presentation slides for chapter 17, part 2 of fundamentals of atmospheric modeling 2 nd edition

Presentation Slides for

Chapter 17, Part 2of

Fundamentals of Atmospheric Modeling 2nd Edition

Mark Z. JacobsonDepartment of Civil & Environmental Engineering

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

April 1, 2005

Solvation and HydrationSolvation

Bonding between solvent and solute in solution

Hydration

When solvent is liquid water, solvation is hydration

Hydration of cations --> lone pairs of electrons on oxygen atom of water attach to cations

Hydration of anions --> water molecule attaches to anion via hydrogen bonding

Water EquationQuantify amount of hydration with empirical water equation

Zdanovskii-Stokes-Robinson (ZSR) equation

Example with two species, x and y (17.64)

mx,a, my,a = molalities of x and y, alone in solution at given relative humidity

mx,m, my,m = molalities of x and y, when mixed together, at same relative humidity

mx,mmx,a

+my,mmy,a

=1

ZSR Equation

Table 17.2

ZSR equation predictions for a sucrose (species x) - mannitol (species y) mixture at two different water activities.

mx,m/mx,a +

Case mx,a my,a mx,m my,m my,m/my,a

1 0.7751 0.8197 0.6227 0.1604 0.999

2 0.9393 1.0046 0.1900 0.8014 1.000

mx,mmx,a

+my,mmy,a

=1

Water EquationGeneralized ZSR equation (17.64)

Polynomial expression for molality of electrolyte alone in solution at a given water activity (17.66)

mk,mmk,ak

∑ =1

mk,a =Y0,k +Y1,kaw +Y2,kaw2 +Y3,kaw

3 +...

Water Equation

Fig. 17.4a

Water activities of several electrolytes at 298.15 K

0 10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60

Molality

Water activityNaNO

3

HNO

3

H

2

SO

4HCl

Wat

er a

ctiv

ity

Water Equation

Fig. 17.4b

Water activities of several electrolytes at 298.15 K

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

Molality

Water activity

NH

4

NO

3

NH

4

Cl

NaCl

(NH

4

)

2

SO

4

Na

2

SO

4

Wat

er a

ctiv

ity

Temp. Dependence of Water ActivityTemperature dependence of binary water activity coefficients under

ambient surface conditions is small.

Polynomial for water activity at reference temperature (17.68)

Temperature dependence of water activity (17.67)

lnaw T( ) =lnaw0 −

mvmk,a2

R*TLT0

∂φL∂mk,a

+TC∂φcP∂mk,a

⎛

⎝ ⎜

⎞

⎠ ⎟

lnaw0 =A0 +A1mk,a

12+A2mk,a +A3mk,a

32+...

Temp. Dependence of Water ActivityCombine (17.67), (17.68), (17.54) (17.69-70)

lnaw T( ) =A0 +A1mk,a12

+A2mk,a +E3mk,a32

+E4mk,a2 +...

El =Al −0.5 l −2( )mv

R*TLT0

Ul −2 +TCVl−2⎛

⎝ ⎜

⎞

⎠ ⎟

Example mHCl= 16 m

T = 273 K---> aw = 0.09

T = 310 K---> aw = 0.11

Practical Use of Water EquationRearrange (17.65) (17.71)

mi,j,a = binary molalities of species alone in solution ci,j,m = hypothetical mol cm-3 of electrolyte pair when mixed in solution with all other components

In a model, ion concentrations known but hypothetical electrolyte concentrations unknown --> find hypothetical concentrations

cw =1mv

ci, j,mmi, j,aj =1

NA

∑⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

i=1

NC

∑

Practical Use of Water Equation

6 mol m-3 of H+, 6 mol m-3 Na+

7 mol m-3 of Cl- , 5 mol m-3 of NO3-

Example 17.1:

Combine ions in a way to satisfy mole balance constraintscH+,m=cHNO3,m+cHCl,m

cNa+,m =cNaNO3,m+cNaCl,m

cCl−,m=cHCl,m+cNaCl,m

cNO3- ,m =cHNO3,m+cNaNO3,m

Case cHCl,m cHNO3,m cNaCl,m cNaNO3,m

1 6 0 1 52 4 2 3 3

Concentrations that satisfy mole balance constraints (Table 17.3)

