Transcript Presentation Slides for Chapter 15 of Fundamentals of Atmospheric Modeling 2nd Edition Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA 94305-4020 [email protected] March 30, 2005

Presentation Slides
for
Chapter 15
of
Fundamentals of Atmospheric Modeling
2nd Edition
Mark Z. Jacobson
Department of Civil & Environmental Engineering
Stanford University
Stanford, CA 94305-4020
[email protected]
March 30, 2005
Coagulation
Process by which particles collide and stick together
Integro-differential coagulation equation
(15.1)


0
0
n 1
  ,nnd  n  ,nd 
t
2
Monomer Size Distribution
Fig. 15.1
Coagulation Over Monomer Distribution
Coagulation equation over monomer size distribution (15.2)
k1

j1
j 1
nk 1
  k j, j nkj n j  nk  k, j n j
t
2
Rewrite in fully implicit finite-difference form

nk,t  nk,th 1 k1
  Pk, j   Lk, j
h
2
j1
j 1
(15.3)
Coagulation Over Monomer Distribution
Finite-difference form
(15.3)

nk,t  nk,th 1 k1
  Pk, j   Lk, j
h
2
j1
Production rate
j 1
(15.4)
Pk,j   kj, j nk j,t n j,t
Loss rate
Lk ,j   k, j nk,t n j,t
-->
Pk ,j  L k  j, j
Rearrange (15.3)
(15.5)
k1

j 1
j1
1
nk,t  nk,th  h k j, j nk j,t n j,t  h k,j nk,t n j,t
2
Semiimplicit Solution Over Monomer
Size Distribution
Write loss rate in semi-implicit form
Lk ,j   k, j nk,t n j,t
(15.6)
--> Lk,j  k, j nk,t n j,th
Substitute (15.6) into (15.3)
k1

j 1
j1
(15.7)
1
nk,t  nk,th  h k j, j nk j,t n j,th  h k, j nk,t n j,t h
2
Rearrange --> semiimplicit solution
(15.8)
Treats number correctly but does not conserve volume
nk,t 
1
nk,t h  h
2
k1
k j, jnk j,t n j,th
j 1

1 h
k,j n j,t h
j 1
Semiimplicit Solution Over Monomer
Size Distribution
Revise to conserve volume, giving up error in number (15.9)
k 1
v k,t h  h
v k,t 
k  j,j vk  j,t n j,t h
j1

1 h
k ,j n j,t h
j1
where vk,t=knk,t
Semiimplicit Solution Over Arbitrary
Size Distribution
Volume of intermediate particle
(15.10)
Vi,j  i   j
Volume fraction of Vi,j partitioned to each model bin k (15.11)
 k 1  Vi,j   k


 k 1  k  Vi, j

fi, j,k  1 fi,j,k 1
1

0

k  Vi, j   k 1
k  NB
k -1  Vi, j  k k  1
Vi, j  k
k  NB
all other cas es
Semiimplicit Solution Over Arbitrary
Size Distribution
Incorporate fractions into (15.9)
(15.12)
k k 1
v k,t 

v k,t h  h    fi, j,k i, j vi,t n j,t h 



j1 i1
NB


1  h  1 fk ,j,k  k ,j n j,t h
j1
Semiimplicit Solution Over Arbitrary
Size Distribution
Final particle number concentration
(15.13)
vk,t
nk,t 
k
Semiimplicit solution for volume concentration
when multiple components
(15.14)
k k 1
v q,k,t 

v q,k ,t h  h    fi, j,k i, j vq,i,t n j,t h 



j1 i1
NB


j1

1  h  1  fk, j,k  k ,j n j,t h

Smoluchowski’s (1918) Solution
Assumes initial monodisperse size distribution, a monomer size
distribution during evolution, and a constant rate coefficient
nk ,t 

