Chapter 17 Fundamentals of Atmospheric Modeling - Stanford University

Overhead Slides for

Chapter 17 of

Fundamentals of Atmospheric Modeling by

Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA 94305-4020 August 21, 1998

Mass Flux to and From a Single Drop Rate of change of mass (g) of single pure liquid water drop dm d ρv = 4πR 2Dv dt dR

(17.1)

Integrate from drop surface to infinity

(

dm = 4πrDv ρv − ρv, r dt

)

(17.2)

Rate of change of heat energy at drop surface due to conduction d Q*r dT = −4πR2 κd dt dR

(17.3)

Integrate from drop surface to infinity d Q*r = 4πrκ d (Tr − T ) dt

(17.4)

Relationship between change in mass and heat energy at surface d Tr dm dQ r* mcW = Le − dt dt dt

(17.5)

Combine (17.4) and (17.5) and assume steady state Le

dm = 4πrκ d (Tr − T ) dt

(17.6)

Mass Flux to and From a Single Drop Combine equation of state at saturation pv, s = ρv,s Rv T with Clausius Clapeyron equation dpv, s ρv,s Le = dT T to obtain dρ v,s L dT dT = e 2 − ρv, s Rv T T

(17.7)

Integrate from infinity to drop surface ln

ρ v,s ( Tr ) = ρ v,s (T )

Le (Tr −T ) T − ln r Rv TTr T

(17.8)

Simplify since T ≈ Tr ρ v,s (Tr ) − ρv, s (T ) Le (Tr −T ) Tr − T = − ρ v,s ( T ) Rv T 2 T

(17.9)

Substitute (17.6) into (17.9) ρ v,s (Tr ) − ρv, s (T )  Le  dm Le = −1 ρ v,s ( T ) 4πrκ d T  RvT  dt

(17.10)

Mass Flux to and From a Single Drop Substitute (17.2) into (17.10)  dm ρ v −ρ v,s ( T )  Le  Le  1 = − 1 +    ρv, s (T )  4πrκd T  Rv T  4πrDv ρv, s (T )  d t (17.11) Mass-flux form of growth equation

(

)

4πrDv pv − pv, s dm = dt Dv Le pv, s  Le  − 1 + RvT κd T  Rv T 

(17.12)

Rewrite mass-flux form equation for trace gases and particle sizes

(

)

4πri Dq,i ′ pq − pq′ ,s,i dm i = Dq,i ′ Le, q p′q, s,i  Le, q mq dt  R*T − 1  + κ ′d,i T  R*T  mq

(17.13)

Fluxes to and From a Single Drop Change in mass as a function of change in radius dm i dr = 4πri2ρ p, i i dt dt

(17.14)

Radius-flux form of growth equation

(

)

Dq,′ i pq − pq′ ,s ,i dri ri = dt Dq,i ′ Le, qρ p, i p′q, s ,i  Le,q mq  R*Tρp ,i −1 +  * κ d′ ,i T R T mq  

(17.15)

Change in mass as a function of change in volume dm i dυ = ρp ,i i dt dt Volume-flux form of growth equation

(

)13

(

)

48π2 υi Dq′ ,i pq − p q, ′ s,i dυi = dt Dq,i ′ Le, q ρ p,i pq′ , s, i  Le, q mq  R*Tρ p, i − 1 +  * κ′d, iT R T mq  

(17.16)

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 ) 3 Dq = 8Adq2 ρa

R*Tm a  m q + ma    2π  mq 

Example 17.1. Calculate diffusion coefficient of carbon monoxide T = 288 K pa

= 1013 mb

ρa

= 0.00123 g cm-3

mq

= 28 g mole-1

Dq

= 0.12 cm2 s-1

(17.17)

Corrected Gas Diffusion Coefficient Corrected diffusion coefficient Dq′ ,i = Dq ω q, i Fq, i

(17.18)

