Separation Processes-I
(ChE-206)
Lecture No. 4
Diffusion
Unimolecular Diffusion
● In UMD, mass transfer of component A occurs through
stagnant(undiffusible) B, resulting in a bulk flow.
● One component is diffusing and other is non-diffusing.
● In a sulphuric acid plant, dry air is required for burning of
sulphur.
● Air is dried in a packed tower.
● Moisture (A) diffuses through a layer or film of air (B) reached
the acid surface and gets absorbed in it.
● Moisture has a source (bulk air) and a sink (acid) while dry air
has a source but no sink.
● Air is non-diffusing, hence its flux will be zero.
● In UMD:
● Therefore:
● Simplify to get:
● The factor (1-xA) accounts for the bulk-flow effect. For a mixture dilute
in A, this effect is small.
● But in an equimolar mixture of A and B, (1-xA) = 0.5 and, because of
bulk flow, the molar mass-transfer flux of A is twice the ordinary
molecular-diffusion flux.
● For the stagnant component, B,
● Thus, the bulk-flow flux of B is equal to but opposite its diffusion
flux.
● At quasi-steady-state conditions (i.e., no accumulation of
species with time) and with constant molar density, in integral
form is:
Concentration Profile
● which upon integration yields
Home Assignment
● Derive this relationship:
● Go through Examples 3.1 and 3.2
Diffusion Co-efficients (Diffusivities)
● Defined for a binary mixture.
● The binary diffusivities, DAB and DBA, are called mutual or binary
diffusion coefficients.
● Other coefficients include DiM , the diffusivity of i in a multicomponent
mixture; Dii, the self-diffusion coefficient; and the tracer or
interdiffusion coefficient.
● The diffusion coefficients, are highest in gases and lowest in solids.
The diffusion coefficients of gases are several orders of magnitude
greater than those of liquids.
● Diffusion coefficients increase with temperature.
● Example: The diffusion coefficient (mass diffusion rate) of carbon
through iron increases by 6000 times as the temperature is raised
from 500°C to 1000°C.
Gases -6 5
>>(5*10 ----1*10- )
DAB -10 -9
2
(m /s)
Liquids>>(10 ----10 )
-14 -10
Solids>>(10 ----10 )
Diffusivity in Gas Mixtures
● As discussed by Poling, Prausnitz, and O’Connell, equations are
available for estimating the value of DAB and DBA in gases at low to
moderate pressures.
● Of greater accuracy and ease of use is the empirical equation of
Fuller, Schettler, and Giddings:
● The theoretical equations based on Boltzmann’s kinetic theory of gases
● Based on the theorem of corresponding states
● A suitable intermolecular energy potential function
● Based on Chapman and Enskog theory
● Predict DAB to be inversely proportional to pressure, to increase
significantly with temperature, and to be almost independent of
composition.
Diffusivity in Nonelectrolyte Liquid
Mixtures
● For liquids, diffusivities are difficult to estimate because of the lack of
a rigorous model for the liquid state.
● An exception is a dilute solute (A) of large, rigid, spherical molecules
diffusing through a solvent (B) of small molecules with no slip at the
surface of the solute molecules.
● The resulting relation, based on the hydrodynamics of creeping flow
to describe drag, is the Stokes–Einstein equation:
● Unfortunately, unlike for gas mixtures, where DAB and DBA, in liquid
mixtures diffusivities can vary with composition.
● This equation is restricted to dilute binary mixtures of not more than
10% solutes.
● Empirical Wilke–Chang equation:
● The Stokes–Einstein and Wilke–Chang equations predict an
inverse dependence of liquid diffusivity with viscosity
● While the Hayduk–Minhas equations predict a somewhat
smaller dependence.
● Liquid diffusivity varies inversely with viscosity raised to an
exponent closer to 0.5 than to 1.0.
Diffusivities of Electrolytes
● For an electrolyte solute, diffusion coefficients of dissolved
salts, acids, or bases depend on the ions.
● However, in the absence of an electric potential, diffusion only
of the electrolyte is of interest.
● Nernst–Haskell equation:
Diffusivities of Biological Solutes in
Liquids
● The Wilke–Chang equation is used3 for solute molecules of
liquid molar volumes up to 500 cm /mol, which corresponds to
molecular weights to almost 600.
● In biological applications, diffusivities of soluble protein
macromolecules having molecular weights greater than 1,000
are of interest.
● Molecules with molecular weights
-6 -9
to 500,000
2
have diffusivities
at 25°C that range from 10 to 10 cm /s, which is three orders
of magnitude smaller than values of diffusivity for smaller
molecules.
● The equation of Geankoplis can be used to estimate protein
diffusivities:
Diffusivity in Solids
● Diffusion in solids takes place by mechanisms that depend on
the diffusing atom, molecule, or ion; the nature of the solid
structure, whether it be porous or nonporous, crystalline, or
amorphous; and the type of solid material, whether it be
metallic, ceramic, polymeric, biological, or cellular.
● Diffusion coefficients in solids cover a range of many orders of
magnitude.
● Despite the complexity of diffusion in solids, Fick’s first law can
be used if a measured diffusivity is available.
● Porous Solids
● Crystalline Solids
● Direct exchange of lattice position, probably by a ring rotation
involving three or more atoms or ions
● Migration by small solutes through interlattice spaces called
interstitial sites
● Migration to a vacant site in the lattice
● Migration along lattice imperfections (dislocations), or gain
boundaries (crystal interfaces)
● Metals
● Important applications exist for diffusion of gases through metals. To
diffuse through a metal, a gas must first dissolve in the metal.
● Hydrogen dissolves in Cu, Al, Ti, Ta, Cr, W, Fe, Ni, Pt, and Pd, but not in
Au, Zn, Sb, and Rh.
● Silica and Glass
● The product of diffusivity and solubility is the permeability, PM.
● Ceramics
● Polymers
● Diffusion through nonporous polymers is dependent on the type of
polymer, whether it be crystalline or amorphous and, if the latter,
glassy or rubbery.
● To compute the mass-transfer flux, Fick’s first law is applied for
different species:
● Gas Species
● Liquid Species
Applications of Fick’s Law
● Mass transfer occurs in:
(1) stagnant or stationary media
(2) fluids in laminar flow
(3) fluids in turbulent flow
● Each requiring a different calculation procedure.
● In one dimension, the molar rate of mass transfer of A in a
binary mixture is given by following equation, which includes
bulk flow and molecular diffusion:
● If A is undergoing mass transfer but B is stationary, nB = 0.
● It is common to assume that c is a constant and xA is small.
● The bulk-flow term is then eliminated and equation becomes
Fick’s law:
● In terms of concentration:
Steady State Diffusion
● For steady-state, one-dimensional diffusion with constant DAB,
the flux equation can be integrated for various geometries.
1. Plane wall with a thickness, z2-z1:
2. Hollow cylinder of inner radius r1 and outer radius r2, with
diffusion in the radial direction outward:
3. Spherical shell of inner radius r1 and outer radius r2, with
diffusion in the radial direction outward:
Unsteady-State Diffusion
● Consider one-dimensional molecular diffusion of species A in
stationary B through a differential control volume with diffusion
in the z-direction only.
● Assume constant diffusivity and negligible bulk flow. The molar
flow rate of species A by diffusion in the z-direction is given by:
Home Assignment
● Go through Examples 3.3-3.8
Book
● Seader, J. D.; Henley, E. J.; Roper, D. K., Separation Process
Principles: Chemical and Biochemical Operations. 3rd Ed.; John
Wiley & Sons, Inc.: 2011.
● Chapter 3