nT,t h 0.5hnT,t h
1  0.5hnT,t h 
Coagulation kernel (rate coefficient)
8k BT

3a

k 1
(15.15)
k 1
(15.16)
Smoluchowski’s (1918) Solution
dn (No.
dn (No. cm
-3 ) /d log
10 DD
cm-3) / d log
10 p p
Comparison of Smoluchowski's solution, an integrated solution,
and three semi-implicit solutions
108
106
4
10
Initial
Smol.
Integrated
SI (1.2)
SI (1.5)
SI (2.0)
102
0
10
0.01
P article diameter (D
, m)
p
0.1
Fig. 15.2
Self-Preserving Solution
Self-preserving size distribution
(15.17)
  
nT,t  h i
i 
ni,t h 
exp

  
p
p 

Solution to coagulation over self-preserving distribution
(15.18)

i  p

ni,t 

2 exp  1  0.5hn

T,t h 
1 0.5hnT,th

nT,thi  p

Self-Preserving Solution
106
4
10
-3
dn (No. cm
) /d log
D
-3
10
dn (No. cm ) / d log10D
pp
Self-preserving versus semi-implicit solutions
2
10
100
0.01
Initial
Analytical
SI (1.5)
SI (2)
0.1
P article diameter (D, m)
p
1
Fig. 15.3
Coagulation Over Multiple Structures
Internal mixing among three externally-mixed distributions
Fig. 15.4
Coagulation Over Multiple Structures
Volume concentration of component q in bin k of distribution N
(15.19)
vq,Nk,t h  h Tq,Nk,t,1  Tq, Nk,t,2
v q,Nk,t 
1 hTq,Nk,t,3

NT 


PN ,M
nMj,t h
Tq, Nk,t,1 
f Ni,Mj, Nk,t h  Ni,Mj,t h v q,Ni,t 





M 1 
j 1
i 1

NT NT 
k 
k 1



QI,M,N
nMj,t h
Tq, Nk,t,2 
fIi,Mj,Nk,t h Ii,Mj,t h v q,Ii,t 




M 1 I 1
j 1
i1

N B  N T
k 

k


Tq, Nk,t,3     1 LN,M 1 fNk ,Mj,Nk,t h  L N,M  Nk,Mj,t h nMj,t h 

M 1

j1 


NT = number of distributions


NB = number of size bins
Coagulation Over Multiple Structures
Total volume concentration in bin k of distribution N
v Nk,t 
(15.21)
NV
 vq,Nk,t
q1
Number concentration in bin k of distribution N
v Nk,t
n Nk,t 
 Nk
(15.22)
Coagulation Over Multiple Structures
Volume fraction of coagulated pair
VIi,Mj   Ii  Mj
partitioned into bin k of distribution N
  Nk 1  VIi,Mj,t h  

 Nk
 Nk 1   Nk,t h  VIi,Mj

fIi,Mj,Nk  1  fIi,Mj,Nk 1

1
0

(15.20)
 Nk  VIi,Mj  Nk 1
k  NB
 Nk -1  VIi,Mj   Nk
VIi,Mj   Nk
k1
all other cas es
k  NB
dn (No. cm-3) / d log10Dp
Coagulation Over Multiple Structures
Fig. 15.5
dn (No. cm-3) / d log10Dp
Coagulation Over Multiple Structures
Fig. 15.5
dn (No. cm-3) / d log10Dp
Coagulation Over Multiple Structures
Fig. 15.5
Particle Flow Regimes
Knudsen number for air
a
Kna,i 
ri
Mean free path of an air molecule
2a
2a
a 