Correction for collision geometry and sticking probability

(

 1.33 + 0.71Kn q−1,i 4 1−α q,i ω q,i = 1 +  + −1 3α q, i  1+ Kn   q, i

)

−1

  Kn q,i   

(17.19)

Knudsen number for condensing vapor Kn q,i =

λq

(17.20)

ri

0 ω q,i →  1

as

Knq, i → ∞

(small particles)

as

Knq, i → 0

(large particles) (17.21)

Mass accommodation (sticking) coefficient, αq,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 32Dq  ma  3Dq λq =  ≈ 3πv q  ma + mq  vq

(17.23)

Ventilation factor Corrects for increased rate of vapor transfer due to eddies sweeping vapor to the surface of large particles 1+ 0.108x2q, i Fq,i =  0.78 + 0.308xq, i 12

x q, i = Re i

Sc1q 3

x q, i ≤ 1.4 x q, i > 1.4

(17.24)

(17.24)

Particle Reynolds number Re i =

2ri Vf ,i νa

Gas Schmidt number Sc q =

νa Dq

(17.25)

Corrected Thermal Conductivity Corrected thermal conductivity of dry air (J cm-1 s-1 K-1 ) κ d′ ,i = κ dω h, iFh,i ,

(17.26)

Correction for collision geometry and sticking probability −1  1.33 + 0.71Kn h ,i 4(1 −α h )   Kn h ,i  ω h,i = 1 +  + −1 3α 1 +Kn h,i   h  

−1

(17.27)

Knudsen number for heat Kn h,i =

λh ri

(17.28)

Mean free path of heat λh =

3Dh va

(17.29)

Corrected Thermal Conductivity Thermal accommodation (sticking) coefficient, αt Fraction of molecules bouncing off surface of a drop that have acquired temperature of drop ≈ 0.96 for water. T −T αh = m Ts − T

(17.30)

Ventilation factor Corrects for increased rate of heat transfer due to eddies 1+ 0.108x2 h,i Fh,i =  0.78 + 0.308xh ,i 12

x h,i = Rei

x h, i ≤ 1.4 x h, i > 1.4

(17.31)

Pr1 3

Prandtl number Pr = ηac p, m κ d

(17.32)

Corrected Surface Vapor Pressure Curvature (Kelvin) effect Increases vapor pressure over small drops Solute effect Decreases vapor pressure over small drops Radiative cooling effect Decreases vapor pressure over large drops

Fig. 17.1. Curvature and solute effects.

Saturation ratio

1.02 Curvature effect 1.01 1 0.99

S* Equilibrium saturation ratio

0.98 0.01

Solute effect r*

0.1 1 Particle radius (µm)

10

Curvature Effect Saturation vapor pressure over curved, dilute surface relative to that over a flat , dilute surface  2σ p m p  2σ p m p  ≈ 1+ = exp *  pq, s ri R*Tρp,i  ri R Tρp, i 

p q,s,i ′

Note exp ( x ) ≈ 1+ x for small values of x

(17.33)

Solute Effect Vapor pressure over flat water surface with solute relative to that without solute (Raoult's Law) p q,s,i ′ pq, s

=

nw nw + ns

(17.34)

Relatively dilute solution nw >> ns p q,s,i ′

n ≈ 1− s pq, s nw

(17.35)

Number of moles of solute in solution i M ns = v s ms Number of moles of liquid water in a drop 3 Mw 4πri ρ w nw = ≈ mv 3mv

(17.36)

Combine terms --> solute effect p q,s,i ′ pq, s

≈ 1−

3mv iv M s

4πri3 ρw ms

(17.37)

Köhler Equation Combine curvature and solute effects --> Sat. ratio at equilibrium S q,i ′ =

pq′ , s, i pq ,s

≈ 1+

2σ pm p riR *Tρp, i

−

3mvi v Ms

(17.38)

