Presentation Slides for
Chapter 15of
Fundamentals of Atmospheric Modeling 2nd Edition
Mark Z. JacobsonDepartment of Civil & Environmental Engineering
Stanford UniversityStanford, CA [email protected]
March 30, 2005
CoagulationProcess by which particles collide and stick together
Integro-differential coagulation equation (15.1)
∂nυ∂t
=12
βυ−υ ,υ nυ−υ nυ dυ 0
υ
∫ −nυ βυ,υ nυ dυ 0
∞
∫
Monomer Size Distribution
Fig. 15.1
υ1 2 x υ 1 3 x υ 1 4 x υ 1
k=1 k=2 k=3 k=4
Coagulation Over Monomer Distribution
Coagulation equation over monomer size distribution (15.2)
Rewrite in fully implicit finite-difference form (15.3)
ΔnkΔt
=12
βk−j, j nk−j njj=1
k−1
∑ −nk βk, jnjj =1
∞
∑
nk,t −nk,t−hh
=12
Pk, jj=1
k−1
∑ − Lk, jj =1
∞
∑
Coagulation Over Monomer Distribution
Production rate (15.4)
Loss rate
Rearrange (15.3) (15.5)
Pk,j =βk−j, jnk−j,tnj,t
Lk,j =βk, jnk,tnj,t Pk,j =Lk−j, j
nk,t =nk,t−h+12
h βk−j, jnk−j,tnj,tj =1
k−1
∑ −h βk,j nk,tnj,tj=1
∞
∑
-->
Finite-difference form (15.3)
nk,t −nk,t−hh
=12
Pk, jj=1
k−1
∑ − Lk, jj =1
∞
∑
Semiimplicit Solution Over Monomer Size Distribution
Write loss rate in semi-implicit form (15.6)
Substitute (15.6) into (15.3) (15.7)
Rearrange --> semiimplicit solution (15.8) Treats number correctly but does not conserve volume
Lk,j =βk, jnk,tnj,t−h
nk,t =nk,t−h+12
h βk−j, jnk−j,tnj,t−hj =1
k−1
∑ −h βk, j nk,tnj,t−hj=1
∞
∑
nk,t =
nk,t−h+12
h βk−j, jnk−j,tnj,t−hj =1
k−1
∑
1+h βk,j nj,t−hj =1
∞
∑
Lk,j =βk, jnk,tnj,t -->
Semiimplicit Solution Over Monomer Size Distribution
Revise to conserve volume, giving up error in number (15.9)
where vk,t=υknk,t
vk,t =
vk,t−h+h βk−j,j vk−j,tnj,t−hj=1
k−1
∑
1+h βk,j nj,t−hj=1
∞
∑
Semiimplicit Solution Over Arbitrary Size Distribution
Volume of intermediate particle (15.10)
Volume fraction of Vi,j partitioned to each model bin k (15.11)
Vi,j =υi +υ j
fi, j,k =
υk+1−Vi,jυk+1−υk
⎛
⎝ ⎜
⎞
⎠ ⎟
υkVi, j
υk ≤Vi, j <υk+1 k <NB
1−fi,j,k−1 υk-1<Vi, j <υk k >1
1 Vi, j ≥υk k =NB
0 all othercases
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
Semiimplicit Solution Over Arbitrary Size Distribution
Incorporate fractions into (15.9) (15.12)
vk,t =
vk,t−h +h fi, j,kβi, jvi,tnj,t−hi=1
k−1
∑⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
j=1
k
∑
1+h 1− fk,j,k( )βk,j nj,t−hj=1
NB
∑
Semiimplicit Solution Over Arbitrary Size Distribution
Final particle number concentration (15.13)
Semiimplicit solution for volume concentrationwhen multiple components (15.14)
nk,t =vk,tυk
vq,k,t =
vq,k,t−h +h fi, j,kβi, j vq,i,tnj,t−hi=1
k−1
∑⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
j=1
k
∑
1+h 1−fk, j,k( )βk,j nj,t−h[ ]j=1
NB
∑
Smoluchowski’s (1918) Solution
Assumes initial monodisperse size distribution, a monomer size distribution during evolution, and a constant rate coefficient
(15.15)
Coagulation kernel (rate coefficient) (15.16)
nk,t =nT,t−h 0.5hβnT,t−h( )
k−1
1+0.5hβnT,t−h( )k+1
β =8kBT3ηa
Smoluchowski’s (1918) Solution
Fig. 15.2
Comparison of Smoluchowski's solution, an integrated solution, and three semi-implicit solutions
10
0
10
2
10
4
10
6
10
8 Initial
Smol.