Practical Use of Water Equation

Cation

Automatic method to recombine ions into hypothetical electrolytes

Execute the following three equations, in succession, for each undissociated electrolyte, i,j

Electrolyte (17.72)

Anion

ci, j,m=minci,mνi

,cj,mν j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

ci,m=ci,m−νici, j,m

cj,m=cj,m−ν jci, j,m

Deliquescence Relative HumidityDeliquescence

Process by which a particle takes up liquid water, lowering its saturation vapor pressure

Deliquescence relative humidity (DRH)The relative humidity at which an initially-dry solid first takes on liquid water during an increase in relative humidity. Above the DRH, the solid may not exist.

Crystallization relative humidity (CRH)The relative humidity at which an initially-supersaturated aqueous electrolyte becomes crystalline upon a decrease in relative humidity.

Deliquescence Relative Humidity

Table 17.4

DRHs and CRHs for several electrolytes at 298 K

In a mixture, the DRH of a solid in equilibrium with the solution is lower than the DRH of the solid alone

Electrolyte DRH(%) CRH(%)

NaCl 75.28 47Na2SO4 84.2 57-59NaHSO4 52.0 <5NH4Cl 77.1 47(NH4)2SO4 79.97 37-40NH4HSO4 40 <5-22NH4NO3 61.83 25-32KCl 84.26 62Oxalic acid 97.3 51.8-56.7

Solid Formation

Consider the equilibrium reaction

Consider the equilibrium reaction

A solid forms when (17.73)

A solid forms when (17.74)

NH4NO3 s( ) NH4++NO3

−

mNH4

+mNO3−γ

NH4+,NO3

−2 >Keq T( )

NH4NO3 s( ) NH3 g( )+HNO3 g( )

pNH3 g( ),spHNO3 g( ),s >Keq T( )

Example Equilibrium ProblemConsider two equilibrium reactions (17.75)

HCl (g)H

+

+ Cl-

HSO4 H

+

+ SO

2-

4

For equilibrium concentrations, solve

equilibrium constant equations

mole balance equations

charge balance equation

water equation

with Newton-Raphson iteration

Example Equilibrium Problem

Equilibrium coefficient equations (17.76)

mH+,eq

mCl- ,eq

γH+,Cl- ,eq2

pHCl,s,eq=Keq T( )

mH+,eq

mSO4

2−,eqγ2H+,SO4

2−,eq3

mHSO4

−,eqγ

H+,HSO4−,eq

2 =Keq T( )

Example Equilibrium Problem

Mole balance equations (17.77)

CHCl(g),eq+cCl-,eq

=CHCl(g),t−h +cCl-,t−h

cHSO4

−,eq+c

SO42−,eq

=cHSO4

−,t−h+c

SO42−,t−h

(17.78)

Example Equilibrium ProblemVapor pressure as a function of mole concentration (17.79)

Charge balance equation (17.80)

Molality as a function of mole concentration

pHCl,s,eq=CHCl(g),s,eqR*T

mCl-,eq

=cCl- ,eq

cw,eqmv

cCl−,eq

+cHSO4

−,eq+2c

SO42−,eq

=cH+,eq

Example Equilibrium ProblemWater equation (17.81)

Hypothetical mole concentration constraints (17.82)

cw,eq =1mv

cH+,Cl- ,m

mH+,Cl- ,a

+cH+,HSO4

−,m

mH+,HSO4

−,a

+c2H+,SO4

2−,m

m2H+,SO4

2−,a

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

cH+,eq

=cH+,Cl-,m

+cH+,HSO4

−,m+2c

2H+,SO42−,m

cCl−,eq

=cH+,Cl-,m

cHSO4

−,eq=c

H+,HSO4−,m

cSO4

2−,eq=c

2H+,SO42−,m

Mass-Flux Iterative MethodSolve each equation iteratively and iterate over all equations

Initialize species concentrations so that charge is conserved

No intelligent first guess required

Solution mass and charge conserving and always converges

Example solution for one equilibrium equation

Equilibrium equation and coefficient relation

νDD+νEE +... νAA +νBB+...

