Post on 23-May-2020
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Cloud droplet formation and growth
Trude Storelvmo Condensation and rain formation 1 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Phase changes
I Phase changes from left to right: increasing molecular order:
I Vapor ↔ liquid (condensation, evaporation)
I Liquid ↔ solid (freezing, melting)
I Vapor ↔ solid (deposition, sublimation)
I Recall: Clausius-Clapeyron equation describes equilibrium
condition for bulk water and its vapor.
I Saturation: equilibrium situation with rates of evaporation and
condensation being equal.
I However, for small droplets, because of the energy barrier
(surface tension), phase transitions do not generally occur at
the equilibrium saturation of bulk water.
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Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Nucleation
I Nucleation: Any process in which a free energy barrier must
be overcome, such as vapor to liquid or liquid to ice transitions.
I Homogeneous nucleation: Cloud droplets form directly from
the vapor phase
I Homogeneous nucleation requires several hundred percent
supersaturation
I Instead cloud droplet form when the ascending air just reaches
equilibrium saturation, because of the presence of CCN.
I Heterogeneous nucleation: Cloud droplets form on nuclei
from the vapor phase
Trude Storelvmo Condensation and rain formation 3 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Kelvin equation I
I Define saturation ratio S = es(r)es(∞) :
I The change in Gibbs free energy associated with the formation
of a droplet of radius r at saturation ratio S at temperature T:
∆G = −4πr3RvT
3αllnS + 4πr2σ (1)
I From ∂∆G∂r = 0. obtain Kelvin equation: vapor pressure in
equilibrium is larger over a droplet with radius r than over a
bulk surface:
es(r) = es(∞)exp(2σ
ρwRvTr) (2)
I where σ is the surface tension = 0.075 N/m.
Def: Surface tension: free energy per unit surface area of the
liquid. Work per unit area required to extend the surface of
liquid at constant temperature
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Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Gibbs free energy (Fig. 9.10 Seinfeld&Pandis)
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Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Kelvin equation II
Critical radii for droplet formation in clean air:
rc =2σ
RvρwTlnS; S =
e
esat(∞)(3)
Saturation ratio Critical radius number of moleculesS rc(µm) n1 ∞ ∞
1.01 0.12 2.47 x 108
1.1 0.0126 2.81 x 105
1.5 2.96 x 10−3 36452 1.73 x 10−3 7303 1.09 x 10−4 183
10 5.22 x 10−4 20
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Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Raoult’s Law I
I Reduction in vapor pressure due to the presence of a
non-volatile solute over a plane surface of water:
e∗
es(∞)= 1− 3imMw
4πMsρw r3= 1− b
r3(4)
where e∗ is the equilibrium vapor pressure over a solution, i is
the degree of ionic dissociation in the solute, Ms is the
molecular weight of the solute, Mw is the molecular weight of
water, m is the mass of solute
I If the vapor pressure of the solute is less than that of the
solvent, the vapor pressure is reduced in proportion to the
amount of solute present.
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Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Kohler curve I
I Combination (multiplication) of Kelvin and Raoult’s equation
(evaluating it for e∗(r)/es(r)) gives the Kohler curve:
Ie∗(r)
es(∞)= (1− b
r3) ∗ exp(
a
r). (5)
with
a =2σ
ρwRvT≈ 3.3 · 10−7
T[m] (6)
I For r not too small, a good approx. is (exp(ar ) ∼ 1 + ar )
e∗(r)
es(∞)= 1 +
a
r− b
r3(7)
I 1. term: surface molecules possess extra energy
2. term: solute molecules displacing surface water molecules
Trude Storelvmo Condensation and rain formation 8 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Kohler curve II
Trude Storelvmo Condensation and rain formation 9 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Kohler curve III
I The critical radius rc and critical supersaturation Sc are given by:
rc =
√3b
a; Sc =
√4a3
27b(8)
I Kohler curve represents equilibrium conditions
I Large particles have large equilibrium radii and may have insufficient
times to grow to their equilibrium size in clouds with strong updrafts.
I As the size of a droplet increases, the equilibrium vapor pressure
above its surface decreases (Kelvin’s equation).
