H. FALLAH ET AL.

58

cles with the retention site, the fixing of particle sites and

the breaking away of previously retained particles [6].

2.2. Mechanisms of Particles Retention

The particle capture in porous media can be caused by

different mechanisms [7].

• Size exclusion (large particles are captured in small

pores and pass through large p ores );

• Electrical forces (London-Van der Waals, double

electrical layer, etc.);

• Gravity segregation;

• Multi particle bridging.

3. Basic Formulation of Particles Transport

through Porous Medium

The model includes transport parameters which are parti-

cle advective velocity, particle longitudinal dispersion

coefficient and filter coefficient. These parameters have

been defined by dimensional analysis using the pertinent

variables of the porous medium system.

3.1. Advective Velocity

The averaged particle velocity in the porous media, has

been found to be either the same or slightly higher than

that of the carrier fluid. This deviation is caused by the

particle’s size. The expected difference can be deter-

mined by analyzing the velocity profiles of both the fluid

and the particles in a pore. The model has been formu-

lated for a capillary tube which has a constant rate with

the following assumption: No interactions between the

particles and the wall, suspension is well-mixed with a

constant concentration across the cross section .There is

no tran sverse flow.

2

0

(0 )rr

1

SO

o

r

UU r

0

(0)

22

1

p

PO

o

o

a

r

UU rr

rr a

S

V

(4)

As the particle travels through a tube, Brownian mo-

tion and shear action will cause the particle to travel

across the entire cross-section of the tube except that the

center-line of the particle will be excluded from the im-

mediate region of the wall due to its radial dimension.

After the particle has traveled far enough longitudi-

nally through the tube, the particle will have spent equal

amounts of time in all radial position across the capillary

tube. Integration of the velocity profile of Equation (4)

over the range of possible radii shown in Equation (4) for

both the particle and fluid yields the higher average ve-

locity for the particle than that for the carrier flu id.

The av erage flu i d velocity, in a capillary tube is:

2

O

S

U

V (5)

The average velocity of a particle,

V in a capillary

tube is:

22

1

11

2

pp

PO

oo

aa

VU rr

(6)

By inspecting Equation (5) and (6), the particles are

expected to have a larger average velocity than the car-

rier fluid velocity. This enhanced velocity of the particle

can be expressed as a fractional difference between the

two average velocities:

2

7

23

pp

PS

Soo

aa

VV

VVrr

V

(7)

This equation shows that as the radius of the particle

increase, the difference between the average particle ve-

locity and the average fluid interstitial velocity also in-

crease. This increase is not unbounded but reaches a

maximum

value for 3

7

p

o

a

r; as 3

7

p

o

a

r the

velocity difference decrease. In physical sense, the pore

radius can be estimated to approximately equal to

one-fifth of media grain diameter (1

5

og

rd); therefore

the largest possible particle to be able to fit through the

porous bed has a radius to this pore radius (

o

ar).

For a particle with

o

ar

, the particles have been

shown to collect on the bed surface in a cake [6]. These

references show that the onset of deep bed filtration oc-

curs for a particle radius

a.less than

one-twentieth of the media grain diameter (1

20

g

ad

5

).

Particles with the radii larger than this will not transport

into the bed, but will collect on the surface. By lettingc

o

dr

, the largest particle which will transport has a

radius equal t o one-fourth of the pore radius (1

4

o

ar).

3.2. Longitudinal Dispersion Coefficient

An important element of any dispersion model is the re-

presentation of the geometry of the porous medium.

Houseworth [8] has thoroughly reviewed such longitudi-

nal dispersion model for solute tracers. Instead of mod-

eling the internal structure of a porous medium, dimen-

sional analysis is used to analyze the problem. In this

study, the effect of mechanisms is expected to scale with

the pertinent transport variables.

The pertinent variables for solute dispersion are:

DL = longitudinal dispersion coefficient (L·T–1)

D = free fluid molecular diffusion coefficient of

Copyright © 2012 SciRes. GM