Chapter 17. Fundamentals of Atmospheric Modeling

Size: px

Start display at page:

Download "Chapter 17. Fundamentals of Atmospheric Modeling"

Rosamond Reynolds
5 years ago
Views:

1 Overhead Slides for Chapter 17 of Fundamentals of Atmospheric Modeling by Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA August 21, 1998

2 Mass Flux to and From a Single Drop Rate of change of mass (g) of single pure liquid water drop dm d t = 4πR 2 D v d ρ v d R (17.1) Integrate from drop surface to infinity dm d t = 4πrD v ( ρ v ρ v,r ) (17.2) Rate of change of heat energy at drop surface due to conduction d Q r * d t = 4πR 2 κ d d T d R (17.3) Integrate from drop surface to infinity d Q r * d t = 4πrκ d ( T r T ) (17.4) Relationship between change in mass and heat energy at surface mc W d T r d t dm = L e d t dq r * (17.5) Combine (17.4) and (17.5) and assume steady state L e d m = 4πrκ d ( T r T ) (17.6)

3 Mass Flux to and From a Single Drop Combine equation of state at saturation p v,s = ρ v,s R v T with Clausius Clapeyron equation dp v,s dt = ρ v,sl e T to obtain dρ v,s ρ v,s = L e dt R v T 2 dt T (17.7) Integrate from infinity to drop surface ln ρ v,s T r ρ v,s T = L e R v ( T r T ) ln T r TT r T (17.8) Simplify since T T r ρ v,s T ρ v,s T r ρ v,s T = L e R v ( T r T ) T 2 T r T T (17.9) Substitute (17.6) into (17.9) ρ v,s T ρ v,s T r ρ v,s T = L e 4πrκ d T L e R v T 1 dm (17.10)

4 Mass Flux to and From a Single Drop Substitute (17.2) into (17.10) ρ v ρ v,s T ρ v,s T L = e L e 4πrκ d T R v T πrD v ρ v,s T dm d t (17.11) Mass-flux form of growth equation dm d t = 4πrD v p v p v,s D v L e p v,s L e κ d T R v T 1 + R v T (17.12) Rewrite mass-flux form equation for trace gases and particle sizes dm i d t = 4πr i D q,i p q p q,s,i L e,q m q R * T 1 + R* T m q D q,i L e,q p q,s,i κ d,i T (17.13)

5 Fluxes to and From a Single Drop Change in mass as a function of change in radius dm i = 4πr i 2 ρ p,i dr i (17.14) Radius-flux form of growth equation dr r i i = D q,i L e,q ρ p,i p q,s,i κ d,i T D q,i p q p q,s,i L e,q m q R * T 1 + R* Tρ p,i m q (17.15) Change in mass as a function of change in volume dm i = ρ p,i dυ i Volume-flux form of growth equation dυ i = ( 48π 2 υ i ) 13 D q,i p q p q,s,i D q,i L e,q ρ p,i p q,s,i L e,q m q κ d,i T R * T 1 + R* Tρ p,i m q (17.16)

6 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 (cm 2 s -1 ) D q = 3 8Ad q 2 ρ a R * Tm a 2π m q + m a m (17.17) q Example Calculate diffusion coefficient of carbon monoxide T = 288 K p a = 1013 mb ρ a = g cm -3 m q = 28 g mole -1 D q = 0.12 cm 2 s -1

7 Corrected Gas Diffusion Coefficient Corrected diffusion coefficient D q,i = D q ω q,i F q,i (17.18) Correction for collision geometry and sticking probability ω q,i = Kn 1 q,i 1 1+ Kn q,i + 4 ( 1 α q,i) 3α q,i 1 Kn q,i (17.19) Knudsen number for condensing vapor Kn q,i = λ q r i (17.20) 0 as Kn q,i (small particles) ω q,i 1 as Kn q,i 0 (large particles) (17.21) Mass accommodation (sticking) coefficient, α q,i Fractional number of gas collisions with particles that results in the gas sticking to the surface. From