4πri3ρ w ms

Simplify Köhler equation S q,i ′ = 1+

a=

a b − 3 ri ri

2σp m p R*Tρp,i

(17.40)

b=

3mvi v Ms 4πρ wms

(17.40)

Set Köhler equation to zero --> Critical radius for growth and critical saturation ratio 3b r* = a

4a 3 S* = 1+ 27b

(17.41)

Table 17.1. 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 S * −1 r* (µm) (%) 0 ∞ 0.019 4.1 0.19 0.41 1.9 0.041 19 0.0041

Ammonium Sulfate S * −1 r* (µm) (%) 0 ∞ 0.016 5.1 0.16 0.51 1.6 0.051 16 0.0051

Radiative Cooling Effect Surface vapor pressure over curved, dilute surface relative to that over a flat , dilute surface p q,s,i ′ pq, s

≈ 1+

Le, q mq Hr,i 4πri R*T 2κ d′ ,i

(17.42)

Radiative cooling rate (W) ∞

( ) ∫0 Qa ( mλ ,α i, λ )(I λ − Bλ)dλ

Hr,i = πri2 4π

(17.43)

Overall Equilibrium Saturation Ratio Overall equilibrium saturation ratio for liquid water pq′ , s, i

2σ pm p

Le,qm qH r, i 3mvi v Ms S q,i ′ = ≈ 1+ − + pq ,s riR *Tρp, i 4πri3ρ w ms 4πri R*T 2 κ′d, i (17.44)

Equilibrium saturation ratio for gases other than liquid water when surface vapor pressures computed separately S q,i ′ =

pq′ , s, i pq ,s

≈ 1+

2σ pm p riR *Tρp, i

(17.45)

Flux to Drop With Multiple Components Volume of a single particle in which one species is growing υi ,t = υ q,i,t + υi ,t −h − υ q,i,t −h

(17.46)

Mass of a single particle in which one species is growing mi,t = ρ p, i, tυ i,t = ρp ,q υq, i, t + ρ p,i, t− hυi, t− h −ρ p, qυ q,i,t −h (17.47) Time derivative of (17.47) dυq,i, t dm i, t dυi, t = ρp ,i ,t = ρp ,q dt dt dt

(17.48)

since dυq, i, t − h dυi ,t −h = =0 dt dt Combine (17.48) and (17.46) with (17.16) Rate of change in volume of one component in one multicomponent particle

dυq ,i , t dt

[ =

(

48π2 υq, i,t + υ i, t− h − υ q,i,t −h

)]

13

(

Dq′ ,i pq − p′q, s, i

)

Dq′ ,i Le,q ρ p,q pq, ′ s,i  Le, q mq  R *Tρp ,q − 1 +  κ′d, iT mq  R*T  (17.49)

Flux to a Population of Drops Volume as a function of volume concentration υ q,i, t =

vq, i, t ni ,t −h

Substitute volumes into (17.49) dv q, i, t dt

=

[

(

23 2 ni,t −h 48π v q, i, t + vi, t− h −v q,i, t− h

)]

13

(

Dq, ′ i pq − p′q,s,i

Dq,i ′ Le,q ρ p,q pq, ′ s, i  Le, q mq  R*Tρp, q − 1 +  κ′d, iT mq  R*T  (17.50)

Partial pressure in terms of mole concentration pq = Cq R*T

(17.51)

Vapor pressure in terms of mole concentration p q,s,i ′ = Cq,s ′ ,i R*T

(17.51)

)

Flux to a Population of Drops Combine (17.50) with (17.51) d vq,i, t dt

23

[

(

= ni, t− h 48π 2 v q, i, t +v i, t− h − vq ,i , t− h

)]

