusion is much slower in liquids as compared to gases. The order of magnitude of diffusion coefficient of gas is 105 times that of the diffusion coefficient of liquids

Diffusion of solutes in liquid is important for industrial processes like separation operations like solvent extraction, gas absorption and distillation.

EQUIMOLAR COUNTER DIFFUSION

From Fick’s law, we know

[pic 30]

For equimolar counter diffusion,

NA= -NB

[pic 31] (eq 1)

where

[pic 32]

Cav= average total concentration of A+B in kg/m3

M1= average molar mass of solution at point 1 in kg mass/kg mol

p1= average density of solution in kg/m3 at point 1

However, equimolar counter diffusion occurs very less in liquids.

DIFFUSION OF A THROUGH NON DIFFUSING B

This is one of the most important cases of diffusion in liquids. Since the liquid B is non diffusing hence the diffusion flux i.e. NB in equation 1 becomes 0.

As in the case of gases we got the equation

[pic 33]

This equation is rewritten in terms of concentration by substituting Cav with P/RT, pA1 with cA1 RT and pBM with PxBM, we get

[pic 34]

Where [pic 35]

Since xA1 + xB1 = xA2 + xB2=1, xBM comes close to 1 and c becomes constant giving,

[pic 36]

Question:

[pic 37]

DIFFUSION COEFFICIENT FOR LIQUIDS

In unsteady state diffusion coefficient is determined experimentally in long capillary tube. Diffusivity determined from concentration profile. Unlike gases DAB does not equal to DBA for liquids.

In a relatively common method a relatively dilute solution and a slightly more concentrated solution are placed with a porous membrane in between. Molecular diffusion takes place through the membrane while the two components are stirred. The effective diffusion length is K1ẟ(K1> 1). In this method, the effective diffusion length is obtained by calibrating with a solute such as KCl having a known diffusivity.

[pic 38]

To derive the equation, quasi-steady-state diffusion in the membrane is assumed.

[pic 39]

Where c is the concentration in the lower chamber at a time t, c' is the concentration in the upper, and e is the fraction of area of the glass open to diffusion. Making a balance on solute A in the upper chamber, where the rate in = rate out + rate of accumulation, making a similar balance on the lower chamber, using volume V = V', and combining and integrating the final equation is

[pic 40]

Where 2eA/K1V is a cell constant that can be determined using a solute of known diffusivity, such