Presentation Slides
for
Chapter 16
of
Fundamentals of Atmospheric Modeling
2nd Edition
Mark Z. Jacobson
Department of Civil & Environmental Engineering
Stanford University
Stanford, CA 94305-4020
jacobson@stanford.edu
March 30, 2005
Mass Flux To and From a Single Drop
Rate of change of mass (g) of single pure liquid water drop
(16.1)
dm
d v
2
 4R Dv
dt
dR
Integrate from drop surface to infinity

dm
 4rDv  v   v,r
dt
(16.2)

Energy change at drop surface due to conduction
d Q*r
dT
2
 4R  a
dt
dR
(16.3)
Mass Flux To and From a Single Drop
Integrate from drop surface to infinite radius
(16.4)
d Q*r
 4r a Tr  T
dt
Relate change in mass and energy at surface
(16.5)
d Tr
dm dQ *r
mcW
 Le

dt
dt
dt
Combine (16.4) and (16.5) and assume steady state
dm
Le
 4r a Tr  T 
dt
(16.6)
Mass Flux To and From a Single Drop
Combine equation of state at saturation
pv,s   v,s Rv T
with Clausius Clapeyron equation
dpv,s
dT

 v,s Le
T
to obtain
(16.7)
d v,s
Le dT dT


2
v,s
Rv T
T
Mass Flux To and From a Single Drop
Integrate from infinite radius to drop surface
 v,s Tr 
ln

 v,s T 
Le Tr  T 
Tr
 ln
Rv TT r
T
Simplify assuming T≈ Tr
 v,s Tr   v,s T 

 v,s T 
Substitute (16.6)
into (16.9) -->
(16.8)
(16.9)
Le Tr  T  Tr  T

2
Rv
T
T
dm
Le
 4r a Tr  T 
dt
 v,s Tr   v,s T 
 dm
Le  Le

 1

 v,s T 
4r a T  Rv T  d t
(16.10)
Mass Flux to and From a Single Drop

dm
 4rDv  v   v,r
dt

(16.2)
 v,s Tr   v,s T 
 dm
Le  Le

 1

 v,s T 
4r a T  Rv T  d t
Substitute (16.2) into (16.10)
 v   v,s T 
 v,s T 
 L
e
 
4r a T
(16.10)
(16.11)
 dm
 Le

1
 1 


RvT
 4rDv v,s T  d t
Rearrange --> Mass-flux form of growth equation


4rDv pv  pv,s
dm

dt
Dv L e pv,s  Le

 1  RvT

 a T  Rv T 
(16.12)
Mass Flux to and From a Single Drop
Mass-flux form of growth equation

(16.12)

4rDv pv  pv,s
dm

dt
Dv L e pv,s  Le

 1  RvT

 a T  Rv T 
Rewrite equation for trace gases and particle sizes


4ri Dq,i
 pq  pq,s,i

dm i

dt
Dq,i
 Le,q p
 R* T
q,s,i Le,q mq
 1 
 *

 R T
 mq
a,i T
(16.13)
Fluxes to and From a Single Drop
Change in mass as a function of change in radius
(16.14)
dm i
dri
2
 4ri  p,i
dt
dt
Radius-flux form of growth equation

(16.15)

Dq,i
 pq  pq,s,i

dri
ri

*
dt
Dq,i
 Le,q  p,i p
L
m
R
T p,i

q,s,i  e,q q
1 
 *

mq
 R T

a,i T
Fluxes to and From a Single Drop
Change in mass as a function of change in volume
dm i
di
  p,i
dt
dt
Volume-flux form of growth equation


(16.16)

13
2
48 i
Dq,i
 pq  pq,s,i


di

*
dt
Dq,i
 L e,q  p,i pq,s,i
  Le,q mq
R
T p,i

 1 
 *

mq
 R T

a,i T
Gas Diffusion Coefficient
Molecular diffusion
Movement of molecules due to their kinetic energy, followed
by collision with other molecules and random redirection.
Uncorrected gas diffusion coefficient (cm2 s-1)
Dq 
5
16 Adq2 a
R*Tm a
2
mq  ma 


 m

q


(16.17)
Collision Diameters and Diffusion
Coefficients of Several Gases
Collision
Diffusion
Diameter
coefficient
Gas
(10-10 m)
(cm2 s-1)
____________________________________________
Air
Ar
CO2
H2
NH3
O2
H2O
3.67
3.58
4.53
2.71
4.32
3.54
3.11
0.147
0.144
0.088
0.751
0.123
0.154
0.234
Table 16.1
Corrected Gas Diffusion Coefficient
Dq,i
  Dq  q,i Fq,L,i
(16.18)
Corrected Gas Diffusion Coefficient
Correction for collision geometry, sticking probability (16.19)