Integrated
SI (1.2)
SI (1.5)
SI (2.0)
0.01 0.1
dn (No. cm
-3
) /d log
10
D
p
Particle diameter (D
p
, μ )m
dn (
No.
cm
-3)
/ d lo
g 10D
p
Self-Preserving Solution
Self-preserving size distribution (15.17)
Solution to coagulation over self-preserving distribution (15.18)
ni,t−h =nT,t−hΔυi
υpexp −
υiυp
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
ni,t =nT,t−hΔυi υp
1+0.5hβnT,t−h( )2 exp −
υi υp1+0.5hβnT,t−h
⎛
⎝ ⎜
⎞
⎠ ⎟
Self-Preserving Solution
Fig. 15.3
Self-preserving versus semi-implicit solutions
10
0
10
2
10
4
10
6
0.01 0.1 1
Initial
Analytical
SI (1.5)
SI (2)
dn (No. cm
-3
) /d log
10
D
p
Particle diameter (D
p
, μ )m
dn (
No.
cm
-3)
/ d lo
g 10D
p
Coagulation Over Multiple Structures
Fig. 15.4
Internal mixing among three externally-mixed distributions
A
B
C
AB = A+B = A+AB = B+AB
AC = A+C = A+AC = C+AC
BC = B+C = B+BC = C+BC
= A+B+C = AB+BC = A+BC = AC+BC = B+AC = AC+ABABC = C+AB = AB+ABC = A+ABC = AC+ABC = B+ABC = BC+ABC = C+ABC = ABC+ABC
Coagulation Over Multiple StructuresVolume concentration of component q in bin k of distribution N
(15.19)
Tq,Nk,t,1 = PN,M nMj,t−h fNi,Mj,Nk,t−hβNi,Mj,t−hvq,Ni,ti =1
k−1
∑⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
j =1
k
∑⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
M=1
NT
∑
Tq,Nk,t,2 = QI,M,N nMj,t−h fIi,Mj,Nk,t−hβIi,Mj,t−hvq,Ii ,ti=1
k
∑⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
j =1
k
∑⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
I=1
NT
∑M=1
NT
∑
Tq,Nk,t,3= 1−LN,M( ) 1−fNk,Mj,Nk,t−h( )+LN,M[ ]βNk,Mj,t−hnMj,t−hM=1
NT
∑⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥ j=1
NB
∑
vq,Nk,t =vq,Nk,t−h +h Tq,Nk,t,1+Tq,Nk,t,2( )
1+hTq,Nk,t,3
NT = number of distributions NB = number of size bins
Coagulation Over Multiple Structures
Total volume concentration in bin k of distribution N (15.21)
Number concentration in bin k of distribution N (15.22)
vNk,t = vq,Nk,tq=1
NV
∑
nNk,t =vNk,tυNk
Coagulation Over Multiple Structures
Volume fraction of coagulated pair
VIi,Mj =υ Ii +υMj
fIi ,Mj,Nk =
υNk+1−VIi,Mj,t−hυNk+1−υNk,t−h
⎛
⎝ ⎜
⎞
⎠ ⎟
υNkVIi ,Mj
υNk ≤VIi ,Mj <υNk+1 k<NB
1−fIi ,Mj,Nk−1 υNk-1 <VIi ,Mj <υNk k>1
1 VIi,Mj ≥υNk k =NB
0 all othercases
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
partitioned into bin k of distribution N (15.20)
Coagulation Over Multiple Structures
Fig. 15.5
10
-6
10
-4
10
-2
10
0
10
2
10
4
10
6
0.01 0.1 1 10 100
A (Spray)
B (Soil)
D (Sulf)
E1 (BC<5% shell)
F (OM)
dn (No. cm
-3
) / d log
10
D
p
Particle diameter (D
p
, μ )m
Number concentration
of each distribution
dn (
No.
cm
-3)
/ d lo
g 10D
p
Coagulation Over Multiple Structures
Fig. 15.5
10
-6
10
-4
10
-2
10
0
10
2
10
4
0.01 0.1 1 10 100
Initial
12 hr
dn (No. cm
-3
) / d log
10
D
p
Particle diameter (D
p
, μ )m
Number concentration
summed over all distributions
dn (
No.
