Chapter 8.

FILTRATION

PART II.Filtration variables, filtration mechanisms

8.1 Filtration variables – input of the filtration process I.

Filtration variables are divided onto three groups:

1. Variables of filter material

2. Variables of filtered particles

3. Variables of filtration process

8.1.1 Variables of filter material:

•Filtration area

•Filter thickness

•Density and surface density of filter

•Uniformity of fibrous material

•Parameters of filter material•surface interactions between the filter material and filtered particles•electrical properties•mechanical characteristics (tenacity, elongation...)•resistance against surrounding factors (heat, solvents...)

•Parameters of fibers•fiber diameter, fiber fineness•shape of fiber cross-section•fiber surface preparations•Mechanical characteristics

•Filter structure•filter density gradient•fiber orientation

8.1.2 Variables of filtered particles

•Particle size

•Distribution of particle size

•Concentration of particles

•Shape and surface of particles

•Particle density

•Electrical properties

8.1.3 Variable of filtration process

•Face velocity (speed of filtered particles in front of filter)

•Viscosity of the flow

•Temperature, pressure, humidity

8.2 Filtration mechanisms of deep filtration(the way how are particles captured)

R

fiber

charge on the fiber surface

diffusional deposition

inertial impaction

direct interception

capture by electrostatic forces

streamlines (air moving trajectory)

Total filtration efficiency

Ec is total efficiency, Er is efficiency of direct interception mechanism represented by parameter Nr, Ei is efficiency of inertial impaction represented by Stokes number Stk, Ed is efficiency of diffusional deposition mechanism represented by Peclet number Pe and Ee is efficiency of electrostatic mechanism represented by the parameter Nq.

NqEPeEStkENEEE edirrcc ,,,

Mechanisms:

• direct interception

• inertial impaction

• diffusional deposition

• capture by electrostatic forces .

8.2.1 Direct interception

Direct interception occurs when airborne particles behave in an entirely passive way with respect to the airflow. Airborne particles follow the streamline, which in steady state are independent of the air velocity. Particle will be captured when it is close to the fiber. This mechanism is independent of air velocity, air viscosity and density. Particle must be small, because inertial effects and external forces are neglected. This type of mechanism is common for simple respirators made from fibers of about 20 m, which operate in filration velocity about 0,04 m/sec. Furthermore interception acts along with other filtration processes.

Parameter of direct interception:

Nr= dp/df

(dp is particle diameter, df is fiber diameter)

df

fiber

streamlines (air moving trajectory)

dp

Relation between parameter Nr and efficiency of direct interceptiom mechanism:

ER Nr2;

the simpliest relation is: ER=NR2/, more exactly:

where =-0,5.ln(c)-0,75 is hydrodynamic factor and m = 2/(3.(1-c)) mR

RR

N

NE

1.

2

8.2.2 Inertial impaction

Any convergence, divergence or curvature of streamlines involves acceleration of the air, and under such conditions a particle may not be able to follow the airflow. What particle does depends upon its mass (inertia) and upon the Stokes drag exerted by the air. Stokes drag is defined as a force which acts on the moving sferical object inside of viscous liquid: F = 3...dp.v (where F is the force, dp is the particle diameter, is the dynamic viscosity and v is the face velocity of the airflow).

fiber

inertial impaction

streamlines (air moving trajectory)

Intensity of the point particle inertia is determined by Stokes number:

where dp is particle diameter, is particle density, U is air face velocity, is air viscosity and df is fiber diameter.

f

p

d

UdSt

..18

..2

Efficiency of inertial impaction Ei depends on the intensity of the point particle inertia. If inertia is negligible then Ei will be zero, if the inertia is infinite then Ei will be 100 %.

Relation between the Stokes number and efficiency of inertial impaction:

For low Stokes number efficiency is lead by direct interception:

Eir=ER+(2.)-2.J.St,

where ER is efficiency of direct interception, is hydrodynamic factor dependent on packing fraction c and J is constant dependent on c and parameter of direct interception Nr.

For high Stokes number efficiency of inertial impaction is defined:

EI=1-(/St),

where is constant dependent on flow field.

8.2.3 Diffussional deposition

The trajektories of individual small particles do not coincide with the streamlines of the fluid because of Brownian motion. With decreasing particle size the intensity of Brownian motion increases and, as a consequence, so does the intensity of diffusion deposition [Pich J,1964]. However the air flow effects on the particles motion too. Thus the real motion of small particles depends on Brownian motion and air flow.

Brownian motion is determined by diffusion coefficient D defined by the Einstein equation:

where kB is Boltzmann constant, K is Kelvin temperature, is air viscosity, dp is particle diameter and Cn is the Cunningham correction, which involve aerodynamic slip flow of particles:

where is mean free path of molecules (at NTP it is 0,065 m) and A, B, Q are constants (A=1,246; B=0,87; Q=0,42) [Brown RC, 1993].

diffusional deposition

streamlines (air moving trajectory)

fiber

p

B

d

TkCnD

...3

..