A{ }νA B{ }νB ...

D{ }νD E{ }νE ...=Keq T( )

Mass-Flux Iterative Method1) Calculate smallest ratio of mole concentration to moles in

denominator and numerator, respectively (17.83)

2) Initialize two parameters

Qd =minCD,0νD

,CE,0νE

⎛

⎝ ⎜

⎞

⎠ ⎟

Qn =mincA,0νA

,cB,0νB

⎛

⎝ ⎜

⎞

⎠ ⎟

z1 =0.5(Qd +Qn) Δx1=Qd −z1

Mass-Flux Iterative MethodAdd mass flux factor (x) to mole concentrations (17.84)

3) Compare ratio of activities to equilibrium coefficient (17.85)

cA,l+1 =cA,l +νAΔxl cB,l+1 =cB,l +νBΔxl

CD,l+1=CD,l −νDΔxl CE,l +1 =CE,l −νEΔxl

F =mA,l+1

νA mB,l +1νB γAB,l+1

νA+νB

pD,l+1νD pE,l+1

νE

1Keq T( )

Mass-Flux Iterative Method4) Cut z in half

5) Check convergence (17.86)

Return to (17.84) until convergence occurs

zl+1 =0.5zl

F =

>1 → Δxl +1=−zl +1<1 → Δxl +1=+zl +1=1 → convergence

⎧

⎨ ⎪

⎩ ⎪

Analytical Equilibrium Iteration MethodSolve most equations analytically but iterate over all equations

Reactions of the form DA

Solve the equilibrium equation (17.87)

Solution for change in concentration (17.88)

Final concentrations

cA,ccD,c

=cA,0+ΔxfincD,0 −Δxfin

=Kr

Δxfin=cD,0Kr −cA,0

1+Kr

cA,c =cA,0+Δxfin cD,c =cD,0 −Δxfin

Analytical Equilibrium Iteration Method

Solve the equilibrium equation (17.89)

Reactions of the form D+EA+B

Solution for change in concentration (17.90)

cA,ccB,ccD,ccE,c

=cA,0 +Δxfin( ) cB,0 +Δxfin( )

cD,0 −Δxfin( ) cE,0−Δxfin( )=Kr

Δxfin=

−cA,0 −cB,0 −cD,0Kr −cE,0Kr

+cA,0 +cB,0 +cD,0Kr +cE,0Kr( )

2

−41−Kr( ) cA,0cB,0−cD,0cE,0( )

2 1−Kr( )

Analytical Equilibrium Iteration Method

Final concentrations

cA,c =cA,0+Δxfin

cB,c =cB,0 +Δxfin

cD,c =cD,0 −Δxfin

cE,c =cE,0 −Δxfin

Analytical Equilibrium Iteration Method

Check if solid can form (17.91)

Reactions of the form D(s)2A+B

If so, solve the equilibrium equation (17.92)

cA,0 +2cD,0( )2

cB,0+2cD,0( )>Kr

cA,c2 cB,c = cA,0 +2Δxfin( )

2cB,0 +Δxfin( )=Kr

Analytical Equilibrium Iteration MethodIterative Newton-Raphson procedure (17.93)

fn x( ) =Δxfin,n3 +qΔxfin,n

2 +rΔxfin,n +s =0

′ f x( ) =3Δxfin,n2 +2qxfin,n +r

q =cA,0 +cB,0

r =cA,0cB,0+0.25cA,02

s =cA,02 cB,0−Kr

Δxfin,n+1=Δxfin,n −fn x( )′ f n x( )

Analytical Equilibrium Iteration Method

Final concentrations

cA,c =cA,0+2Δxfin

cB,c =cB,0 +Δxfin

cD,c =cD,0 −Δxfin

Equilibrium Solver Results

Fig. 17.4

Aerosol composition versus NaCl concentration when the relative humidity was 90%. Other initial conditions were H2SO4(aq) = 10 g m-3, HCl(g) = 0 g m-3, NH3(g) = 10 g m-3, HNO3(g) = 30 g m-3, and T = 298 K.

