For the mass conservation, the equation of continuity for binary mixture is
 n
 t
A

(
 
J
A
)

R
A where J
A
is the flux for species A and is determined by the transport mechanisms.
1. Convection: Transported by the bulk flow of fluid, thus, in the same direction as bulk flow of fluid
J p

V p
C p

VC p
where V p
and V are the velocities of particle and fluid, respectively. For the bulk flow, both are generally assumed to be the same without the effect of inertia force. For convection only, the continuity equation is
 n
 t
A

V p

C p

R
A
because
 
V p

0 .
2. Brownian diffusion: Transported by the thermal random motion or Brownian motion of particle, thus, transported in all direction with the same magnitudes
J p
 
D p

C p
or J px
 
D p dC p dx
where D p
is the Brownian diffusivity of particle and it is the same in every direction. The Brownian diffusivity is generally assumed to be the same in each system except for large variations in temperature and concentration. The Brownian diffusivity is only state dependent for different size of particles as the following equation:

D p
 k T
6
B
 d p
[ 2

10

4
Pd p
( 12 .
64

4 .
02 e

1095 Pd p )] where k
B
= the Boltzmann’s constant = 1.32*10
-16
erg/molecule K
P = pressure = 76 cmHg

= viscosity of air = 1.85*10
-4
g/cm sec
T = temperature
D p

1 .
136 * 10

7 d p
[ 2

1
76 d p
( 12 .
64

4 .
02 e

8 .
322 d p )] where d p
is unit of
 m
Comparison of Brownian diffusivities for particles at different sizes and gaseous
SO
2
in the ambient atmosphere
Particle size (
 m) 0.1 1 10
Diffusivity (cm
2
/sec) 4.42*10
-6
2.46*10
-7
2.29*10
-8
SO
2
(g)
0.12
Therefore, the Brownian diffusivity of particles is much smaller than that of gaseous species in the ambient atmosphere with the difference of at least four orders of magnitudes. The significant differences in Brownian diffusivities are used to separate the particle and gas samplings, like the annular denuder and honeycomb. In order to sample particle and gaseous species simultaneously, particles are collected with filter and the gaseous species are sampled in the following air stream by various methods in the traditional filter pack system.
That is, particles are collected first and then gaseous species in the filter pack system. However, significant error may occur for some chemical species, like
NH
4
NO
3
, due to the sampling artifacts of evaporization, condensation, and so on.
For annular denuder or honeycomb, gaseous species are collected first and then particles and their evaporation effects sampled together. Note that in order to have Brownian diffusion as the only effective transport mechanism, the flow needs to be in laminar flow.
For Brownian diffusion only, the continuity equation is
 n
A
 t

D p
 2
C p

R
A and for both convection and diffusion, the continuity equation is
 n
 t
A

V p

C p

D p
 2
C p

R
A
and it is called the convective diffusion equation.
Therefore, the effect of another mechanism is to add additional term for representing its contribution if there are no interactions among various transport mechanisms.

Example : The particle transport rate to circular vertical tube in laminar flow as an approximation to the annular denuder at steady state.
The governing equation for particle transport is
V p

C
 z p 
D p
1 r

 r

 r

C
 r p

 where V p

V since
U r

C p
 r

C p
 t


U

0 due to the steady state assumption,

C p
 

0 due to and U r

U


0 (no flow radial and circular direction), the Brownian diffusion in z-axis direction is neglected since it is much smaller than the convective term, and the Brownian diffusion in circular direction is zero due to the symmetrical condition.
The velocity distribution for laminar flow in vertical circular tube is
V ( r )

2 V av
[ 1
 r
(
R
)
2
] where R is the radius of tube, r is the radial distance from center, and V av
is the average velocity across the section. Thus, the governing equation is
2 V av




1
 r
R
2




C p
 z

D p
1 r

 r

 r

C p
 r

 with the boundary conditions C p
( r , 0 )

C o
, C p
( R , z )

0 , and

C
 r p r

0

0 .
3. Eddy diffusion: Transportation of particles by turbulent eddy or in turbulent flow, thus, transport rates vary in different directions due to the differences in

eddy diffusivities.
J p
   p

C p
or J px
   p dC p dx
where
 p
is the eddy diffusivity of particle and it is different is different directions due to the non-isotropic flow pattern and the effect of gravity. Various methods have been used to estimate the magnitudes of eddy diffusivity. Empirical eddy diffusivities have been derived from wind tunnel studies by Sehmel (1970):
Vertical tube:



0 .
011 ( zu
*

)
1 .
1
(

Horizontal surface: and






0 .
018 ( zu
*

140 for p u
*
2

)
1 .
1
)
2 .
0
(
 z
  p u
*
2

350
)
1 .
7
for z
 
350
Because the Brownian diffusion is always present in any situation, like the effect of gravity on the earth, Brownian diffusion as well as eddy diffusion should be included simultaneously. However, the effect of Brownian diffusion is generally much smaller than that of eddy diffusion. Thus, Brownian diffusion can be neglected in the presence of eddy diffusion. That is,
J p
 
( D p
  p
)

C p
   p

C p
4. External forces: Two most common external forces are discussed in the followings: gravity and electrical forces. Note that migration velocity or drift velocity is defined as the velocity at balance between the force field and the drag force.
I. Graivity: The drift velocity or terminal settling velocity is always downward to the surface, that is, in the negative z-direction. The terminal settling velocity as discussed before is
V s

 p d p
2
18

C c
( 1


 p
) g
  p g
For the transport mechanisms of gravity and Brownian together, the transport flux in the z-direction is J z
 
D p

C p
 z

V s
C p
and the continuity equation becomes

C p
 t
  

( V s
C p
)
  
( D p

C p
) or


C p
 t

V s

C p
 z

D p
 2
C p
 x
2

 2
C p
 y
2

 2
C p
 z
2
in Cartesian coordinate and constant D p
Example : Find the steady state deposition rate to a flat surface by Brownian diffusion and gravitational sedimentation.
From the continuity equation, the governing equation is
V s

C p
 z

D p
 2
C
 z
2 p 
0 or flux

J
 
V s
C p

D p dC p dz
 const
 k
1
Assume that the concentration boundary layer thickness is b. The boundary conditions are then C p
( z

0 )

0 and C p
( z
 b )

C o
. Thus,
J

[ 1


V s
C o exp(

V s b / D p
)]
and
J
 
V s
C o
for sedimentation only ( V s b

D p
)
J

D p
C o
for Brownian diffusion only ( V s b

D p
) b
Note that the dimension of ( D p
/b ) is the same as that of velocity and it represents the transport velocity of Brownian diffusion.
II. Electric force: The external force in an electrical field is
F e
 n e eE where F e
is the electric force, n e
is the number of charge on particle, e is the elementary electric charge = 1.6*10
-19
coulomb = 4.8*10
-10
stat coulomb or esu, and E is the electric field in unit of volt/cm or stat volt/cm.
Note that 1 coulomb = 3*10
9
stat coulomb and 1 volt = 1/300 stat volt.
Based on the definition of migration velocity and mobility, the electric migration velocity is
V e
 n e eE
3
 d p
/ C c and the electric mobility is
Z

V e
E
 n e e
3
 d p
/ C c
 n e e f

The mechanisms for particle charging includes:
Collisions with ions, which can be classified as diffusion charging and field charging based on the charging mechanisms;
Direct ionization; and
Static electrification, which can be classified as contact charging and frictional charging based on the charging mechanisms.
(A). Diffusional charging: The mechanism for the transport of ions to particles is due to the random thermal motion. Assume that the thermal motion of particle is negligible with respect to ions and complete capture on collision, the accumulation rate of charge on particles is
(1) Free molecular range dn e dt

( V p
C ) A


 kT m i


1 / 2 n i
 exp
 

2 K e e
2 n e d p kT

4 d p
2
, where n e
is the charge number on particle, m i
is the molecular weight of ions, n i

is the concentrations of ions far from particles, and K e
is a constant of proportionality that depends on the system of units used. For example, the Coulomb’s law for the electrostatic force between two charged particles is F e

K e q
1 q
2 r
2
and K e
is 9.0*10
9
N-m
2
/C
2
in SI system and is one in the cgs system.
Conversion Factors and Constants for SI and cgs Electrostatic Units
Quantity
Charge
Current
SI
1 C
1 A cgs
3.0*10
9
stC
3.0*10
9
stA
Potential difference
K e
Charge of an electron, e
Electrons per unit charge
Electrical mobility
Field strength
1 V 0.0033 stV
9.0*10 9 N-m
6.3*10
18
C
2
-1
1 m
2
/V-sec
/C
1.60*10
-19
C
2 1
4.80*10
-10
stC
2.1*10
9
stC-1
3.0*10
6
cm2/stV-sec
1 N/C or V/m 3.3*10 -5 dyn/stC or stV/cm
Thus, n e
 d p kT
2 K e e
2 ln



1


2 n i

K e d p e
2


1 m i kT


1 / 2 t



For N i t > 10
12
sec/m
3
, the above equation is accurate to within a factor of two for particles from 0.07 to 1.5
 m.

