Presentation Slides for
Chapter 16of
Fundamentals of Atmospheric Modeling 2nd Edition
Mark Z. JacobsonDepartment of Civil & Environmental Engineering
Stanford UniversityStanford, CA [email protected]
March 30, 2005
Mass Flux To and From a Single DropRate of change of mass (g) of single pure liquid water drop
(16.1)
Integrate from drop surface to infinity (16.2)
Energy change at drop surface due to conduction (16.3)
dmdt
=4πR2DvdρvdR
dmdt
=4πrDv ρv −ρv,r( )
dQr*
dt=−4πR2κa
dTd R
Mass Flux To and From a Single DropIntegrate from drop surface to infinite radius (16.4)
Relate change in mass and energy at surface (16.5)
Combine (16.4) and (16.5) and assume steady state (16.6)
dQr*
dt=4πrκa Tr −T( )
mcWdTrdt
=Ledmdt
−dQr
*
dt
Ledmdt
=4πrκa Tr −T( )
Mass Flux To and From a Single DropCombine equation of state at saturation
with Clausius Clapeyron equation
to obtain (16.7)
pv,s =ρv,sRvT
dpv,sdT
=ρv,sLe
T
dρv,sρv,s
=LeRv
dT
T2 −dTT
Mass Flux To and From a Single DropIntegrate from infinite radius to drop surface (16.8)
Simplify assuming T≈ Tr (16.9)
Substitute (16.6)
lnρv,s Tr( )ρv,s T( )
=LeRv
Tr −T( )TTr
−lnTrT
ρv,s Tr( )−ρv,s T( )
ρv,s T( )=
LeRv
Tr −T( )
T2 −Tr −T
T
ρv,s Tr( )−ρv,s T( )
ρv,s T( )=
Le4πrκaT
LeRvT
−1⎛
⎝ ⎜
⎞
⎠ ⎟
dmdt
Ledmdt
=4πrκa Tr −T( )
into (16.9) -->
(16.10)
Mass Flux to and From a Single Drop
Substitute (16.2) into (16.10) (16.11)
Rearrange --> Mass-flux form of growth equation (16.12)
ρv −ρv,s T( )
ρv,s T( )=
Le4πrκaT
LeRvT
−1⎛
⎝ ⎜
⎞
⎠ ⎟ +
14πrDvρv,s T( )
⎡
⎣ ⎢
⎤
⎦ ⎥
dmdt
dmdt
=4πrDv pv −pv,s( )
DvLepv,sκaT
LeRvT
−1⎛
⎝ ⎜
⎞
⎠ ⎟ +RvT
ρv,s Tr( )−ρv,s T( )
ρv,s T( )=
Le4πrκaT
LeRvT
−1⎛
⎝ ⎜
⎞
⎠ ⎟
dmdt
dmdt
=4πrDv ρv −ρv,r( ) (16.2)
(16.10)
Mass Flux to and From a Single DropMass-flux form of growth equation (16.12)
Rewrite equation for trace gases and particle sizes (16.13)
dmdt
=4πrDv pv −pv,s( )
DvLepv,sκaT
LeRvT
−1⎛
⎝ ⎜
⎞
⎠ ⎟ +RvT
dmidt
=4πri ′ D q,i pq − ′ p q,s,i( )
′ D q,i Le,q ′ p q,s,i′ κ a,iT
Le,qmq
R*T−1
⎛
⎝ ⎜
⎞
⎠ ⎟ +
R*Tmq
Fluxes to and From a Single Drop
Change in mass as a function of change in radius (16.14)
Radius-flux form of growth equation (16.15)
dmidt
=4πri2ρp,i
dridt
ridridt
=′ D q,i pq − ′ p q,s,i( )
′ D q,i Le,qρp,i ′ p q,s,i′ κ a,iT
Le,qmq
R*T−1
⎛
⎝ ⎜
⎞
⎠ ⎟ +
R*Tρp,i
mq
Fluxes to and From a Single DropChange in mass as a function of change in volume
Volume-flux form of growth equation (16.16)
dmidt
=ρp,idυidt
dυidt
=48π2υi( )
13′ D q,i pq − ′ p q,s,i( )
′ D q,iLe,qρp,i ′ p q,s,i
′ κ a,iT
Le,qmq
R*T−1
⎛
⎝ ⎜
⎞
⎠ ⎟ +
R*Tρp,i
mq
Gas Diffusion CoefficientMolecular 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) (16.17)
Dq =5
16Adq2ρa
R*Tma2π
mq +mamq
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Collision Diameters and Diffusion Coefficients of Several Gases
Table 16.1
Collision DiffusionDiameter coefficient
Gas (10-10 m) (cm2 s-1)____________________________________________
Air 3.67 0.147Ar 3.58 0.144CO2 4.53 0.088H2 2.71 0.751NH3 4.32 0.123O2 3.54 0.154H2O 3.11 0.234
Corrected Gas Diffusion Coefficient
(16.18)′ D q,i =Dqωq,iFq,L,i
Corrected Gas Diffusion CoefficientCorrection for collision geometry, sticking probability (16.19)
ωq,i = 1+1.33+0.71Knq,i
−1
1+Knq,i−1 +
4 1−αq,i( )
3αq,i
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ Knq,i
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
−1
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.