 1.33  0.71Kn 1 4 1 

q,i
q,i

 q,i  1 

1
3 q,i
 1 Kn q,i

 
Knudsen number for condensing vapor
q
Kn q,i 
ri
0
 q,i  
1
Kn


1


q,i 


(16.20)
as
Kn q,i  
(s mall particles )
as
Kn q,i  0
(large particles )
Mass accommodation (sticking) coefficient, q,i
Fractional number of gas collisions with particles that results in
the gas sticking to the surface. From 0.01 - 1.0.
Corrected Gas Diffusion Coefficient
Mean free path of a gas molecule
q 
ma
Adq2 a
(16.23)

64Dq  m
ma
a 



ma  mq
5vq 
m

m
q 
 a
Ventilation factor
(16.24)
Corrects for enhanced vapor transfer to a large-particle surface
due to eddies sweeping vapor to the surface
2

1
0.108x

q,i
Fq,L,i  

0.78  0.308 x q,i
12 13
x q,i  Re i Sc q
x q,i  1.4
x q,i  1.4
Corrected Gas Diffusion Coefficient
Particle Reynolds number
Re i 
2 ri V f ,i
a
Gas Schmidt number
(16.25)
a
Sc q 
Dq
Corrected Thermal Conductivity
Corrected thermal conductivity of air
(16.26)

a,i  a h,iFh,L,i
Correction to conductivity for collision geometry and sticking
probability
(16.27)
1
 1.33  0.71Kn 1



4
1





h,i
h
Kn h,i 
 h,i  1  

1
3 h 

1

Kn


h,i

 

Corrected Thermal Conductivity
Knudsen number for energy
(16.28)
h
Kn h,i 
ri
Thermal mean free path
(16.29)
3Dh
h 
va
Corrected Thermal Conductivity
Thermal accommodation (sticking) coefficient
Fraction of molecules bouncing off surface of a drop that have
acquired temperature of drop ≈ 0.96 for water.
(16.30)
Tm  T
h 
Ts  T
Ventilation factor
(16.31)
Corrects for enhanced energy transfer to drop surface due to
eddies
2

x h,i  1.4
1 0.108 x h,i
Fh,L,i  

x h,i  1.4
0.78  0.308 x h,i
12 13
x h,i  Re i Pr
Corrected Saturation Vapor Pressure
Curvature (Kelvin) effect
Increases saturation vapor pressure over small drops.
Solute effect (Raoult’s Law)
The saturation vapor pressure of a solvent containing solute is
reduced to that of the pure solvent multiplied by the mole
fraction of the solvent in solution.
Radiative cooling effect
Decreases saturation vapor pressure over large drops
1.02
Saturation ratio
Saturation ratio
Curvature and Solute Effects
Curvature effect
1.01
1
0.99
S*
Equilibrium
saturation
ratio
0.98
0.01
Solute effect
r*
0.1
1
Particle radius (m)
10
Fig. 16.1
Curvature Effect
Saturation vapor pressure over a curved, dilute surface relative to
that over a flat, dilute surface
(16.33)
 2 p m p 
2  pm p
  1 
 exp  *
*
r R T 
pq,s
r
R
p,i 
i Tp,i
 i
pq,s,i

Note that exp(x)≈1+x for small x
Curvature Effect
Surface tension of water containing dissolved organics (16.34)
 p   w a  0.0187T ln 1 628.14mC 
Surface tension of water containing dissolved inorganic ions
(16.35)
 p   w a 1.7mI
Solute Effect
Vapor pressure over flat water surface with solute relative to that
without solute (Raoult's Law)
(16.36)
pq,s,i

nw

pq,s
nw  ns
Relatively dilute solution: nw >>ns
pq,s,i

ns
 1
pq,s
nw
Number of moles of solute in solution
iv Ms
ns 
ms
(16.37)
Solute Effect
Number of moles of liquid water in a drop
(16.38)
3
Mw 4ri  w
nw 

mv
3m v
Combine terms --> solute effect
pq,s,i

pq,s
 1
(16.39)
3mv iv M s
3
4ri w ms
Köhler Equation
Combine curvature and solute effects --> Sat. ratio at equilibrium
p
2 m
3m i M (16.40)
q,s,i
p p
vv s
Sq,i
 