a v a
va
Thermal speed of an air molecule
va 
(15.23)
(15.24)
(15.25)
8k BT
M
Particle Reynolds number
Re i  2ri V f ,i  a
(15.26)
Particle Flow Regimes
T = 292 K, pa = 999 hPa, and p = 1.0 g cm-3
105
1
10
Knudsen
number
Reynolds
number
10-3
2 -1
Diffusion coef. (cm s )
-7
10
-1 1
10
0.01
0.1
1
10
100 1000
Particle Diameterm)
(
Fig. 15.6
Particle Flow Regimes
Continuum regime
Kna,i« 1 --> ri » a and particle resistance to motion is due to
viscosity of the air.
Free molecular regime
Kna,i » 10 --> ri « a and particle resistance to motion is due to
inertia of air molecules hit by particles.
Example 15.2
T
--->
va
--->
a
--->
a
--->
a
--->
Kna,i
= 288 K
ri = 0.1 m
= 4.59 x 104 cm s-1
= 1.79 x 10-4 g cm-1 s-1
= 0.00123 g cm-3
= 6.34 x 10-6 cm
= 0.63 --> continuum regime
Coagulation Kernel
Coagulation kernel (rate coefficient)
Brownian diffusion
Convective Brownian diffusion enhancement
Gravitational collection
Turbulent inertial motion
Turbulent shear
Van der Waals forces
Viscous forces
Fractal geometry
Diffusiophoresis
Thermophoresis
Electric charge
Kernel = product of coalescence efficiency and collision kernel
(15.27)
 i, j  Ec oal,i,j Ki, j
Brownian Diffusion Kernel
Brownian motion
Irregular motion of particle due to random bombardment by
gas molecules
Continuum regime Brownian collision kernel (cm3 partic. s-1)
(15.28)
B
Ki, j  4 ri  rj Dp,i  Dp, j



Particle diffusion coefficient
(15.29)
k BT
Dp,i 
Gi
6ri a
Cunningham slip-flow correction to particle resistance to motion
(15.30)


Gi  1  Kna,i A   Bexp C Kna,i

Brownian Diffusion Kernel
Free molecular regime Brownian collision kernel (cm3 partic. s-1)
(15.31)
2
B
2
Ki, j   ri  rj
v p,i
 v 2p, j


Particle thermal speed
(15.32)
v p,i 
8k B T
M p,i
Interpolate between continuum and free molecular regimes
(15.33)
4 ri  rj D p,i  D p, j
B
Ki, j 
4 D p,i  D p, j
ri  rj

2
ri  r j  2i   2j
v p,i
 v 2p, j ri  rj







Brownian Diffusion Kernel
Mean distance from center of a sphere reached by particles
leaving the sphere's surface and traveling a distance p,i
(15.34)
2ri   p,i 

3
i

32
2
2
 4ri   p,i
6ri  p,i
Particle mean free path (cm)
 p,i 
 2ri
(15.34)
8Dp,i
vp,i
Brownian Diffusion Enhancement
Eddies created in the wake of a large, falling particle enhance
diffusion to the particle surface
Brownian diffusion enhancement collision kernel
K B 0.45Re1/3 Sc1/3
j
p,i
DE  i, j
Ki, j   B
1/ 2 Sc1/3
K
0.45Re

j
p,i
 i, j
Particle Schmidt number
a
Scp,i 
D p,i
(15.35)
Re j  1; rj  ri
Re j  1; rj  ri
(15.36)
Gravitational Collection
Collision and coalescence when one particle falls faster than and
catches up with another
Differential fall speed collision kernel

(15.37)

2
GC
Ki, j  Ec oll,i,j  ri  rj Vf ,i  V f ,j
Collection (coalescence) efficiency
60EV,i, j  EA,i, j Re j
Ecoll,i, j 
60 Re j
Ecoll,i,j simplifies to
EVi,j when Rej « 1 (viscous flows)
EAi,j when Rej » 1 (potential flows)
(15.38)
rj  ri
Gravitational Collection

2


 0.75ln 2Sti, j
 1 
EV ,i,j  
Sti, j  1.214



0
EA,i,j 
(15.39)
Sti, j  1.214


Sti, j  1.214
2
S ti,j
S ti,j  0.5
2
Stokes number
Sti, j  V f ,i V f , j  V f ,i rj g
for rj≥ri
Turbulent Inertia and Shear
Collision kernel due to turbulent inertial motion
Collision between drops moving relative to air
TI
Ki, j 
34
 d