cm
-3)
/ d lo
g 10D
p
Coagulation Over Multiple Structures
Fig. 15.510
-6
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
10
10
0.01 0.1 1 10 100
A (Spray)
B (Soil)
D (Sulf)
E1 (BC<5% shell)
F (OM)
AB (Spray-soil)
AD (Spray-sulf)
AE (Spray-BC)
AF (Spray-OM)
BD (Soil-sulf)
BE (Soil-BC)
BF (Soil-OM)
DE (Sulf-BC)
DF (Sulf-OM)
EF (BC-OM)
MX
dn (No. cm
-3
) / d log
10
D
p
Particle diameter (D
p
, μ )m
Number concentration
of each distribution
dn (
No.
cm
-3)
/ d lo
g 10D
p
Particle Flow RegimesKnudsen number for air (15.23)
Mean free path of an air molecule (15.24)
Thermal speed of an air molecule (15.25)
Particle Reynolds number (15.26)
Kna,i =λari
λa =2ηaρav a
=2νav a
v a =8kBTπM
Rei =2riVf,i νa
Particle Flow Regimes
Fig. 15.6
T = 292 K, pa = 999 hPa, and p = 1.0 g cm-3
10
-11
10
-7
10
-3
10
1
10
5
0.01 0.1 1 10 100 1000
Particle Diameter ( μ )m
Knudsen
number
. (Diffusion coef cm
2
s
-1
)
Reynolds
number
Particle Flow RegimesContinuum 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 = 288 K ri = 0.1 μm
---> va = 4.59 x 104 cm s-1 ---> a = 1.79 x 10-4 g cm-1 s-1
---> a = 0.00123 g cm-3
---> a = 6.34 x 10-6 cm---> Kna,i = 0.63 --> continuum regime
Coagulation KernelCoagulation kernel (rate coefficient)
Brownian diffusionConvective Brownian diffusion enhancementGravitational collectionTurbulent inertial motionTurbulent shearVan der Waals forcesViscous forcesFractal geometryDiffusiophoresisThermophoresisElectric charge
Kernel = product of coalescence efficiency and collision kernel(15.27)
βi, j =Ecoal,i,j Ki, j
Brownian Diffusion KernelBrownian motion
Irregular motion of particle due to random bombardment by gas molecules
Continuum regime Brownian collision kernel (cm3 partic. s-1) (15.28)
Particle diffusion coefficient (15.29)
Cunningham slip-flow correction to particle resistance to motion(15.30)
Ki, jB =4π ri +rj( ) Dp,i +Dp, j( )
Dp,i =kBT
6πriηaGi
Gi =1+Kna,i ′ A + ′ B exp− ′ C Kna,i( )[ ]
Brownian Diffusion KernelFree molecular regime Brownian collision kernel (cm3 partic. s-1)
(15.31)
Particle thermal speed (15.32)
Interpolate between continuum and free molecular regimes(15.33)
Ki, jB =π ri +rj( )
2v p,i
2 +v p, j2
v p,i =8kBTπM p,i
Ki, jB =
4π ri +rj( ) Dp,i +Dp, j( )
ri +rj
ri +r j + δi2 +δ j
2+
4 Dp,i +Dp, j( )
v p,i2 +v p, j
2 ri +rj( )
Brownian Diffusion KernelMean distance from center of a sphere reached by particles leaving the sphere's surface and traveling a distance p,i
(15.34)
Particle mean free path (cm) (15.34)
δi =2ri +λp,i( )
3− 4ri
2 +λp,i2
( )3 2
6riλp,i−2ri
λp,i =8Dp,iπv p,i
Brownian Diffusion EnhancementEddies created in the wake of a large, falling particle enhance diffusion to the particle
surface
Particle Schmidt number (15.36)
Brownian diffusion enhancement collision kernel (15.35)
Ki, jDE =
Ki, jB 0.45Rej
1/3Scp,i1/3 Rej ≤1; rj ≥ri
Ki, jB 0.45Rej
1/2Scp,i1/3 Rej >1; rj ≥ri
⎧ ⎨ ⎪
⎩ ⎪
Scp,i =νaDp,i
Gravitational CollectionCollision and coalescence when one particle falls faster than and catches up with another
Collection (coalescence) efficiency (15.38)
Differential fall speed collision kernel (15.37)
Ki, jGC =Ecoll,i,j π ri +rj( )
2Vf,i −Vf,j
Ecoll,i, j =60EV,i, j +EA,i, j Rej
60+Rejrj ≥ri
Ecoll,i,j simplifies to EVi,j when Rej « 1 (viscous flows)EAi,j when Rej » 1 (potential flows)
Gravitational Collection
Stokes number
(15.39)
EV,i,j =1+
0.75ln 2Sti, j( )
Sti, j −1.214
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
−2
Sti, j >1.214
0 Sti, j ≤1.214
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
EA,i,j =Sti,j
2
Sti,j +0.5( )2
Sti, j =Vf,i Vf, j −Vf,i rj g for rj≥ri
Turbulent Inertia and ShearCollision kernel due to turbulent inertial motion
Collision between drops moving relative to air (15.40)
Collision kernel due to turbulent shearCollisions due to spatial variations in turbulent velocities of drops moving with air (15.41)
k = dissipation rate of turbulent energy per gram (cm2 s-3)
Ki, jTI =
πεd3 4
gνa14 ri +rj( )
2Vf,i −Vf,j
Ki, jTS =
8πεd15νa
⎛
⎝ ⎜
⎞
⎠ ⎟
12
ri +r j( )3
Comparisons of Coagulation KernelsCoagulation kernels when particle of (a) 0.01 μm and (b) 10 μm in radius coagulate at 298
K.