 1

*
3
pq,s
ri R Tp,i 4ri  w ms
Simplify Köhler equation
a
2 p m p
*
R Tp,i
(16.42)
a b
Sq,i
  1  3
ri r
i
3mviv Ms
b
4 wm s
Set Köhler equation to zero -->
(16.43)
Critical radius for growth and critical saturation ratio
3b
r* 
a
S*  1 
4a 3
27 b
Köhler Equation
Critical radii / supersaturations for water drops containing sodium
chloride or ammonium sulfate at 275 K
Solute mass (g)
0
10-18
10-16
10-14
10-12
Sodium chloride
r* (m) S*-1 (%)
0
∞
0.019
4.1
0.19
0.41
1.9
0.041
19
0.0041
Ammonium sulfate
r* (m) S*-1 (%)
0
∞
0.016
5.1
0.16
0.51
1.6
0.051
16
0.0051
Table 16.2
Radiative Cooling Effect
Saturation vapor pressure over a drop that radiatively heats/cools
relative to one that does not
(16.44)
pq,s,i

pq,s
 1
Le,q mq Hr,i
* 2
4ri R T 
d,i
Radiative cooling rate (W)
 

2
Hr,i  ri 4 Qam ,  i, I   B d
0

(16.45)
Overall Equilibrium Saturation Ratio
Overall equilibrium saturation ratio for liquid water
Sq,i
 
pq,s,i

pq,s
 1
2 p m p
*
ri R Tp,i

3mviv Ms
(16.46)
Le,q m q H r,i

3
* 2
4ri w ms 4ri R T 
d,i
Equilibrium saturation ratio for gases other than liquid water
(16.47)
pq,s,i

2 p m p
Sq,i
 
 1
*
pq,s
ri R Tp,i
Flux to Drop With Multiple Components
Volume of a single particle in which one species is growing
(16.48)
i,t  q,i,t  i,t h  q,i,t h
Mass of a single particle in which one species is growing
(16.49)
mi,t  p,i,ti,t  p,q q,i,t  p,i,thi,th p,qq,i,t h
Time derivative of (16.47)
dq,i,t
dm i,t
di,t
  p,i,t
  p,q
dt
dt
dt
since
di,t h dq,i,t h

0
dt
dt
(16.50)
Flux to Drop With Multiple Components
Combine (16.50) and (16.48) with (16.16)
Rate of change in volume of one component in one
multicomponent particle
 


(16.51)
13
2
48 q,i,t  i,t h  q,i,t h
Dq,i
 pq  p
dq,i,t
q,s,i

*
dt
Dq,i
 Le,q  p,q p
L
m
R
T p,q

q,s,i  e,q q
 1 
 *

mq
 R T

a,i T

Flux to a Population of Drops
Volume as a function of volume concentration
 q,i,t 
v q,i,t
ni,t h
Substitute volumes into (16.49)
 
(16.52)


13
23
2
D
dv q,i,t ni,t h 48 v q,i,t  vi,t h  v q,i,t h
q,i pq  p
q,s,i

*
dt
Dq,i
 Le,q  p,q p
L
m
R
T p,q

q,s,i  e,q q
 1 
 *

mq
 R T

a,i T

Flux to a Population of Drops
Partial pressure in terms of mole concentration
(16.53)
*
pq  Cq R T
Vapor pressure in terms of mole concentration
pq,s,i
  Cq,s,i
 R* T
(16.53)
Flux to a Population of Drops
Combine (16.52) with (16.53)
d vq,i,t
dt
 
(16.54)