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Concentration (

g m

-3

)

NO

3

-

H

2

( ) 0.1O aq x

SO

4

2-

NH

4

+

(NaCl concentration g m

-3

)

Cl

-

Con

cent

rati

on (g

m-3)

Equilibrium Solver Results

Fig. 17.5

Aerosol composition versus relative humidity. Initial conditions were H2SO4(aq) = 10 g m-3, HCl(g) = 0 g m-3, NH3(g) = 10 g m-3, HNO3(g) = 30 g m-3, and T = 298 K.

0

5

10

15

20

25

0 20 40 60 80 100

Concentration (

g m

-3

)

NH

4

NO

3

( )s

NO

3

-

( )Relative humidity percent

SO

4

2-

NH

4

+

H

2

( ) 0.1O aq x

(NH

4

)

2

SO

4

( )s

Con

cent

rati

on (g

m-3)

Dissolutional GrowthSaturation vapor pressure of gas q over particle size i (17.95)

Saturation vapor pressure as function of gas mole concentration (17.96)

Molality as function of particle mole concentration (17.97)

pq,s,i =mq,iHq

pq,s,i =Cq,s,i R*T

mq,i =cq,i

mvcw,i

Dissolutional GrowthSubstitute (17.95) and (17.97) into (17.96) (17.98)

where (17.99)

Cq,s,i =pq,s,i

R*T=

mq,i

R*THq=

cq,i

mvcw,i R*THq

=cq,i

′ H q,i

′ H q,i =mvcw,i R*THq

Dissolutional GrowthCondensational growth equations (16.67)

(16.68)

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

∑

Dissolutional GrowthSubstitute (17.98)

--> Dissolutional growth equations (17.100)

(17.101)

dcq,i,tdt

=kq,i,t−h Cq,t − ′ S q,i,t−hcq,i,t′ H q,i,t−h

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

dCq,tdt

=− kq,i,t−h Cq,t − ′ S q,i,t−hcq,i,t′ H q,i,t−h

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ i=1

NB

∑

Analytical Predictor of DissolutionIntegrate (17.100) for final aerosol concentration (17.102)

Mole balance equation (17.103)

Substitute (17.102) into (17.103) (17.104)

cq,i,t =′ H q,i,t−hCq,t

′ S q,i,t−h+ cq,i,t−h−

′ H q,i,t−hCq,t′ S q,i,t−h

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ exp−

h ′ S q,i,t−hkq,i,t−h′ H q,i,t−h

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Cq,t + cq,i,ti=1

NB

∑ =Cq,t−h+ cq,i,t−hi =1

NB

∑

Cq,t =

Cq,t−h + cq,i,t−h 1−exp−h ′ S q,i,t−hkq,i,t−h

′ H q,i,t−h

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ i=1

NB

∑

1+′ H q,i,t−h′ S i,q,t−h

1−exp −h ′ S q,i,t−hkq,i,t−h

′ H q,i,t−h

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ i=1

NB

∑

Growth During DissociationGrowth equation for hydrochloric acid (17.105)

Total dissolved chlorine (17.106)

Find saturation mole concentration from equilibrium expressions(17.107) HClHCl(aq)

(17.108) HCl(aq)H++Cl-

dcCl,i,tdt

=kHCl,i,t−h CHCl,t − ′ S HCl,i,t−hCHCl,s,i,t( )

cCl,i,t =cHCl aq( ),i,t +cCl-,i,t

Growth During DissociationEquilibrium coefficient relations (17.107)

(17.108)

Equilibrium coefficient relations in terms of mole concentration (17.109)

(17.110)

mHCl aq( ),i

pHCl,s,i=HHCl

molkg atm

mH+,imCl-,iγi,H+ Cl-2

mHCl aq( ),i=KHCl

molkg

CHCl,s,i =cCl,i′ K HCl,i

′ K HCl,i = HHCl 1+KHCl mvcw,i( )

2R*T

cH+,iγi,H+ Cl-2

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Dissolution of Acids/BasesSubstitute saturation mole concentration into growth equation (17.111)