10
-17
10
-15
10
-13
10
-11
10
-9
10
-7
0.01 0.1 1 10
Coagulation kernel (cm
3
particle
-1
s
-1
)
Radius of second particle ( μ )m
Total
Brownian
. Diff
enhancement
Settling
.Turb
inertia
.Turb
shear
Ker
nel (
cm3 p
arti
cle-1
s-1)
Fig. 15.7
10
-17
10
-15
10
-13
10
-11
10
-9
10
-7
0.01 0.1 1 10
Coagulation kernel (cm
3
particle
-1
s
-1
)
Radius of second particle ( μ )m
Total
Brownian
. Diff
enhancement
Settling
. Turb inertia
. Turb shear
Ker
nel (
cm3 p
arti
cle-1
s-1)
Van der Waals/Viscous ForcesVan 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 (15.42)
Ki, jV =Ki, j
B VE,i, j −1( ) =Ki,jB
Wc,i,j 1+4 Dp,i +Dp, j( )
v p,i2 +v p, j
2 ri +rj( )
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
1+Wc,i, j
Wk,i,j
4 Dp,i +Dp,j( )
v p,i2 +v p, j
2 ri +rj( )
−1
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪
Van der Waals/Viscous Forces
Free-molecular regime correction (15.43)
Free-molecular regime correction (15.44)
Wk,i, j =−1
2 ri +rj( )2kBT
dEP,i,j r( )
dr+r
d2EP,i, j r( )
dr2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
×
exp−1
kBTr2
dEP,i, j r( )
dr+EP,i, j r( )
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪
ri +rj
∞∫ r2dr
Wc,i, j =1
ri +rj( )Di, j
∞
Dr,i,jr( )exp
EP,i, j r( )
kBT
⎡
⎣ ⎢
⎤
⎦ ⎥ ri +rj
∞∫
dr
r2
Van der Waals/Viscous Forces
Van der Waals interaction potential (15.46)
Particle pair Knudsen number (15.47)
EP,i, j r( ) =−AH6
2rirj
r2 − ri +rj( )2 +
2rirj
r2 − ri −rj( )2 +ln
r2 − ri +rj( )2
r2 − ri −rj( )2
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Knp =λp,i2 +λ p, j
2
ri +rj
Van der Waals/Viscous Forces
Fig. 15.8
Van der Waals/viscous correction factor
0
1
2
3
4
5
6
0.01 0.1 1 10 100 1000
1
2
5
10
50
250
Correction factor
Particle Knudsen number
r
2
/r
1
Cor
rect
ion
fact
or
Fractal GeometryFractals
Particles of irregular, fragmented shape
Number of spherules in aggregate (15.49)
Fractal (outer) radius of agglomerate (15.48)
rf,i =rsNi1D f
Ns,i =υiυs
Fractal Geometry
Area-equivalent radius (15.51)
Mobility radius (15.50)
rm,i =MAXrf,i
ln rf,i rs( )+1,rf,i
Df −1
2⎛
⎝ ⎜
⎞
⎠ ⎟ 0.7
,rA,i
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
rA,i =rs MAX Ns,i2 3
,MIN 1+23
Ns,i −1( ),13
Df Ns,i2 Df⎡
⎣ ⎢ ⎤ ⎦ ⎥
⎧ ⎨ ⎩
⎫ ⎬ ⎭
Fractal Geometry
Brownian collision kernel modified for fractals (15.52)
Ki, jB =
4π rc,i +rc, j( ) Dm,i +Dm, j( )
rc,i +rc, j
rc,i +rc, j + δm,i2 +δm, j
2+
4 Dm,i +Dm, j( )
v p,i2 +v p, j
2 rc,i +rc, j( )
Modified Brownian Collision Kernels
Fig. 15.9
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
0.01 0.1 1
Spherical, no van der Waals
Spherical, with van der Waals
Fractal, no van der Waals
Fractal, with van der Waals
Collision kernel
(cm
3
partic.