(

1 3 eff mq Dq,i ,t − h C − Cq, ′ s, i, t − h ρ p,q q, t

(17.52) Effective diffusion coefficient eff Dq,i ,t − h =

Dq′ ,i

(17.53)

mq Dq,i ′ Le,q Cq, ′ s,i, t− h  Le, q mq  − 1  +1 * κ d,i ′ T  R T 

Simplify effective diffusion coefficient for gases other than water eff Dq,i ′ = Dq ω q,i Fq ,i = ,t − h ≈ Dq,i

Dq Fq, i

(

 1.33 + 0.71Kn −1 4 1−α q, i q,i 1+ + −1  1+ Knq, 3α q, i i

) Kn q,i 

(17.54) Corresponding gas-conservation equation d Cq ,t dt

ρ p, q N B d vq,i,t =− mq dt

∑

i= 1

(17.55)

)

Integrate Growth Equations Matrix of partial derivatives v q,1,t v q,1,t v q,2, t v q,3,t Cq, t

(17.73) v q,2, t

v q,3,t

Cq, t

 ∂2v q,1,t ∂2v q,1,t  1− hβs 0 0 −hβs ∂v ∂t ∂Cq ,t ∂t  q ,1,t  ∂2v q,2, t ∂ 2vq,2,t  0 1− hβs 0 −hβs  ∂vq ,2,t ∂t ∂Cq ,t ∂t  ∂ 2vq,3,t ∂2vq,3,t  0 0 1 − hβs −hβs  ∂vq,3,t ∂t ∂Cq ,t ∂t  2 2 2 2  −hβ ∂ Cq, t −hβ ∂ Cq, t −hβ ∂ Cq, t 1 − hβ ∂ Cq ,t s s s s  ∂v ∂t ∂v ∂t ∂v ∂t ∂C q, t∂t q,1,t q,2,t q,3,t  (17.56)

2 3 1 3 eff mq 1  ni,t −h  =   48π 2 Dq,i ,t − h C − Cq,′ s, i, t − h ∂v q, i, t ∂t 3  vq,i ,t  ρ p,q q, t (17.57) 2 ∂ vq ,i ,t 1 3 eff mq 23 2 = ni, t− h 48π vq,i,t + vi,t −h − vq ,i ,t −h Dq,i ,t −h ∂Cq,t ∂t ρ p,q (17.58) 2 2 ∂ Cq, t ρ p,q ∂ vq ,i , t =− , (17.59) ∂vq, i, t∂t mq ∂v q,i,t ∂t

∂ 2 vq ,i ,t

(

[

(

ρ p,q N B ∂2 vq,i, t =− ∂Cq,t ∂t mq ∂C q, t∂t i= 1 ∂ 2 Cq,t

∑

)

(

)]

(17.60)

)

            

Integrated Solution to Growth Equations Table 17.2. Reduction in array space and in the number of matrix operations before and after the use of sparse-matrix techniques to solve growth ODEs. NB + 1 = 17.

Order of matrix No. init. matrix spots filled % of initial positions filled

Growth / Evaporation Initial After Sparse Matrix Reductions 17 17 289 49 100 17

No. fin. matrix spots filled % of final positions filled

289 100

49 17

No. operations decomp. 1 No. operations decomp. 2 No. operations backsub. 1 No. operations backsub. 2

1496 136 136 136

16 16 16 16

Analytical Predictor of Condensation Assume radius in growth term constant during time step Change in particle volume concentration d vq,i, t dt

23

(

= ni, t− h 48π 2v i, t− h

)

(

1 3 eff mq Dq, i,t − h C − Cq′ ,s,i,t −h ρp,q q, t

)

(17.61) Define mass transfer coefficient

(

)

1 3 eff 23 2 eff k q, i, t − h = ni,t −h 48π vi, t− h Dq, i,t − h = ni,t −h 4πri, t − hDq,i ,t −h

(17.62) Effective surface vapor mole concentration C ′q, s,i, t− h = S q,i,t ′ −hC q, s,i ,t −h

(17.63)