Mole balance equation (17.112)

dcCl,i,tdt

=kHCl,i,t−h CHCl,t − ′ S HCl,i,t−hcCl,i,t′ K HCl,i,t−h

⎛

⎝ ⎜

⎞

⎠ ⎟

CHCl,t + cCl,i,ti =1

NB

∑ =CHCl,t−h + cCl,i,t−hi=1

NB

∑

Dissolution for Dissociating SpeciesIntegrate (17.111) for final aerosol concentration (17.113)

Substitute (17.113) into (17.112) (17.114)

cCl,i,t =′ K HCl,i,t−hCHCl,t

′ S Cl- ,i,t−h

+ cCl,i,t−h −′ K HCl,i,t−hCHCl,t

′ S HCl,i,t−h

⎛

⎝ ⎜

⎞

⎠ ⎟ exp−

hkHCl,i,t−h ′ S HCl,i,t−h′ K HCl,i,t−h

⎛

⎝ ⎜

⎞

⎠ ⎟

CHCl,t =

CHCl,t−h + cCl,i,t−h 1−exp−hkHCl,i,t−h ′ S HCl,i,t−h

′ K HCl,i,t−h

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ i=1

NB

∑

1+′ K HCl,i,t−h′ S HCl,i,t−h

1−exp−hkHCl,i,t−h ′ S HCl,i,t−h

′ K HCl,i,t−h

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ i=1

NB

∑

Solve for Ammonia/AmmoniumCharge balance equation (17.115)

where (17.116)

Mole balance equation (17.117)

cNH4+,i,t +cH+,i,t +c±,i,t =0

c±,i,t =−cNO3-,i,t −cCl-,i,t−cHSO4

- ,i,t −2cSO42- ,i,t + z

q∑ cq,i,t−h

CHCl,t =CNH3,t + cNH3 aq( ),i,t +cNH4+,i,t( )

i=1

NB

∑

=CNH3,t−h+ cNH3 aq( ),i,t−h +cNH4+,i,t−h( )

i =1

NB

∑ =Ctot

Solve for Ammonia/AmmoniumEquilibrium expressions (17.118) NH3(g)NH3(aq)

(17.119) NH3(aq)+H+NH4+

Equilibrium coefficient expressions (17.118)

(17.119)

mNH3 aq( ),i

pNH3

=HNH3 mol

kg atm

mNH4+,iγi,NH4

+

mNH3 aq( ),imH+,iγi,H+=KNH3

kgmol

Solve for Ammonia/AmmoniumNH4

+/H+ activity coefficient relationship (17.120)

Equilibrium coefficient relations in terms of mole concentration (17.121,2)

γi,NH4+

γi,H+=

γi,NH4+γi,NO3

−

γi,H+γi,NO3−

=γ

i,NH4+ NO3

−2

γi,H+ NO3

−2

=γi,NH4

+γi,Cl-

γi,H+γi,Cl-=

γi,NH4

+ Cl-2

γi,H+ Cl-2

cNH3 aq( ),i

CNH3

= ′ H NH3,imolmol ′ H NH3,i =HNH3R*Tmvcw,i

cNH4+,i

cNH3 aq( ),icH+,i= ′ K NH3,i

cm3

mol′ K NH3,i =KNH3

1mvcw,i

γi,H+

γi,NH4+

Solve for Ammonia/AmmoniumIon concentration in each size bin (17.124)

Substitute into mole-balance equation (17.125)

cNH4+,i,t =

−c±,i,tCNH3,t ′ H NH3,i,t−h ′ K NH3,i,t−h

CNH3,t ′ H NH3,i,t−h ′ K NH3,i,t−h +1

CNH3,t +

CNH3,t ′ H NH3,i,t−h

−c±,i,tCNH3,t ′ H NH3,i,t−h ′ K NH3,i,t−hCNH3,t ′ H NH3,i,t−h ′ K NH3,i,t−h +1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ i =1

NB

∑ −Ctot=0

Solve for Ammonia/AmmoniumIterate for ammonia gas concentration (17.126)

where (17.128)

CNH3,t,n+1=CNH3,t,n−fn CNH3,t,n( )