8 Corrected Gas Diffusion Coefficient Mean free path of a gas molecule λ q = 32D q 3πv q m a m a + m 3D q (17.23) q v q Ventilation factor Corrects for increased rate of vapor transfer due to eddies sweeping vapor to the surface of large particles F q,i = x 2 q,i x q,i x q,i x q,i > 1.4 (17.24) x q,i = Re i 1 2 Scq 1 3 (17.24) Particle Reynolds number Re i = 2r i V f,i ν a Gas Schmi number Sc q = ν a D q (17.25)

9 Corrected Thermal Conductivity Corrected thermal conductivity of dry air (J cm -1 s -1 K -1 ) κ d,i = κ d ω h,i F h,i, (17.26) Correction for collision geometry and sticking probability ω h,i = Kn 1 h,i 1 1 +Kn h,i α h 3α h 1 Kn h,i (17.27) Knudsen number for heat Kn h,i = λ h r i (17.28) Mean free path of heat λ h = 3D h v a (17.29)

10 Corrected Thermal Conductivity Thermal accommodation (sticking) coefficient, α t Fraction of molecules bouncing off surface of a drop that have acquired temperature of drop 0.96 for water. α h = T m T T s T (17.30) Ventilation factor Corrects for increased rate of heat transfer due to eddies F h,i = x 2 h,i x h,i x h,i x h,i > 1.4 (17.31) x h,i = Re i 1 2 Pr 1 3 Pranl number Pr = η a c p,m κ d (17.32)

11 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 Curvature and solute effects. Saturation ratio S* Equilibrium saturation ratio Curvature effect r* Solute effect Particle radius (µm)

12 Curvature Effect Saturation vapor pressure over curved, dilute surface relative to that over a flat, dilute surface p q,s,i = exp 2σ p m p p q,s r i R * Tρ 1+ 2σ p m p p,i r i R * (17.33) Tρ p,i Note exp( x) 1+ x for small values of x

13 Solute Effect Vapor pressure over flat water surface with solute relative to that without solute (Raoult's Law) p q,s,i = p q,s n w n w + n s (17.34) Relatively dilute solution n w >> n s p q,s,i p q,s 1 n s n w (17.35) Number of moles of solute in solution n s = i v M s m s Number of moles of liquid water in a drop n w = M w m v 4πr 3 i ρw (17.36) 3m v Combine terms --> solute effect p q,s,i p q,s 1 3m v i v M s 4πr i 3 ρw m s (17.37)

14 Köhler Equation Combine curvature and solute effects --> Sat. ratio at equilibrium S q,i = p q,s,i p q,s 1+ 2σ p m p r i R * Tρ p,i 3m v i v M s 4πr i 3 ρw m s (17.38) Simplify Köhler equation S q,i = 1+ a b r i r3 (17.40) i a = 2σ p m p R * Tρ p,i b = 3m v i v M s 4πρ w m s (17.40) Set Köhler equation to zero --> Critical radius for growth and critical saturation ratio r* = 3b a S* = 1+ 4a3 27b (17.41) Table Critical radii / supersaturations for water drops containing sodium chloride or ammonium sulfate at 275 K. Sodium Chloride Ammonium Sulfate Solute Mass (g) r * (µm) S * 1 (%) r * (µm) S * 1 (%)

15 Radiative Cooling Effect Surface vapor pressure over curved, dilute surface relative to that over a flat, dilute surface p q,s,i 1+ L e,q m q H r,i p q,s 4πr i R * T 2 (17.42) κ d,i Radiative cooling rate (W) H r,i = πr2 ( i )4π Q a ( m λ,α i,λ )( I λ B λ )dλ (17.43) 0

16 Overall Equilibrium Saturation Ratio Overall equilibrium saturation ratio for liquid water S q,i = p q,s,i p q,s 1+ 2σ p m p r i R * Tρ p,i 3m v i v M s 4πr i 3 ρw m s + L e,q m q H r,i 4πr i 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 = p q,s,i p q,s 1+ 2σ p m p r i R * Tρ p,i (17.45)

