3. Diffusion of solutes in a porous solid
Solute molecules can diffuse through the pores present in porous solids. In order for
this to happen, the pores have to be filled with some liquid medium. Therefore no
diffusion takes place through the solid material itself. All it does is hold the liquid
medium in place. However, the solid material can have an influence on the diffusion
within the liquid medium. It can for instance increase the effective diffusion path
length of the solute if the pores are tortuous in nature. When the pores have dimension
of the same order of magnitude as the solute, the pore wall can cause hindrance to
diffusion. An example of un-hindered diffusion in a porous medium is the transport of
sodium chloride through a microfiltration membrane (which has micron sized pores)
while an example of hindered diffusion is the transport of albumin through an
ultrafiltration membrane (which has nanometer sized pores). The steady state equation
for unhindered diffusion of a solute from point 1 to 2 within a slab of porous solid is
given by:
Where
The hindered diffusion of a solute through a porous solid from point 1 to 2 is given
by:
Where Deff = effective hindered diffusivity (m2/s).
The effective hindered diffusivity of a solute in a pore can be obtained by:
Where
ds = solute diameter (m)
dp = pore diameter (m)
It should be appreciated that when the pore size is very large as compared to particle
or molecular size, the term in the parenthesis will essentially equal 1, and thus:
Deff = D
Which is of course the case for unhindered diffusion.
Example
Glucose is diffusing at 25 degrees celsius in water within a porous medium having a
porosity of 0.5, tortuosity of 1.8 and average pore diameter of 8.6x 10-3 microns.
Determine the steady state flux of glucose between two points within the medium
separated by a distance of 1 mm and having concentrations 1.5 g/L and 1.51 g/L
respectively.
Solution
Table 2.2, gives the molecular weight of glucose as being 180 kg/kgmole. The
diffusivity of glucose at 25 degrees centigrade can be determined using Poison
correlation:
The size of glucose can be obtained from appropriate tables. The effective diffusivity
of glucose in the porous structure can be obtained using the equation:
The steady state flux of glucose can be obtained using equation:
The molar concentration of glucose = (0.01/180) mol/L
Convective mass transfer
Convective transport occurs when a constituent of the fluid is carried along with the
fluid. The amount carried past a plane of unit area perpendicular to the velocity (the
flux) is the product of the velocity and the concentration
NA = kcA(C0 – CA), or
NA = kcA∆CA
Where
NA is the flux in convective mass transfer
kcA is the convective mass transfer coefficient
(C0 – CA) is the solute concentration difference between two points
Convective mass transfer is observed in flowing fluids can be emphasized for two
situations:
1. Transport of a solute in a liquid flowing past a solid surface (see Fig. 3.5)
2. transport of a solute in a liquid flowing past another immiscible liquid (see
Fig. 3.6).
An example of the first type is the transfer of urea from blood towards the surface of a
dialyser membrane in haemodialysis.
An example of the second type is the transfer of penicillin G within filtered aqueous
media flowing past an organic solvent in a liquid-liquid extractor.
When a liquid flows past a solid surface a stagnant boundary liquid layer is formed
close to the surface. Similarly when two liquids flow past one another, two boundary
liquid layers are generated on both sides of the interface.
Within these boundary layers, the transport of solute mainly takes place by molecular
diffusion. If the flow of liquid is laminar, the transfer of solute in the directions
indicated in Figs. 3.5 and 3.6 would be by molecular diffusion. However, if the flow
were turbulent in nature, mass transfer would take place by a combination of
molecular diffusion and eddy diffusion. This is referred to as convective mass
transfer.
Another argument of convective mass transfer can be mentioned for a solute band
progresses along a column, where the solute molecules are continually transferring
from the mobile phase into the stationary phase and back from the stationary phase
into the mobile phase. This transfer process is not instantaneous, because a finite time
is required for the molecules to traverse (by diffusion) through the mobile phase in
order to reach, and enter the stationary phase. Thus, those molecules close to the
stationary phase will enter it almost immediately, whereas those molecules some
distance away from the stationary phase will find their way to it a significant interval
of time later. However, as the mobile phase is moving, during this time interval while
they are diffusing towards the stationary phase boundary, they will be swept along the
column and thus dispersed away from those molecules that were close and entered it
rapidly. The dispersion resulting from the resistance to mass transfer in the mobile
phase is depicted in the figure below. The diagram shows 6 solute molecules in the
mobile phase and those closest to the surface (1 and 2) enter the stationary phase
immediately. During the period, while molecules 3 and 4 diffuse through the mobile
phase to the interface, the mobile phase moves on. Thus, when molecules 3 and 4
reach the interface they will enter the stationary phase some distance ahead of the first
two. Finally, while molecules 5 and 6 diffuse to the interface the mobile phase moves
even further down the column until molecules 5 and 6 enter the stationary phase
further ahead of molecules 3 and 4. Thus, the 6 molecules, originally relatively close
together, are now spread out in the stationary phase. This explanation is a little oversimplified, but gives a correct description of the mechanism of mass transfer
dispersion.
The same phenomenon occurs when solutes transfer between two immiscible moving
liquids in contact with one another.