1 4 ri  rj
g a
(15.40)
 Vf ,i  V f ,j
2
Collision kernel due to turbulent shear
Collisions due to spatial variations in turbulent velocities of
drops moving with air
(15.41)
12
TS 8 d 
Ki, j  

15

a 
ri  r j 
3
k = dissipation rate of turbulent energy per gram (cm2 s-3)
Comparisons of Coagulation Kernels
10
-13
10
10-15
Diff.
enhancem ent
Turb.
shear
Turb.
inertia
Settling
10-17
0.01
0.1
1
10
Radius of second particlem)
(
-9
10
Settling
10-11
Turb. inertia Brownian
-13
10
Kernel
-11
Brownian
particle
-1 -1
3
-1
(cm particle s-1s) )
-9
10
Total
10-7
3
Total
Coagulation kernel (cm
10-7
3
Kernel (cm3 particle s )
particle
-1 -1
-1 -1 s )
Coagulation kernels when particle of (a) 0.01 m and (b) 10 m
in radius coagulate at 298 K.
-15
Turb. shear
10
Diff.
enhancem ent
-17
10
0.01
0.1
1
10
Radius of second particlem)
(
Fig. 15.7
Van der Waals/Viscous Forces
Van der Waals forces
Weak dipole-dipole attractions caused by brief, local charge
fluctuations in nonpolar molecules having no net charge
Viscous forces
Two particles moving toward each other in viscous medium
have diffusion coefficients smaller than the sum of the two
Van der Waals/viscous collision kernel



1 
W
 c,i,j 


V
B
B 
Ki, j  Ki, j VE,i, j  1  K i,j 

Wc,i, j
 1 
Wk ,i,j




(15.42)

4 Dp,i  Dp, j


2
2
v p,i  v p, j ri  rj 




4 Dp,i  Dp,j




2
v 2p,i  v p,
j ri  rj





1




Van der Waals/Viscous Forces
Free-molecular regime correction
(15.43)
 dE

r d2 EP,i, j r
P,i,j


 
r
2

dr

dr
 

 2
1


Wk,i, j 

r dr
2
r r
2 ri  rj kB T i j   1  r dEP,i, j r
 
exp 
 EP,i, j r 

k T 2
dr

 
  B 




Free-molecular regime correction
Wc,i, j 
Di,j
(15.44)
1
EP,i, j r  dr
ri  rj 
r exp 

2
ri r j Dr,i,j
k
T
r
 B




Van der Waals/Viscous Forces
Van der Waals interaction potential
(15.46)




2 

2
r  ri  rj 
2ri rj
2ri rj
AH 
EP,i, j r  

 ln
2
2
2 

6 r2  r  r
2
2
r  ri  rj
r  ri  rj 

i
j





Particle pair Knudsen number
Kn p 

(15.47)
2p,i  2p, j
ri  rj
Van der Waals/Viscous Forces
Correction factor
Van der Waals/viscous correction factor
Fig. 15.8
Fractal Geometry
Fractals
Particles of irregular, fragmented shape
Fractal (outer) radius of agglomerate
(15.48)
r f ,i  rs Ni1 D f
Number of spherules in aggregate
i
Ns,i 
s
(15.49)
Fractal Geometry
Mobility radius
(15.50)
0.7


rf ,i


 D f  1
rm,i  MAX
, rf ,i 
 , rA,i 
2 

ln
r
r

1


f
,i
s




Area-equivalent radius
(15.51)
 2 3
1
2 Df
 2
rA,i  rs MAXNs,i , MIN 1 Ns,i  1 , D f Ns,i
 3

3






Fractal Geometry
Brownian collision kernel modified for fractals
B
Ki, j 


4 rc,i  rc, j Dm,i  Dm, j
rc,i  rc, j
rc,i  rc, j  2m,i  2m, j


(15.52)