(2) Continuum range
Assuming radial symmetry, the flux of ions to a spherical particles is
Ji
 
D i

 dn i dr
 d
 dr n i


, where D i
is the diffusivity of ions,
 is the electrostatic potential field in the gas surrounding the particle, and n i
is the concentration of ions. In the quasi-steady state, the rate of ion transported to the particle is independent of r ; that is, F i
= 4
 r
2
J i
(r) = constant. Thus,
F i
 a p

 1 r
2
4

D i n i

 exp kT
 dr
. For
  ie
2 r
and F i
 dn e dt
, dn e dt

4

D i n i
 n e e
2 kT


  exp
2 n e e
2 d p kT

1




The implicit solution of n e
as function of time is m



1
1 mm !
2 ie
2 d p kT m

4

D i e
2 n i
 t kT
(B). Field charging: The transport of ions to particle is determined by the effect of electric field. The number of charges acquired by a particle during time period t in an electric field E with an ion number concentration
N i
is n e
( t )


3


2


1

K
 e

K eZ e i eZ
N i i
N t i t




Ed p
2
4 K e e

 where

is the relative permittivity of particle, Z i
is the mobility of ion, which is approximately 1.5 cm/Volt-sec.
Untimately, the charge builds up to the point where no incoming field lines converge on the particle and no ions can reach the particle. At this maximum charge condition, the particle is at saturation charge and it is n e
( t )


3


2







Ed p
2
4 K e e



For liquid droplets, they tend to break down to smaller sizes as charges accumulate, the limiting charge number is n e , R

1 e
2
 d p
3
, where

is the surface tension and it is called the Rayleigh limit. Therefore, the final droplet size is dp

32

E
2
.
The Boltzmann equilibrium charge distribution or the steady-state, stationary or bipolar equilibrium charge distribution represents the charge distribution of an aerosol in charge equilibrium with bipolar ions. For equal concentrations of positive and negative ions, a reasonable first approximation for normal air, the fraction of particles f n
of a given size having n positive (or negative) elementary units of charge is f n
 exp( K e n e
2 e
2 n


  exp( K e n e
2 e
2
/ d p kT )
/ d p kT )
For particles larger than 0.5
 m, the above equation becomes identical to the equation for normal distribution and can be rewritten as f n

K e e
2
 d p kT
1 / 2 exp

K e n e
2 e
2 d p kT

0 .
240
1
 d p exp

0 .
058 n e
2 d p for d p
> 0.01
 m in the ambient atmosphere and d p
is in unit of
 m.
The average charge number is n e
 n



0 n * f n

2 .
37 d p
where d p
is in unit of
 m.
Equilibrium half-time: the time needed for half of the equilibrium charge to be attained is t
1 / 2

0 .
693 C p
4 q i
, where C p
is the aerosol concentration and q i
is the ion production rate. For the ambient atmosphere, n is 5*10
4 particle/cm
3
and q i
is about 10 ions/cm
3
-sec, the half-time is about 15 minutes.
5. Phoretic effects: Phoretic effects are the indirect effects on particle transport and are generally more significant for smaller particles. Two different effects are discussed in the followings: diffusiophoresis and thermophoresis.
I. Diffusiophoresis: The diffusiophoresis is the effect of transported heavier particles on the lighter particles.

(A) small particle
V d
  i
 i
V i
and
 i



 k

 molar fraction of species k
For binary mixture with
  i
V i

0

8 k
 k



8 k


 k m k m k
, where
 k

 n k n k
is the
V d
 

1 m
1 m
1

 
2 m
2 m
2
D
 
1
Thus, particles move in the same direction as that of the diffusion flux of the heavier component.
If V
2

0 in the binary mixture,
V d
 

1
2


1
1 D
 
1
, where

1

1


1

8

1

1

8

1 m
1
 
1 m
1

8

2 m
2
(B) large particles
V d
 

1 m
1 m
1

 
2 m
2 m
2
D
 
1

If V
2

0 , V d
 

1 m
1 m
1
 
2 m
2

D
2
 
1

II. Thermophoresis: transport of particle due to a temperature gradient with the resulting transport direction from high temperature to low temperature.
(A) small particle, d p
 
(mean free path)
V th
 
4 1
3
 
T

8


T
 
5 1
1

8
 trans
P

T where

is the accommodation coefficient and is about 0.9,

is the kinematic viscosity, and
  trans
15 k

4 m
where m is the molecular mass.