.2

.

...2

1pdB

p

eQAd

Cn

Coefficient of diffusional deposition:

Capture of particles by a diffusional deposition will depend on the relation between the diffusional motion and the convective motion of the air past the fiber. Dimensionless coefficient of diffusional deposition ND is defined:

where df is fiber diameter, U is air flow velocity and Pe is named „Peclet number“.

Diffusional capture efficiency:

According to Fokker-Planck equation was aproximated relation between the ND (or 1/Pe) and diffusional capture efficiency

ED = 2,9 . -1/3 . Pe-2/3

where is hydrodynamic factor ( = -0,5. ln(c)-0,75 by Kuwabara) [Brown RC, 1993].

Previous equation was verified by experiments with model filters with the some and observed functional dependance was the some with little different numerical coefficient:

ED = 2,7 . Pe-2/3

When we calculate with the slip flow (see chapter 9) the resulting capture efficiency is bigger.

Ud

D

PeN

fD .

1

8.2.4 Electrostatic forces:

Both the particles and the fibers in the filter may carry electric charges. Deposition of particles on the fibers may take place because of the forces acting between charges or induced forces. [Pich J, 1964]. The capture of oppositely charged particles is given by coulomb forces. The capture of neutral particles comes about by the action of polarisation forces. We can define three cases of interaction between particle and fiber. Used equations were derived from Coulomb´s law.

fiber

charge on the fiber surface

capture by electrostatic forces

streamlines (air moving trajectory)

1. Charged particle, charged fiberwhere q is the particle charge, Q is fiber charge per unit lenght of fiber and x is the distance between fiber and particle.

2. Charged fiber, neutral particleswhere D1 is the dielectric constant of the particle and dp is particle diameter.

3. Charged particles, neutral fiberwhere D2 is dielectric constant of the fiber and df is fiber diameter.

x

qQF

..21

3

3

1

122 .

2

1..4

x

d

D

DQF p

1

1.

.4 2

22

2

3

D

D

dx

qF

f

Coefficient of electrostatic mechanism, efficiency of electrostatic mechanism

We can interpret this parameter as a ratio of electrostatic forces to drag forces. From this parameter were derived equations for efficiency [Pich J, 1964].

B is mechanical mobility of the particle, U0 is the velocity far form the fiber, df is fiber diameter, dp is particle diameter and is viscosity

Coefficient of electrostatic mechanism

Efficiency of electrostatic mechanism

Charged fiber and charged particle

Charged fiber and neutral particle

Carged paricle and neutral fiber

00 .....3

..4

.

...4

Udd

qQ

Ud

BqQN

fpfQq

..

..

2

1.

.3

4

03

22

1

10 Ud

Qd

D

DN

f

pQ

1

1.

....3 2

2

02

2

0

D

D

Udd

qN

fpq

0....3

..4.

Udd

qQNE

fpQqQq

2

1

02

1.

Reln2

2qQq NE

31

0

31

0 .2

3QQ NE

8.3 Filtration variables vs.capture efficiency of filtration mechanisms

Efficiency of each filtration mechanisms

Relations how some filtration variables increase or decrease or not affect the efficiency of each filtration mechanisms

filter density

fiber diameter

particle diameter

particle mass

face velocity

viscosity of air

relative charge

direct interception

- - - - -

inertial impaction

? -

diffusional deposition

- -

electrostatic deposition

- -

32

1

dp

d

e

p

d p

Efficiency of each filtration mechanisms

Numeric relations between the filter variables and capture efficiency of each mechanisms

filter density

c

fiber diameter

df

particle diameter

dp

particle mass

face velocity

U

viscosity of air

relative chargeq, Q

direct interception

- 

1/df2 dp

2 - - - -

inertial impaction

1/(ln c)21/df or

1 – k.df

dp2 or

1-1/dp2

or 1-k/

U or1-k/U

1/ -

diffusional deposition

1/(ln c)1/3

1/df2/3 - 1/U2/3 1/2/3 -

electrostatic deposition

- 1/df

1/dp or

dp2/3 or

1/dp1/2

-1/U or1/U1/3 or1/U1/2

1/q.Q or Q2/3 orq

8.4 Filtration mechanism of flat filtration – „Sieve effect“

Es = 1 for dp dpore; ; Es= 0 for dp < dpore,

where Es is efficiency of sieve effect and dpore is pore diameter.

Relation between fiber and pore diameter according to Neckar [Neckar B., 2003]:

()

where q is fiber shape factor (zero for cylindrical fibers), c is packing factor, a and k are constats related to filter structure (usually a is ½).

For cylindrical fibers with hexagonal structure is k = 2-1/2.

.dpore fd

a

fpore c

c

q

kdd

1.

1.