′ f n CNH3,t,n( )

′ f n CNH3,t,n( ) =

1+

′ H NH3,i,t−h −c±,i,t ′ H NH3,i,t−h ′ K NH3,i,t−h

CNH3,t,n ′ H NH3,i,t−h ′ K NH3,i,t−h +1

+c±,i,tCNH3,t,n ′ H NH3,i,t−h ′ K NH3,i,t−h( )

2

CNH3,t,n ′ H NH3,i,t−h ′ K NH3,i,t−h +1( )2

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

i=1

NB

∑

Simulations of Growth/Dissociation

Fig. 17.7

Initial distributions for simulation

10

-1

10

0

10

1

10

2

10

-2

10

-1

10

0

10

1

10

2

10

3

10

4

0.1 1 10

dM (

g m

-3

) / d log

10

D

p

( . dN No cm

-3

) / d log

10

D

p

(Particle diameter D

p

, )m

Soildust

( )NaCl s

.Number conc

( ) a Initial

dM (g

m-3)

/ dlo

g 10 D

pdN

(No. cm

-3) / dlog10 D

p

Simulations of Growth/Dissociation

0

5

10

15

20

25

30

0 2 4 6 8 10 12

Summed concentration (

g m

-3

)

( )Time from start h

H

2

0.1O x

NH

4

+

NO

3

-

Cl

-

( )S VI

( ) =5b h s

Na

+

Aerosol concentrations, summed over all sizes, during nonequilibrium growth plus internal aerosol equilibrium at RH=90 percent when h=5 s.

Sum

med

con

cent

rati

on (g

m-3)

Simulations of Growth/Dissociation

0

5

10

15

20

25

30

0 2 4 6 8 10 12

Summed concentration (

g m

-3

)

( )Time from start h

H

2

0.1O x

NH

4

+

NO

3

-

Cl

-

( )S VI

( ) =300c h s

Na

+

Sum

med

con

cent

rati

on (g

m-3)

Same as previous slide, but h=300 s

Nonequilibrium Growth of SolidsGas-solid equilibrium reactions (17.129)

NH4NO3(s)NH4(g)+HNO3(g)

Solids can form when (17.131)

NH4Cl(s)NH4(g)+HCl(g) (17.130)

(17.132)

pNH3pHNO3 >KNH4NO3

pNH3pHCl >KNH4Cl

Nonequilibrium Growth of Solids

Gas-solid equilibrium coefficient relation (17.133)

(17.134)

CNH3,s,tCHNO3,s,t =KNH4NO3 R*T( )−2

CNH3,s,tCHCl,s,t =KNH4Cl R*T( )−2

Nonequilibrium Growth of SolidsGrowth equations for gases that form solids (solids formed during

operator-split equilibrium calculation)

dcNO3−,i,t

dt=kHNO3,i,t−h CHNO3,t − ′ S HNO3,i,t−hCHNO3,s,t( )

dcCl−,i,tdt

=kHCl,i,t−h CHCl,t − ′ S HCl,i,t−hCHCl,s,t( )

Simulations of Solid GrowthTime-dependent aerosol concentrations, summed over all sizes, during nonequilibrium growth

plus internal aerosol equilibrium at RH=10 percent when h=5 s.

Fig. 17.8

1

10

0 2 4 6 8 10 12

Summed concentration (

g m

-3

)

( )Time from start h

NH

4

NO

3

( )s

NaNO

3

( )s

(NH

4

)

2

SO

4

( )s

Na

2

SO

4

( )s

NH

4

( )Cl s

( ) =5a h s

( )NaCl s

Sum

med

con

cent

rati

on (g

m-3)

Simulations of Solid Growth

Fig. 17.8

1

10

0 2 4 6 8 10 12

Summed concentration (

g m

-3

)

( )Time from start h

NH

4

NO

3

( )s

NaNO

3

( )s

(NH

4

)

2

SO

4

( )s

Na

2

SO

4

( )s

NH

4

( )Cl s

( ) =300b h s

( )NaCl s

Same as previous slide, but h=300 sS

umm

ed c

once

ntra

tion

(g

m-3)