(B) large paraticle,
V th where d p
 
and k p 
10 k
 
2 k
2 k

 k p
 k
P

T

0 .
2 , k and k p
are the thermal conductivities for gas and particle, respectively.
6. Inertia effect: If the inertia effect is significant, the particle velocity is not the same as the gas velocity. Therefore, the particle velocity has to be determined first and it is then used to compute the particle trajectory. Thus, Lagrangian approach is generally used in this situation. That is, the force balance as shown for single particle dynamics is used to describe the transport of particles.

Eulerian and Lagrangian approaches:
Eulerian method: Coordinate system is fixed with respect to the same point in all dimensions (spatial and temporal).
Lagrangian method: Coordinate system is defined with respect to particle, that is, traveled with particle.

The Einstein-Stokes equation is to relate the properties for Eulerian and Lagrangian approaches in particle transport and dynamics: diffusivity and friction factor.
To find the average position
2 x for a group of particles which are at x=0 at the beginning and are moved by Brownian diffusion:
For Eulerian approach, the diffusion equation based on mass conservation is
 n
 t

D
 2 n
 x
2 with the initial condition boundary conditions
 n
 x n x

0
( x

, 0 )
0

N o
and
and n (

, t )

0 .
Therefore, the concentration distribution is n ( x , t )

2
N o

Dt exp


 x
2
4 Dt


The mean square displacement is x 2 
1
N o


  x 2 n ( x , t ) dx

2 Dt
For Lagragian approach, the force balance for each individual particle is m p dV p dt
 
3
 d p
V p

F
B
( t ) where F
B
(t) is the fluctuating force on particle from random thermal motion.
Assume that
1. The F
B
(t) is independent of Vp;
2. The F
B
(t) fluctuates much faster with respect to time compared with V p
.
Thus, V p
is assumed to be constant when considering the variations of F
B
(t).
3. The average of F
B
(t) over a lot of particles is zero or F
B
( t )

0 over a lot of particles.
Let A

F
B
( t )
and m p
B

3
 d p m p
 f m p
and multiply the equation by x . Then the above equation becomes x dV p dt

BxV p
 xA or d ( xV p
)

BxV p dt

V p
2  xA .

xV p
 e

Bt t

0
V p
2 e
B
 d
  e

Bt t

0 xAe
B
 d

Averaging over all particles xV p




 xV p n ( x , t ) dx

V p
2
B

1
 e

Bt


1
2 d ( x
2
) dt x
2
2

V p
2 t
B

V p
2
B
 e

Bt 
1


V p
2 t
B
 m p
V p
2 t f
Combining the Eulerian and Lagrangian approaches,
D
 x
2
2
 m p
V p
2 f
 kT f which is called the Einstein-Stokes equation.

1. Flat plate with laminar boundary layer
For flow over a flat plate, the boundary layer is laminar if the Reynolds number
Re

 xU

 xU


5 * 10
5
Because the particle diffusivity is generally much smaller than the kinematic viscosity of fluid, the concentration boundary layer thickness is thinner than the momentum boundary layer thickness or
 c
  m
due to Dp
 
.
The governing equation for the particle transport or the convective diffusion equation is u
 n
 x
 v
 n
 y

 2 n
D
 y
2 where u and v are the x- and y-components of particle velocity and they are the same as those of fluid, n is the particle concentration, and D is the particle diffusivity. Because the particle velocity is the same as the fluid velocity, u

0 .
332

U and v

0 .
083
 xU

1 / 2

2
U , where
 
U
 x

1 / 2 y . The boundary conditions are n ( 0 , y )
 n o
, n ( x , 0 )

0 and n ( x ,

)
 n o
.
The governing can be solved for the concentration profiles and the local diffusional flux to the plate is
J x
( x )
 
D
 n
 y y

0

0 .
339
D
 x   xU


1 / 2 
D

1 / 3 n o
The average diffusional flux from the edge to x is
J av

1
L

L
0
J x
( x ) dx

0 .
678

D

L
 
LU

1 / 2 
D
1 / 3 n o
 k av n o
. Thus k av
L
D

0 .
678
LU

1 / 2 
D

1 / 3
, where k av
is the average transport coefficient
2. Flat plate with turbulent boundary layer