1 3 eff
mq
23
2
 ni,th 48 v q,i,t  v i,th  vq,i,th
Dq,i,t h
Cq,t  C
q,s,i,th
 p,q
Effective diffusion coefficient
(16.55)
Dq,i

eff
Dq,i,t  h 
mq Dq,i
 Le,q Cq,s,i,t


h Le,q mq
 1  1
 *

 R T

a ,i T

Flux to a Population of Drops
Simplify effective diffusion coefficient for non-water gases (16.56)
eff
Dq,i,t  h  Dq,i
  Dq  q,i Fq,L,i

Dq Fq,L,i
Corresponding gas-conservation equation
d Cq,t
dt

1.33  0.71Kn 1 4 1 
q,i
q,i

1

1
 1 Kn q,i
3 q,i

p,q N B d vq,i,t

mq
dt

i1
Kn


(16.57)
q,i
Matrix of Partial Derivatives for Growth
ODEs
vq,1,t

2 v q,1,t
vq,1,t 
1  h s
vq,2,t
vq,3,t
vq,4,t

vq,1,t t


0



0


2

 C q,t
 h s
v q,1,t t


vq,2,t
vq,3,t
vq,4,t


0
0
h s
Cq,t t 

2
2
 v q,2,t
 vq,2,t 
1  h s
0
h s
vq,2,t t
Cq,t t 

2
2
 vq,3,t
 vq,3,t 
0
1  h s
h s

v q,3,t t
Cq,t t 
2
2
2
 C q,t
 C q,t
 Cq,t 
h s
h s
1  h s

v q,2,t t
vq,3,t t
C q,t t 

2 v q,1,t
(16.58)
Partial Derivatives For Matrix
(16.58)
23


mq
1 ni,t h
2 1 3 eff
 

48
Dq,i,t h
Cq,t  Cq,s,i,t

h


v q,i,t t 3  vq,i,t 
 p,q
(16.60)
2
 vq,i,t
1 3 eff
mq
23
2
 ni,t h 48 vq,i,t  vi,t h  vq,i,t h
Dq,i,t h
Cq,t t
 p,q
 2 vq,i,t
 

 

2 Cq,t
 p,q 2 vq,i,t

v q,i,t t
mq v q,i,t t
 2 Cq,t
Cq,t t

 p,q N B  2 vq,i,t
mq
 C
i1
q,t t
(16.61)
(16.62)

Effect of Sparse-Matrix Reductions
When Solving Growth ODEs
Condensation between gas phase and 16 size bins: NB + 1 = 17
Quantity
Order of matrix
Initial fill-in
Final fill-in
Decomp. 1
Decomp. 2
Backsub. 1
Backsub. 2
Without
Reductions
17
289
289
1496
136
136
136
With
Reductions
17
49
49
16
16
16
16
Table 16.3
Analytical Predictor of Condensation
(APC) Solution For Solving Growth
Assume radius in growth term constant during time step
Change in particle volume concentration
d vq,i,t
dt


(16.63)

1 3 eff
mq
23
2
 ni,th 48 v i,th
Dq,i,th
Cq,t  Cq,s,i,t

h
p,q
Define mass transfer coefficient



(16.64)
1 3 eff
23
2
eff
k q,i,th  ni,t h 48 vi,th
Dq,i,th  ni,t h 4ri,t h Dq,i,t h
APC Solution
Effective surface vapor mole concentration
(16.65)
C
 h Cq,s,i,t h
q,s,i,th  Sq,i,t
Volume concentration of a component
v q,i,t 
mq cq,i,t
 p,q
Uncorrected surface vapor mole concentration
Cq,s,i,t h 
pq,s,t  h
*
R T
(16.66)
APC Solution
Substitute conversions into (16.63) and (16.57)
dc q,i,t
dt

 k q,i,t h Cq,t  Sq,i,t
 h Cq,s,i,t h
(16.67)

(16.68)
dCq,t
dt

NB
 h Cq,s,i,t 
kq,i,t hCq,t  Sq,i,t
i1
Integrate (16.67) for final aerosol concentration

(16.69)
c q,i,t  c q,i,t h  hk q,i,t h Cq,t  Sq,i,t
 h Cq,s,i,t h

APC Solution
Aerosol mole concentration
(16.69)

c q,i,t  c q,i,t h  hk q,i,t h Cq,t  Sq,i,t
 h Cq,s,i,t h
Mole balance equation
Cq,t 
(16.70)
NB
NB
i1
i1
 cq,i,t  Cq,t h   cq,i,t h  Ctot
Substitute (16.69) into (16.70)
NB
(16.71)