ωq,i →0 as Knq,i → ∞ (smallparticles)
1 as Knq,i → 0 (largeparticles)
⎧ ⎨ ⎩
Knudsen number for condensing vapor (16.20)
Knq,i =λqri
Corrected Gas Diffusion CoefficientMean free path of a gas molecule (16.23)
Ventilation factor (16.24)Corrects for enhanced vapor transfer to a large-particle surface due to eddies sweeping vapor to the surface
λq =ma
πAdq2ρa
mama +mq
=64Dq5πv q
mama +mq
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Fq,L,i =1+0.108xq,i
2 xq,i ≤1.4
0.78+0.308xq,i xq,i >1.4
⎧ ⎨ ⎪
⎩ ⎪
xq,i =Rei12
Scq13
Corrected Gas Diffusion CoefficientParticle Reynolds number
Gas Schmidt number (16.25)
Rei =2riVf,i
νa
Scq =νaDq
Corrected Thermal ConductivityCorrected thermal conductivity of air (16.26)
Correction to conductivity for collision geometry and sticking probability (16.27)
′ κ a,i =κaωh,iFh,L,i
ωh,i = 1+1.33+0.71Knh,i
−1
1+Knh,i−1 +
41−αh( )3αh
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ Knh,i
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
−1
Corrected Thermal ConductivityKnudsen number for energy (16.28)
Thermal mean free path (16.29)
Knh,i =λhri
λh =3Dhv a
Corrected Thermal ConductivityThermal accommodation (sticking) coefficient
Fraction of molecules bouncing off surface of a drop that have acquired temperature of drop ≈ 0.96 for water. (16.30)
Ventilation factor (16.31)Corrects for enhanced energy transfer to drop surface due to eddies
αh =Tm−TTs−T
Fh,L,i =1+0.108xh,i
2 xh,i ≤1.4
0.78+0.308xh,i xh,i >1.4
⎧ ⎨ ⎪
⎩ ⎪
xh,i =Rei12
Pr13
Corrected Saturation Vapor PressureCurvature (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 effectDecreases saturation vapor pressure over large drops
Curvature and Solute Effects
Fig. 16.1
0.98
0.99
1
1.01
1.02
0.01 0.1 1 10
Saturation ratio
Particle radius ( μ )m
Curvature effect
Solute effectEquilibrium
saturation
ratio
*r
*S
Sat
urat
ion
rati
o
Curvature EffectSaturation vapor pressure over a curved, dilute surface relative to that
over a flat, dilute surface (16.33)
′ p q,s,ipq,s
=exp2σpmp
ri R*Tρp,i
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ ≈1+
2σpmp
riR*Tρp,i
Note that exp(x)≈1+x for small x
Curvature EffectSurface tension of water containing dissolved organics (16.34)
Surface tension of water containing dissolved inorganic ions (16.35)
σ p =σw a −0.0187T ln 1+628.14mC( )
σ p =σw a +1.7mI
Solute EffectVapor pressure over flat water surface with solute relative to that
without solute (Raoult's Law) (16.36)
Relatively dilute solution: nw >>ns (16.37)
Number of moles of solute in solution
′ p q,s,ipq,s
=nw
nw +ns
′ p q,s,ipq,s
≈1−nsnw
ns =ivMsms
Solute EffectNumber of moles of liquid water in a drop (16.38)
Combine terms --> solute effect (16.39)
nw =Mwmv
≈4πri
3ρw3mv
′ p q,s,ipq,s
≈1−3mvivMs4πri
3ρwms