17 Flux to Drop With Multiple Components Volume of a single particle in which one species is growing υ i,t = υ q,i,t + υ i,t h υ q,i,t h (17.46) Mass of a single particle in which one species is growing m i,t = ρ p,i,t υ i,t = ρ p,q υ q,i,t + ρ p,i,t h υ i,t h ρ p,q υ q,i,t h (17.47) Time derivative of (17.47) dm i,t = ρ p,i,t dυ i,t = ρ p,q dυ q,i,t (17.48) since dυ i,t h = dυ q,i,t h = 0 Combine (17.48) and (17.46) with (17.16) Rate of change in volume of one component in one multicomponent particle dυ q,i,t = [ 48π 2 ( υ q,i,t + υ i,t h υ q,i,t h )] 1 3 D q,i L e,q ρ p,q p q,s,i κ d,i T D q,i p q p q,s,i L e,q m q R * T 1 + R* Tρ p,q m q (17.49)

18 Flux to a Population of Drops Volume as a function of volume concentration υ q,i,t = v q,i,t n i,t h Substitute volumes into (17.49) dv q,i,t = 2 3 n i,t h [ 48π 2 ( v q,i,t + v i,t h v q,i,t h )] 1 3 D q,i L e,q ρ p,q p q,s,i κ d,i T D q,i p q p q,s,i L e,q m q R * T 1 + R* Tρ p,q m q (17.50) Partial pressure in terms of mole concentration p q = C q R * T (17.51) Vapor pressure in terms of mole concentration p q,s,i = C q,s,i R * T (17.51)

19 Flux to a Population of Drops Combine (17.50) with (17.51) d v q,i,t d t 2 3 = n i,t h [ 48π 2 ( v q,i,t +v i,t h v q,i,t h ) ] 1 3 eff D q,i,t h m q C ρ q,t C q,s,i,t h p,q (17.52) Effective diffusion coefficient eff D q,i,t h = D q,i m q D q,i L e,q C q,s,i,t h κ d,i T L e,q m q R * T (17.53) Simplify effective diffusion coefficient for gases other than water eff D q,i,t h D q,i = D q ω q,i F q,i = D q F q,i Kn 1 q,i α q,i 1+ Kn 1 q,i 3α q,i Kn q,i (17.54) Corresponding gas-conservation equation d C q,t d t = ρ p,q m q N B d v q,i,t (17.55) i=1

20 Integrate Growth Equations Matrix of partial derivatives (17.73) v q,1,t v q,2,t v q,3,t C q,t v q,1,t v q,2,t v q,3,t C q,t 2 v 1 hβ q,1,t s v q,1,t t 2 C q,t hβ s v q,1,t t 0 1 hβ s 2 v q,2,t v q,2,t t 0 0 hβ s 2 v q,1,t C q,t t 0 hβ s 2 v q,2,t C q,t t hβ s 2 v q,3,t v q,3,t t hβ s hβ s 2 C q,t v q,2,t t hβ s 2 C q,t v q,3,t t 1 hβ s 2 v q,3,t C q,t t 2 C q,t C q,t t 2 v q,i,t v q,i,t t = n i,t h 3 v 48π 2 q,i,t 2 v q,i,t C q,t t = n 2 3 i,t h 2 C q,t v q,i,t t = ρ p,q m q 1 3 eff D q,i,t h (17.56) m q C ρ q,t C q,s,i,t h p,q [ 48π 2 ( v q,i,t + v i,t h v q,i,t h )] 1 3 eff D q,i,t h (17.57) m q ρ p,q (17.58) 2 v q,i,t v q,i,t t, (17.59) 2 C q,t C q,t t = ρ p,q m q N B 2 v q,i,t (17.60) C q,t t i=1

21 Integrated Solution to Growth Equations Table Reduction in array space and in the number of matrix operations before and after the use of sparse-matrix techniques to solve growth ODEs. N B + 1 = 17. Growth / Evaporation Initial After Sparse Matrix Reductions Order of matrix No. init. matrix spots filled % of initial positions filled No. fin. matrix spots filled % of final positions filled No. operations decomp No. operations decomp No. operations backsub No. operations backsub