I The curves for droplets containing fixed masses of salt approach the
Kelvin curve as they increase in size, since the droplets become
increasingly dilute solutions.
Trude Storelvmo Condensation and rain formation 10 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Kohler curve IV
I Solution effect dominates for small particles: Small solution
droplets are in equilibrium with the vapor for RH < 100%. If
RH increases, droplet will grow until it reaches equilibrium
again.
I If droplet grows beyond rc , its equilibrium saturation ratio falls
below Sc . Then vapor will diffuse to the droplet. It will now
grow even if RH decreases → activated drop.
I Drops experiencing S > Sc can thus be activated
(S = e∗(r)es(∞) − 1).
I If cloud droplet is not activated and grows slightly, then S of
air adjacent to drop needs to be higher than that of ambient
air to maintain that state. Since it is not, the drop will shrink
again and vice versa.
I The higher S , the more and the smaller aerosols are activated.
Trude Storelvmo Condensation and rain formation 11 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Trude Storelvmo Condensation and rain formation 12 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Droplet growth by condensation
I At the early stages of a cloud droplets development, it grows
by diffusion of water molecules from the vapor onto its surface.
I Droplet growth equation:
rdr
dt=
(S − 1)− ar + b
r3
Fk + Fd(9)
I Fk = thermodynamic term:(
LRvT− 1)
LρlKT ; “-1” can be
neglected
I Fd = vapor diffusion term: ρlRvTDes
.
I Equation (9) cannot be solved analytically. For sufficiently
large droplets it can be approximated by neglecting the solution
and curvature effects on the drop’s equilibrium pressure:
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Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Droplet growth equation
rdr
dt=
S − 1
Fk + Fd(10)
Figure: Droplet growth equation for different r0 at 0.5% supersaturation
Trude Storelvmo Condensation and rain formation 14 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Droplet growth by condensation
I The cloud droplet radius increases with time according to:
r(t) =√r20 + 2ξt (11)
where ξ = (S − 1)/(Fk + Fd).
I ξ depends on temperature and pressure. The larger T , the lower p,
the higher ξ
I The parabolic form of (11) leads to a narrowing of the drop-size
distribution as growth proceeds.
I Consider 2 cloud droplets with initial radii of r1(0) and r2(0) with
r2 > r1. From (11) it follows:
r2(t)− r1(t) =r22 (0)− r2
1 (0)
r2(t) + r1(t)(12)
because the difference between the squares of the initial radii remains
constant, at any time t, the difference in radii becomes smaller.
Trude Storelvmo Condensation and rain formation 15 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Growth of droplet population
I Condensation growth is controlled by ambient S , thus we need
to examine the vapor budget of a developing cloud
I Assume that the vapor is provided by saturated air that is
cooled in ascent and that it is lost by condensation on the
growing cloud droplet, so the rate of change of the saturation
ratio S is given by:
dS
dt= P − C = Q1
dz
dt− Q2
dχ
dt(13)
where:
P = production, C = condensationdzdt = w = vertical air velocitydχdt : condensation rate [kg/kgs]
Q1 = 1T
[εLg
RdcpT− g
Rd
], and Q2 = ρ
[RdTεes
+ εL2
pTcp
]Trude Storelvmo Condensation and rain formation 16 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Trude Storelvmo Condensation and rain formation 17 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Initiation of rain in nonfreezing (=warm) clouds
I Rain formation by coalescence is the dominant precipitation
formation process in warm clouds in the tropics and in
midlatitude cumuli whose tops may extend to subfreezing T .
I Need to explain how raindrops can be created by condensation
and coalescence in 20 min (∼ time between first appearance of
Cu and appearance of rain).
I 50-fold increase in radius (from 10 µm cloud drop with 105 l−1
to a 0.5 mm rain drop with 1 l−1) is achieved by collision and
coalescence if some drops exceed 20µm.
I Smaller drops have small collision cross section and slow
settling speeds → small chance to collide (collision rate ∼ r4)
I A raindrop of 1 mm results from ∼ 105 collisions
I Cloud droplets can grow by condensation to r = 20µm in 10
min if S ≥ 0.5% or u ≥ 5m/s (developing Cu).