C q,t h  h  k q,i,t h Sq,i,t
 h C q,s,i,t h
C q,t 

i1
NB
1 h  k q,i,t h
i1

APC Growth Simulation
dv 
( m3
dv (m3
cm-3
cm-3)
) / d log
D
10
/ d log10Dp p
Comparison of APC growth solution, when h = 10 s, with an exact
solution. Both solutions lie almost on top of each other.
10
9
10 7
Final
10 5
10 3
Initial
10 1
10 -1
0.1
1
10
Particle diameter (D p , m)
100
Fig. 16.2
Solving Homogen. Nucl. with Cond.
Homog. nucleation rate converted to mass transfer rate (16.74)

q 1 
J hom,q
k q,hom,1,t h 




mq C q,t h  Sq,1,t

C
h q,s,1,t h 
Sum nucleation, condensation transfer rates in first bin (16.73)
kq,1,th  kq,cond,1,th  kq,hom,1,t h
Final number concentration in first bin after nucleation (16.74)

m q kq,hom,1,t h 
n1,t  n1,t h  MAX  c q,1,t  cq,1,t h
, 0
 q 1 kq,1,t h






cm-3)
dn (No.
-3
dn (No. cm ) / d log
D
10 D
/ d log
10 p p
Homogeneous Nucleation with
Condensation Simultaneously
10 10
10
8
10
6
10
4
102
0.001
Initial
After 8 seconds
0.01
0.1
Particle diameter (D , m)
p
1
Fig. 16.3
Effect of Coagulation on Condensation
dn (No. cm ) / d log
-3
D
-3
10
dn (No. cm ) / d log10Dp p
Growth plus coagulation pushes particles to larger sizes than does
growth alone or coagulation alone
Growth only
10 5
Init.
10
3
10 1
10-1
0.01
Growth
+ coag.
Coag. only
0.1
1
Particle diameter (D p , m)
10
Fig. 16.4
Effect of Coagulation on Condensation
3 cm-3
) / d log
dv 
( m 3 -3
dv (m cm ) / d log1010DDpp
Growth plus coagulation pushes particles to larger sizes than does
growth alone or coagulation alone
103
10
2
10
1
10
0
10
Growth
only
Initial
Coag. only
Growth + coag.
-1
0.01
0.1
1
Particle diameter (D , m)
p
10
Fig. 16.4
Growth With Different Size Structures
dn (No. cm ) / d log
-3
dn (No. cm-3) / d log10D
D
10 pp
Comparison of full-moving (FM) with moving-center (MC) results for
growth-only and growth plus coagulation cases shown in Fig. 16.4(a)
6
10
5
10
104
103
Growth only (FM )
Growth
only
(M C)
2
10
101
0
10
-1
10
0.01
Growth
+ coag.
(FM )
Growth
+ coag.
(M C)
0.1
1
Particle diameter (D p , m)
10
Fig. 16.5
Ice Crystal Growth
Rate of mass growth of a single ice crystal

(16.76)

4 i Dv,i
 pq  pv,I
 ,i
dm i

dt
Dv,i
 L s p

v,I,i  Ls
 1  RvT


 Rv T 
a,i T
Ice Crystal Growth
Electrical capacitance of crystal (cm)
a c,i 2

a e
 c,i c,i
a e
 c,i c,i

 i  a c,i ln

a e
 c,i c,i

a c,i ec,i

a c,i 
(16.77)
s phere


ln 1  ec,i a c,i b c,i

prolate s pheroid
s in 1 ec,i
oblate s pheroid
4a2c,i b 2c,i 
ln1  ec,i  1 ec,i 
1 e
2s
in

c,i 
needle
column
hexagonal plate
thin plate
ac,i = length of the major semi-axis (cm)
bc,i = length of the minor semi-axis (cm)
2
2
ec,i  1 b c,i a c,i
Ice Crystal Growth
Effective saturation vapor pressure over ice
pv,I,i
  Sv,i
 pv,I
Ventilation factor for falling oblate spheroid crystals
1  0.14x 2
Fq,I,i , Fh,I,i  
0.86  0.28x
x= xq,i for ventilation of gas
x= xh,i for ventilation of energy
(16.78)
x  1.0
x  1.0