4 Dm,i  Dm, j


2
v p,i
 v 2p, j rc,i  rc, j

Modified Brownian Collision Kernels
partic.
-1 -1-1 -1
3
3
(cm
Kernel (cm particle
s ) s )
-4
10
-5
10
10-6
Spherical, no van der Waals
Spherical, with van der Waals
Fractal, no van der Waals
Fractal, with van der Waals
-7
10
-8
10
Volume-equivalent diameter of
first particle=10 nm
10-9
0.01
0.1
1
Volume-equivalent diameter of second particle
m)(
Fig. 15.9
partic.
-1 -1-1 -1
(cm
3 3
s ) s )
Kernel (cm particle
Modified Brownian Collision Kernels
-7
10
Volume-equivalent diameter of first particle=100 nm
Spherical, no van der Waals
Spherical, with van der Waals
Fractal, no van der Waals
Fractal, with van der Waals
-8
10
-9
10
0.01
0.1
1
Volume-equivalent diameter of second particle
m)(
Fig. 15.9
dn (No.
-3
cm-3)
dn (No. cm
) / d log
D
/ d log1010Dpp
Effect on Aerosol Evolution
5
4 10
3.5 105
5
3 10
2.5 105
2 105
1.5 105
1 105
5 104
0 100
Sum of all distributions
Spheres, no van der Waals
Box Model
0.01
0.1
P article diameter (D
, m)
p
8s
1m
2m
3m
5m
10 m
15 m
20 m
30 m
45 m
1
Fig. 15.10
-3
dn (No. cm
) / d log
D
1010D
dn (No. cm-3) / d log
pp
Effect on Aerosol Evolution
4 105
Sum of all distributions
Fractal, with van der Waals
Box model
3 105
8s
1m
2m
3m
5m
10 m
5
2 10
1 105
0 100
0.01
0.1
P article diameter (D
, m)
p
1
Fig. 15.10
Diffusiophoresis/Thermophoresis/Charge
Diffusiophoresis
Flow of aerosol particles down concentration gradient of gas
due to bombardment of particles by the gas as it diffuses down
same gradient
Thermophoresis
Flow of aerosol particles from warm to cool air due to
bombardment of particles by gases in presence of temperature
gradient.
Electric charge
Opposite-charge particles attract due to Coulomb forces
Diffusiophoresis/Thermophoresis/Charge
Collision kernel for diffusiophoresis, thermophoresis, charge,
other kernels
Ki, j 

exp 4BP,i Ci, j

4BP,i Ci, j

TI  KTS   1
Ki,Bj  Ki,DE

K
j
i,j
i, j 
Mobility
(15.54)
Vf ,i
V f ,i
Dp,i
Gi
BP,i 



FG
FD 6ari kBT
Particle diffusion coefficient
Dp,i  BP,i k BT
(15.57)
Diffusiophoresis/Thermophoresis/Charge
Diffusiophoresis, thermophoresis, charge terms
Th
Df
(15.58)
e
Ci, j  Ci, j  Ci,j  Ci, j
Th
Ci, j  

(15.59)
 

51 3Kna,i  p  2a  5 pKna,i pa
12ri a  a  2.5 pKna,i  a rj T  Ts, j Fh,L, j


(15.60)
0.74Dv md rj  v   v,s Fv,L, j
Df
Ci, j  6a ri
Gi mv a
Ci,e j  Q i Q j
(15.61)
Collision Efficiency for Cloud-Aerosol
Coagulation
0
Collision efficiency
10
 =100, q=2
-1
10
d
10-2
 =0, q=0
d
-3
10
-4
10
 =0, q=2
d
 =100,q=0
d
r
=42 m
large
-5
10
0.001
0.01
0.1
1
Radius of aerosol particlem)
(
10
Fig. 15.11
3
partic.
-1
-1
Kernel (cm3 particle-1 s-1) s )
Collision Kernel for Cloud-Aerosol
Coagulation
10-2
-3
10
Total
-4
10
Grav.
-5
10
Turb. shear
-6
10
-7
Turb. Inert.
10
Br. Dif. Enhanc.
10-8
Brownian
r =42 m
-9
large
10
10-10
0.001
0.01
0.1
1
10
Radius of small particlem)
(
Fig. 15.12