Volume concentration of a component v q, i, t = mqcq,i ,t ρ p, q Uncorrected surface vapor mole concentration Cq, s, i, t − h = pq,s,t − h R* T

(17.64)

Analytical Predictor of Condensation Substitute conversions into (17.61) and (17.55) dc q, i, t dt dCq, t dt

(

= kq ,i ,t −h Cq, t − S q,i ′ ,t −hC q, s,i ,t − h

)

(17.65)

NB

=−

∑ [kq,i,t −h (Cq, t − Sq,′ i, t− h Cq ,s ,i,t )]

(17.66)

i =1

Integrate (17.65) for final aerosol concentration

(

c q, i, t = c q, i, t − h + hkq, i, t− h Cq ,t − S q, ′ i,t − h Cq, s,i, t− h

)

(17.67)

Mass balance equation Cq, t +

NB

NB

∑ ( cq ,i ,t ) = C q, t− h + ∑ ( cq ,i ,t −h ) = Ctot

i =1

(17.68)

i =1

Substitute (17.67) into (17.68) Cq, t − h + h Cq, t =

NB

∑ (kq, i, t− h Sq,′ i, t− h Cq , s, i, t − h)

i =1

NB

1+ h

∑ kq,i ,t −h

i =1

(17.69)

APC Growth Simulation

dV /d log 10Dp V ( in µm3 cm -3)

Fig. 17.2. Comparison of APC solution, when h = 10 s, to an exact solution of condensational growth. Both solutions lie almost exactly on top of each other. 109 107 5

Final

10

1000

Initial

10 0.1 0.1

1 10 Time from start (h)

100

Effect of Coagulation on Condensation

dN /d log 10Dp (N in particles cm-3 )

Figs. 17.3 a and b. Growth plus coagulation pushes particles to larger sizes than does growth alone or coagulation alone.

Growth only

105 Init.

103

Growth + coag. Coag. only

101

3 -3 dV /dlog 10Dp (V in µm cm )

10-1 0.01

0.1 1 Particle diameter (µm)

10

103 Growth only

102 101 100

Initial

10-1 0.01

Coag. only Growth + coag.

0.1 1 Particle diameter (µm)

10

Moving-Center vs. Full-Moving Size Structure

dV /d log 10Dp V ( in µm3 cm -3 )

Fig. 17.4. Comparison of moving-center (MC) particle size structure to full-moving (FM) size structure for the growth / coagulation and growth-only cases shown in fig. (17.3 a). 106 105 104 103

Growth only (FM) Growth only (MC)

2

10 101

100 10-10.01

Growth + coag. (FM)

Growth + coag. (MC)

0.1 1 Particle diameter (µm)

10

Ice Crystal Growth Rate of mass growth of a single ice crystal

(

)

4πχi Dv,i ′ pq − pv,′ I ,i dm i = dt Dv, ′ iLs p′v,I,i  Ls  − 1 + RvT  κ d, ′ iT  Rv T 

(17.71)

Electrical capacitance of crystal (cm) a 2  c,i ac,i ec, i  ac,i ec, i  χ i = a c,i ln  ac,i ec, i  a c,i ec, i  a c,i π

[(

)

ln 1 +ec, i a c,i b c,i sin −1 ec, i

]

(4a 2c,i b c,i2 ) ln[ (1 +ec, i ) (1− ec,i ) ] (2sin −1 ec, i)

sphere prolate spheroid oblate spheroid needle column hexagonal plate thin plate (17.72)

a c,i b c,i

= length of the major semi-axis (cm) = length of the minor semi-axis (cm)

ec,i = 1− b 2c,i a2c,i

Ice Crystal Growth

Effective saturation vapor pressure over ice p v,I ′ ,i = Sv,i ′ pv, I

Ventilation factor for falling oblate spheroid crystals 1 + 0.14x 2 Fq,i ,Fh,i =  0.86 + 0.28x x = x q, i for ventilation of gas x = x h,i for ventilation of heat

x