Köhler EquationCombine curvature and solute effects --> Sat. ratio at equilibrium
(16.40)
Simplify Köhler equation (16.42)
Set Köhler equation to zero --> (16.43)Critical radius for growth and critical saturation ratio
′ S q,i =′ p q,s,ipq,s
≈1+2σpmp
riR*Tρp,i
−3mvivMs
4πri3ρwms
′ S q,i =1+ari
−b
ri3
a =2σpmp
R*Tρp,ib =
3mvivMs4πρwms
r* =3ba S* =1+
4a3
27b
Table 16.2
Critical radii / supersaturations for water drops containing sodium chloride or ammonium sulfate at 275 K
Köhler Equation
Sodium chloride Ammonium sulfateSolute mass (g) r* (μm) S*-1 (%) r* (μm) S*-1 (%)0 0 ∞ 0 ∞10-18 0.019 4.1 0.016 5.110-16 0.19 0.41 0.16 0.5110-14 1.9 0.041 1.6 0.05110-12 19 0.0041 16 0.0051
Radiative Cooling EffectSaturation vapor pressure over a drop that radiatively heats/cools
relative to one that does not (16.44)
Radiative cooling rate (W) (16.45)
′ p q,s,ipq,s
≈1+Le,qmqHr,i
4πri R*T2 ′ κ d,i
Hr,i = πri2
( )4π Qa mλ,αi,λ( ) Iλ −Bλ( )dλ0
∞∫
Overall Equilibrium Saturation RatioOverall equilibrium saturation ratio for liquid water (16.46)
Equilibrium saturation ratio for gases other than liquid water (16.47)
′ S q,i =′ p q,s,ipq,s
≈1+2σpmp
riR*Tρp,i
−3mvivMs
4πri3ρwms
+Le,qmqHr,i
4πri R*T2 ′ κ d,i
′ S q,i =′ p q,s,ipq,s
≈1+2σpmp
riR*Tρp,i
Flux to Drop With Multiple ComponentsVolume of a single particle in which one species is growing
(16.48)
Time derivative of (16.47) (16.50)
Mass of a single particle in which one species is growing (16.49)
since
υi,t =υq,i,t +υi,t−h −υq,i,t−h
mi,t =ρp,i,tυi,t =ρp,qυq,i,t +ρp,i,t−hυi,t−h −ρp,qυq,i,t−h
dmi,tdt
=ρp,i,tdυi,tdt
=ρp,qdυq,i,t
dt
dυi,t−hdt
=dυq,i,t−h
dt=0
Flux to Drop With Multiple Components
Combine (16.50) and (16.48) with (16.16) (16.51)Rate of change in volume of one component in one multicomponent particle
dυq,i,tdt
=48π2 υq,i,t +υi,t−h−υq,i,t−h( )[ ]
13′ D q,i pq − ′ p q,s,i( )
′ D q,i Le,qρp,q ′ p q,s,i′ κ a,iT
Le,qmq
R*T−1
⎛
⎝ ⎜
⎞
⎠ ⎟ +
R*Tρp,q
mq
Flux to a Population of DropsVolume as a function of volume concentration
Substitute volumes into (16.49) (16.52)
υq,i,t =vq,i,tni,t−h
dvq,i,tdt
=ni,t−h
23 48π2 vq,i,t +vi,t−h −vq,i,t−h( )[ ]13
′ D q,i pq − ′ p q,s,i( )
′ D q,i Le,qρp,q ′ p q,s,i′ κ a,iT
Le,qmq
R*T−1
⎛
⎝ ⎜
⎞
⎠ ⎟ +
R*Tρp,q
mq
Flux to a Population of DropsPartial pressure in terms of mole concentration (16.53)
Vapor pressure in terms of mole concentration (16.53)
pq =CqR*T
′ p q,s,i = ′ C q,s,i R*T
Flux to a Population of DropsCombine (16.52) with (16.53) (16.54)
Effective diffusion coefficient (16.55)
dvq,i,tdt
=ni,t−h23
48π2 vq,i,t +vi,t−h−vq,i,t−h( )[ ]13
Dq,i,t−heff mq
ρp,qCq,t − ′ C q,s,i,t−h( )
Dq,i,t−heff =
′ D q,imq ′ D q,i Le,q ′ C q,s,i,t−h
′ κ a,iT
Le,qmq
R*T−1
⎛
⎝ ⎜
⎞
⎠ ⎟ +1
Flux to a Population of DropsSimplify effective diffusion coefficient for non-water gases (16.56)