-1
s
-1
)
Volume-equivalent diameter of second particle ( μ )m
- Volume equivalent diameter of
=10 first particle nmKer
nel (
cm3 p
arti
cle-1
s-1)
Modified Brownian Collision Kernels
Fig. 15.9
10
-9
10
-8
10
-7
0.01 0.1 1
Spherical, no van der Waals
Spherical, with van der Waals
Fractal, no van der Waals
Fractal, with van der Waals
Collision kernel
(cm
3
partic.
-1
s
-1
)
Volume-equivalent diameter of second particle ( μ )m
- =100 Volume equivalent diameter of first particle nm
Ker
nel (
cm3 p
arti
cle-1
s-1)
Effect on Aerosol Evolution
Fig. 15.10
0 10
0
5 10
4
1 10
5
1.5 10
5
2 10
5
2.5 10
5
3 10
5
3.5 10
5
4 10
5
0.01 0.1 1
8 s
1 m
2 m
3 m
5 m
10 m
15 m
20 m
30 m
45 m
dn (No. cm
-3
) / d log
10
D
p
Particle diameter (D
p
, μ )m
Sum of all distributions
, Spheres no van der Waals
Box Model
dn (
No.
cm
-3)
/ d lo
g 10D
p
Effect on Aerosol Evolution
Fig. 15.10
0 10
0
1 10
5
2 10
5
3 10
5
4 10
5
0.01 0.1 1
8 s
1 m
2 m
3 m
5 m
10 m
dn (No. cm
-3
) / d log
10
D
p
Particle diameter (D
p
, μ )m
Sum of all distributions
, Fractal with van der Waals
Box model
dn (
No.
cm
-3)
/ d lo
g 10D
p
Diffusiophoresis/Thermophoresis/ChargeDiffusiophoresis
Flow of aerosol particles down concentration gradient of gas due to bombardment of particles by the gas as it diffuses down same gradient
ThermophoresisFlow of aerosol particles from warm to cool air due to bombardment of particles by gases in presence of temperature gradient.
Electric chargeOpposite-charge particles attract due to Coulomb forces
Mobility (15.54)
Collision kernel for diffusiophoresis, thermophoresis, charge, other kernels
Diffusiophoresis/Thermophoresis/Charge
Particle diffusion coefficient (15.57)
Ki, j =4πBP,iCi, j
exp 4πBP,i Ci, j Ki, jB +Ki, j
DE +Ki,jTI +Ki, j
TS[ ]
⎛ ⎝
⎞ ⎠
−1
BP,i =Vf,iFG
=Vf,iFD
=Gi
6πηari=
Dp,ikBT
Dp,i =BP,ikBT
(15.59)
Diffusiophoresis, thermophoresis, charge terms (15.58)
Diffusiophoresis/Thermophoresis/Charge
(15.60)
(15.61)
Ci, j =Ci, jTh+Ci,j
Df +Ci, je
Ci, jTh =−
12πriηa κa+2.5κpKna,i( )κarj T∞−Ts, j( )Fh,L, j
51+3Kna,i( ) κp +2κa +5κpKna,i( )pa
Ci, jDf =−6πηari
0.74Dvmdrj ρv −ρv,s( )Fv,L, j
Gi mvρa
Ci, je =QiQ j
Collision Efficiency for Cloud-Aerosol Coagulation
Fig. 15.11
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
0.001 0.01 0.1 1 10
Collision efficiency
Radius of aerosol particle ( μ )m
d
=0, =0q
d
=0, =2q
d
=100, =2q
d
=100, =0q
r
large
=42 μ m
Col
lisi
on e
ffic
ienc
y
Collision Kernel for Cloud-Aerosol Coagulation
Fig. 15.12
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
0.001 0.01 0.1 1 10
Collision kernel (cm
3
partic.
-1
s
-1
)
Radius of small particle ( μ )m
r
large
=42 μ mBrownian
. . .Br Dif Enhanc
Total
. Turb shear
. .Turb Inert
.Grav
Ker
nel (
cm3 p
arti
cle-1
s-1)