Trude Storelvmo Condensation and rain formation 18 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Droplet growth by collision, coalescence
I Collision may occur due to gravitational effects once some
drops are larger than others
I Define the coalescence efficiency as number of coalescence
events / number of collisions
I The growth of droplets by the collision-coalescence process is
governed by the collection efficiency = collision efficiency *
coalescence efficiency
I Assume a coalescence efficiency=1, so that collision efficiency
= collection efficiency
I The problem then is to determine collision rates among a
droplet population
Trude Storelvmo Condensation and rain formation 19 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Droplet terminal velocity (u)
I Assuming equilibrium between the drag force of a sphere in a
viscous fluid and the gravitational force:
FD =π
2r2u2ρCD = 6πµru
(CDRe
24
)=
4
3πr3gρl = Fg (14)
where CD is the drag coefficient, µ the dynamic viscosity, r the
droplet radius and Re = 2ρurµ the Reynolds number. This gives:
u =2
9
r2gρl
µCDRe
24
(15)
I For r < 40µm, CDRe
24 ≈ 1, so that
u =2
9
r2gρlµ
= k1r2 (16)
where k1 ∼ 1.19 x 106 cm−1 s−1 (Stokes law)
Trude Storelvmo Condensation and rain formation 20 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
RaindropsI For spherical raindrops 0.6 mm < r < 2 mm.
u = k2
√r (17)
where k2 = 2200√
ρoρ [√cm/s] and ρo = 1.2 kg/m3.
I In the intermediate range (40 µ m< r < 0.6 mm), an approx.
formula for the fall speed with k3 = 8000 s−1 is given by:
u = k3r (18)
Trude Storelvmo Condensation and rain formation 21 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Collision efficiency
E (R, r) =x2o
(R + r)2(19)
I xo = impact parameter within which a
collision is certain to occur. Outside xodrops will be deflected out of the path.
I E = fraction of drops with r colliding
with collector drops over the path swept
out by the collector drop
I or: E = probability that a collision will
occur with a drop located at random in
the swept volume.
Trude Storelvmo Condensation and rain formation 22 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Collision efficiency
Collision efficien-
cies for small col-
lector drops from
3 sets of theoret-
ical calculations.
E vs. r/R.
Trude Storelvmo Condensation and rain formation 23 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Growth equation
I Suppose a drop with radius R falls with terminal velocity u(R)
through a population of smaller droplets. During unit time it
sweeps out droplets of radius r from a volume given by:
π(R + r)2[u(R)− u(r)] (20)
I The number of drops with radii between r and r+dr then is:
π(R + r)2[u(R)− u(r)]n(r)E (R, r)dr (21)
where E(R,r) is the collection efficiency and n(r) is the number
concentration of droplets with radius r.
I If r and R < 100µm → coalescence efficiency = 1 and the
collection efficiency = collision efficiency.
Trude Storelvmo Condensation and rain formation 24 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Growth equation
I The increase in the droplet radius is then given as:
dR
dt=π
3
∫ R
o(R + r
R)2n(r)[u(R)− u(r)]r3E (R, r)dr (22)
I If droplets are much smaller than the collector drop u(r) ∼ 0
and R + r ∼ R, so that
dR
dt=π
3
∫ R
on(r) u(R) r3 E (R, r)dr =
EMu(R)
4ρl(23)
where E is the average collection efficiency for the droplet
population and M is the cloud liquid water content.
I These equations describe drop growth as a continuous
collection process, regarding the cloud as a continuum. Growth
actually occurs by the discrete events of droplet capture.
Trude Storelvmo Condensation and rain formation 25 / 26
Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation
Marine case: Onset
of coalescence reduces
drop number. S in-
creases because the few
drops cannot consume
the excess vapor. New
CCN are activated, but
quickly consumed by
coalescence.
Continental case: Coa-
lescence causes a slight
reduction in drop num-
ber, but not enough to
cause a sudden rise in S .
Trude Storelvmo Condensation and rain formation 26 / 26