Corresponding gas-conservation equation (16.57)
Dq,i,t−heff ≈ ′ D q,i =Dqωq,iFq,L,i =
DqFq,L,i
1+1.33+0.71Knq,i
−1
1+Knq,i−1 +
4 1−αq,i( )
3αq,i
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ Knq,i
dCq,tdt
=−ρp,qmq
dvq,i,tdt
i=1
NB
∑
Matrix of Partial Derivatives for Growth ODEs
(16.58)
1−hβs∂2vq,1,t∂vq,1,t∂t
0 0 −hβs∂2vq,1,t∂Cq,t∂t
0 1−hβs∂2vq,2,t
∂vq,2,t∂t0 −hβs
∂2vq,2,t
∂Cq,t∂t
0 0 1−hβs∂2vq,3,t∂vq,3,t∂t
−hβs∂2vq,3,t∂Cq,t∂t
−hβs∂2Cq,t
∂vq,1,t∂t−hβs
∂2Cq,t
∂vq,2,t∂t−hβs
∂2Cq,t
∂vq,3,t∂t1−hβs
∂2Cq,t
∂Cq,t∂t
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
vq,1,t vq,2,t vq,3,t vq,4,t
vq,1,t
vq,2,t
vq,3,t
vq,4,t
Partial Derivatives For Matrix(16.58)
∂2vq,i,t∂vq,i,t∂t
=13
ni,t−hvq,i,t
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2 3
48π2( )
13Dq,i,t−h
eff mqρp,q
Cq,t − ′ C q,s,i,t−h( )
∂2vq,i,t∂Cq,t∂t
=ni,t−h2 3
48π2 vq,i,t +vi,t−h −vq,i,t−h( )[ ]13
Dq,i,t−heff mq
ρp,q
∂2Cq,t∂vq,i,t∂t
=−ρp,qmq
∂2vq,i,t∂vq,i,t∂t
∂2Cq,t∂Cq,t∂t
=−ρp,qmq
∂2vq,i,t∂Cq,t∂t
i=1
NB
∑
(16.60)
(16.61)
(16.62)
Table 16.3
Condensation between gas phase and 16 size bins: NB + 1 = 17
Effect of Sparse-Matrix Reductions When Solving Growth ODEs
Without With Quantity Reductions ReductionsOrder of matrix 17 17Initial fill-in 289 49Final fill-in 289 49 Decomp. 1 1496 16Decomp. 2 136 16Backsub. 1 136 16Backsub. 2 136 16
Analytical Predictor of Condensation (APC) Solution For Solving Growth
Assume radius in growth term constant during time step
Define mass transfer coefficient (16.64)
Change in particle volume concentration (16.63)
dvq,i,tdt
=ni,t−h23
48π2vi,t−h( )13
Dq,i,t−heff mq
ρp,qCq,t − ′ C q,s,i,t−h( )
kq,i,t−h =ni,t−h23
48π2vi,t−h( )13
Dq,i,t−heff =ni,t−h4πri,t−hDq,i,t−h
eff
APC Solution
Volume concentration of a component (16.66)
Effective surface vapor mole concentration (16.65)
Uncorrected surface vapor mole concentration
′ C q,s,i,t−h = ′ S q,i,t−hCq,s,i,t−h
vq,i,t =mqcq,i,tρp,q
Cq,s,i,t−h =pq,s,t−h
R*T
APC Solution
(16.68)
Substitute conversions into (16.63) and (16.57) (16.67)
Integrate (16.67) for final aerosol concentration (16.69)
dcq,i,tdt
=kq,i,t−h Cq,t − ′ S q,i,t−hCq,s,i,t−h( )
dCq,tdt
=− kq,i,t−h Cq,t − ′ S q,i,t−hCq,s,i,t( )[ ]i=1
NB
∑
cq,i,t =cq,i,t−h +hkq,i,t−h Cq,t − ′ S q,i,t−hCq,s,i,t−h( )
APC Solution
Mole balance equation (16.70)
Substitute (16.69) into (16.70) (16.71)
Cq,t + cq,i,t( )i=1
NB
∑ =Cq,t−h + cq,i,t−h( )i=1
NB
∑ =Ctot
Cq,t =
Cq,t−h +h kq,i,t−h ′ S q,i,t−hCq,s,i,t−h( )i=1
NB
∑
1+h kq,i,t−hi=1
NB
∑
Aerosol mole concentration (16.69)
cq,i,t =cq,i,t−h +hkq,i,t−h Cq,t − ′ S q,i,t−hCq,s,i,t−h( )
Fig. 16.2
Comparison of APC growth solution, when h = 10 s, with an exact solution. Both solutions lie almost on top of each other.
