Presentation Slides for

Chapter 16of

Fundamentals of Atmospheric Modeling 2nd Edition

Mark Z. JacobsonDepartment of Civil & Environmental Engineering

Stanford UniversityStanford, CA 94305-4020jacobson@stanford.edu

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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