22 Analytical Predictor of Condensation Assume radius in growth term constant during time step Change in particle volume concentration d v q,i,t d t 2 3 = n i,t h ( 48π 2 v i,t h ) 1 3 eff D q,i,t h m q C ρ q,t C q,s,i,t h p,q (17.61) Define mass transfer coefficient 2 3 k q,i,t h = n i,t h ( 48π 2 v i,t h ) 1 3 eff eff D q,i,t h = n i,t h 4πr i,t h D q,i,t h (17.62) Effective surface vapor mole concentration C q,s,i,t h = S q,i,t h C q,s,i,t h (17.63) Volume concentration of a component v q,i,t = m q c q,i,t ρ p,q (17.64) Uncorrected surface vapor mole concentration C q,s,i,t h = p q,s,t h R * T

23 Analytical Predictor of Condensation Substitute conversions into (17.61) and (17.55) dc q,i,t dc q,t = k q,i,t h ( C q,t S q,i,t h C q,s,i,t h ) (17.65) N B [ ] = k q,i,t h C q,t S q,i,t h C q,s,i,t (17.66) i=1 Integrate (17.65) for final aerosol concentration c q,i,t = c q,i,t h + hk q,i,t h ( C q,t S q,i,t h C q,s,i,t h ) (17.67) Mass balance equation N B N B C q,t + c q,i,t = C q,t h + c q,i,t h = C tot (17.68) i=1 i=1 Substitute (17.67) into (17.68) C q,t = N B C q,t h + h k q,i,t h S q,i,t h C q,s,i,t h i=1 N B 1+ h k q,i,t h i =1 (17.69)

24 APC Growth Simulation Fig 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. d V /d log D V 10 ( in µm 3 cm -3 p ) Final Initial Time from start (h)

25 Effect of Coagulation on Condensation Figs a and b. Growth plus coagulation pushes particles to larger sizes than does growth alone or coagulation alone. d N /d log 10 D ( N in particles cm -3 p ) Init. Growth only Growth + coag. Coag. only Particle diameter (µm) d V /dlog 10 D ( V in µm 3 cm -3 p ) Growth only Initial Coag. only Growth + coag Particle diameter (µm)

26 Moving-Center vs. Full-Moving Size Structure Fig 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). d V /d log D 10 V ( in µm 3 cm -3 p ) Growth only (MC) Growth + coag. (FM) Growth only (FM) Growth + coag. (MC) Particle diameter (µm)

27 Ice Crystal Growth Rate of mass growth of a single ice crystal dm i d t = 4πχ i D v,i p q p v,i,i D v,i L s p v,i,i L s κ d,i T R v T 1 + R v T (17.71) Electrical capacitance of crystal (cm) a c,i 2 a c,i e c,i a c,i e c,i sin 1 e c,i 2 2 χ i = a c,i ln 4a c,i b c,i a c,i e c,i ln 1 +e c,i a c,i e c,i 2sin 1 e c,i a c,i π sphere ln[ ( 1 +e c,i )a c,i b c,i ] prolate spheroid oblate spheroid needle [ ( 1 e c,i ) ] column hexagonal plate thin plate (17.72) a c,i b c,i e c,i = = length of the major semi-axis (cm) = length of the minor semi-axis (cm) 1 b2 c,i a2 c,i

28 Ice Crystal Growth Effective saturation vapor pressure over ice p v,i,i = S v,i p v,i Ventilation factor for falling oblate spheroid crystals F q,i,f h,i = x2 x < x x 1.0 (17.73) x = x q,i for ventilation of gas x = x h,i for ventilation of heat

Köhler Curve. Covers Reading Material in Chapter 10.3 Atmospheric Sciences 5200 Physical Meteorology III: Cloud Physics

Köhler Curve. Covers Reading Material in Chapter 10.3 Atmospheric Sciences 5200 Physical Meteorology III: Cloud Physics Köhler Curve Covers Reading Material in Chapter 10.3 Atmospheric Sciences 5200 Physical Meteorology III: Cloud Physics Review of Kelvin Effect Gibbs Energy for formation of a drop G = G &'()*+, G ).'+