APC Growth Simulation
10
-1
10
1
10
3
10
5
10
7
10
9
0.1 1 10 100
dv (
μ
m
3
cm
-3
) / d log
10
D
p
(Particle diameter D
p
, μ )m
Initial
Final
dv (μm
3 cm
-3)
/ d lo
g 10D
p
Solving Homogen. Nucl. with Cond.
Sum nucleation, condensation transfer rates in first bin (16.73)
Homog. nucleation rate converted to mass transfer rate (16.74)
Final number concentration in first bin after nucleation (16.74)
kq,hom,1,t−h =ρqυ1mq
J hom,qCq,t−h− ′ S q,1,t−hCq,s,1,t−h
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
kq,1,t−h =kq,cond,1,t−h+kq,hom,1,t−h
n1,t =n1,t−h+MAX cq,1,t −cq,1,t−h( )mq
ρqυ1
kq,hom,1,t−hkq,1,t−h
,0⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Fig. 16.3
Homogeneous Nucleation with Condensation Simultaneously
10
2
10
4
10
6
10
8
10
10
0.001 0.01 0.1 1
Initial
After 8 seconds
dn (No. cm
-3
) / d log
10
D
p
Particle diameter (D
p
, μ )m
dn (
No.
cm
-3)
/ d lo
g 10D
p
Fig. 16.4
Growth plus coagulation pushes particles to larger sizes than does growth alone or coagulation alone
Effect of Coagulation on Condensation
10
-1
10
1
10
3
10
5
0.01 0.1 1 10
dn (No. cm
-3
) / d log
10
D
p
Particle diameter (D
p
, μ )m
.Init
. Coag only
Growth only
Growth
+ .coag
dn (
No.
cm
-3)
/ d lo
g 10D
p
Fig. 16.4
Growth plus coagulation pushes particles to larger sizes than does growth alone or coagulation alone
Effect of Coagulation on Condensation
10
-1
10
0
10
1
10
2
10
3
0.01 0.1 1 10
dv (
μ
m
3
cm
-3
) / d log
10
D
p
(Particle diameter D
p
, μ )m
Initial
. Coag only
Growth
only
+ .Growth coagdv (μm
3 cm
-3)
/ d lo
g 10D
p
Fig. 16.5
Comparison of full-moving (FM) with moving-center (MC) results for growth-only and growth plus coagulation cases shown in Fig. 16.4(a)
Growth With Different Size Structures
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
0.01 0.1 1 10
dn (No. cm
-3
) / d log
10
D
p
Particle diameter (D
p
, μ )m
Growth
+ .coag
( )MC
Growth
+ .coag
( )FM
Growth
only
( )MC
( )Growth only FM
dn (
No.
cm
-3)
/ d lo
g 10D
p
Ice Crystal Growth
Rate of mass growth of a single ice crystal (16.76)
dmidt
=4πχi ′ D v,i pq − ′ p v,I ,i( )
′ D v,iLs ′ p v,I,i′ κ a,iT
LsRvT
−1⎛
⎝ ⎜
⎞
⎠ ⎟ +RvT
Ice Crystal GrowthElectrical capacitance of crystal (cm) (16.77)
ac,i = length of the major semi-axis (cm)
bc,i = length of the minor semi-axis (cm)
χi =
ac,i 2 sphere
ac,iec,i ln 1+ec,i( )ac,i bc,i[ ] prolatespheroid
ac,iec,i sin−1ec,i oblatespheroid
ac,i ln 4ac,i2 bc,i
2( ) needle
ac,iec,i ln 1+ec,i( ) 1−ec,i( )[ ] column
ac,iec,i 2sin−1ec,i( ) hexagonalplate
ac,i π thinplate
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
ec,i = 1−bc,i2 ac,i
2
Ice Crystal GrowthEffective saturation vapor pressure over ice
Ventilation factor for falling oblate spheroid crystals (16.78)
x= xq,i for ventilation of gas
x= xh,i for ventilation of energy
′ p v,I ,i = ′ S v,i pv,I
Fq,I,i, Fh,I,i =1+0.14x2 x <1.0
0.86+0.28x x ≥1.0
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