1. Generating cloud drops from CCN
Wallace & Hobbs (1977)

2. Cloud Drops and Equilibrium Considerations: review
We discussed how to compute the equilibrium vapor pressure over a pure water drop, or a solution droplet, as a function of composition (which is in turn dependent on the amount and nature of soluble material in the CCN) and curvature
This led us to an equation for the “Köhler curve”, which expresses vapor pressure as a function of the droplet size
Non-unique relationship (above S=1) if generated for a solution droplet, except at the critical diameter (sc)
Unique relationship if computed for a pure water drop (or insoluble, wettable particle)
How does the drop move along the x-axis (i.e., gain or lose water)?
Condensation / evaporation
Direction is dictated by equilibrium (system wants to move toward equilibrium)
But the rate of movement is no longer in the realm of equilibrium thermo, but closer to kinetics

3. Droplet Growth By Condensation
From our Kohler Curve discussions, we have seen that an “activated” droplet will grow by vapor condensing on its surface, provided the vapor pressure of the ambient air exceeds the vapor pressure adjacent to (over, or at the surface of) the droplet. For droplets exceeding ~1 µm in radius, the curvature term is ~negligible and hence the droplet behaves nearly as a “flat” surface of water (no curvature).
Droplet Growth Ambient air is above water saturation
Droplet Evaporation Ambient air is below water saturation
Our task here is to derive an expression for the rate that a small droplet grows by vapor condensing on its surface
traditionally, the rate is written either in terms of the rate of change of the drop radius or of the drop’s water mass

4. Consider the following situation:
Droplet of mass m and radius r is embedded within a steady-state field of water vapor r
To calculate the droplet growth rate by condensation, surround this droplet with an imaginary sphere of radius R, write a mass balance, and use the Fickian Diffusion Law to compute the flux of water vapor through the imaginary spherical surface, towards the droplet’s surface.
For steady-state conditions and no “storage” (accumulation) of water vapor in the region around the droplet, this vapor flux must be equivalent to the growth rate of the droplet. R
Droplet density
Drop mass
The unsteady-state diffusion of species A (here, water) to the surface of a stationary particle of radius Rp is described by
where c(r,t) is the molar concentration of A (moles vol-1), and JA,r is the flux of A (units: moles area-1 time-1) at any radial position r. (The equation arises from a mass balance in a spherical shell around the particle.) Fick’s Law for our problem can be well approximated by
(Cartesian)
DV is the diffusivity of vapor in air

5. Combining, we get the governing equation,
We need one initial condition, and two boundary conditions, to solve this. We use
We can find a solution for this full problem (see Chapter 11 of the text by Seinfeld and Pandis), and putting in typical values for the kinds of problems we’ll be trying to solve, we find out that the transient is very short. So it is valid to assume we get to a steady-state situation very quickly, and that we can set the time derivative to zero. The steady-state solution is,
(*)
The total flow of A (moles time-1) toward the particle is
So, take the derivative of the solution (*), plug into the equation for JA,r, and find
What happens as r = Rp and as r gets very large?
The transferred moles time-1 increase as Rp increases, and are proportional to the difference between the surface concentration and the far-field concentration
“Maxwellian flux” (1877)

6. Which we could also express in terms of the vapor density (mass vol-1)
Now write a mass (or here, mole) balance on the growing (or evaporating) droplet:
This yields an equation for the time rate of change of the drop radius,
Which we could also express in terms of the vapor density (mass vol-1)
Or as a rate of change of the droplet mass,
The diffusivity of water vapor can be expressed as
r∞ and rw,∞ are the same thing; w used to indicate water vapor here; eq means the equilibrium value over the drop
DV in units of cm2 s-1, T in K, p in atm

7. Fig 15.13, Seinfeld and Pandis
For very small droplets, we should modify the droplet growth equation to include noncontinuum effects. We can accomplish this by defining a modified diffusivity, that here includes a “water accommodation coefficient” ac:
The figure below shows the impact of this correction. For ac=1, the correction is less than 25% for drops larger than 1 µm and less than 5% for those > 5 µm.
Recommended values, ac:
0.045 (P & K)
1 (many authors)
See recent work by R. Shaw
Fig 15.13, Seinfeld and Pandis

8. Go back to the equations for the rates of change or drop mass or radius:
Both rates are proportional to the difference between the environmental vapor pressure and that over the droplet
The rate of change of the radius with time is inversely proportional to the radius
 The radii of the smallest drops grow (or evaporate) the fastest
The rate of change of the droplet mass is proportional to the radius
 The masses of the largest drops increase (or decrease) the fastest
Also, there is something a bit strange about these final equations. We derived them initially for a fixed radius Rp (using a time-independent steady-state profile around the drop). But now we have developed an equation for how Rp changes with time!
This treatment implies that the vapor concentration profile near the drop achieves steady state before appreciable growth occurs. Diffusion is much faster than the time rate of change of Rp, so we can assume that the adjustment of the vapor profile is very fast compared with the changes brought about by the rate of the drop growth.

9. The “R-squared law” arises if one can assume that are constant
The “R-squared law” arises if one can assume that are constant. Then we can easily integrate the equation
to get
How does the characteristic time scale with Rp,0?
What does this mean for growth rates of different-sized drops?

10. The growth rate is not strictly given by the equations we just derived, since heat is released at the droplet’s surface owing to release of latent heat of condensation, Lc
The heat liberated will cause the droplet to warm, thereby increasing the vapor pressure adjacent to the droplet, reducing the vapor pressure gradient and therefore reducing the growth rate.
eV,s  with heating, therefore gradient is reduced.
The heat can be released either toward the particle or toward the exterior gas phase. As mass transfer continues, the particle surface temperature changes until the rate of heat transfer balances the rate of heat generation (or consumption, if evaporation). The formation of the external T and vapor conc. profiles must be related by a steady-state energy balance, to determine the steady-state surface temperature at all times.
The droplet growth rate is thus a function of the rate that heat can be conducted away from the droplet – by heat conduction (the energy analogy to vapor diffusion).
To derive an expression for the change in temperature, construct a balance between heat released by condensation and conduction of heat away from the droplet:
Please note here r=r∞ and rr is at the surface (called rs above)
rate that latent heat
is released
heat conduction term

11. By analogy to diffusion,
can be written as,
Where
is the thermal conductivity of air.
is the radial temperature gradient. (-) sign is needed since heat flow in opposite sense to gradient.
Integrating (6),
For steady-state conditions,
Equation (7) reduces to the condition,
vapor & thermal fields around droplet satisfy
this relationship
particle “wet bulb temperature”

12. The next step is to combine equation (7) and the dRp/dt eqn with the Clausius Clapeyron equation and the Ideal Gas Law to write in terms of the supersaturation of the environment.
After much algebra,
Assumes pure water, no curvature
molecular weight of water
All other variables have been
previously defined k
is the saturation ratio of the environment. All of the Ts in this equation refer to the environmental temperature.
This equation is written in the simpler form as,
Please also see pages of the Seinfeld and Pandis text for a complete, and clear, derivation of this growth equation

13. molecular weight of salt
For droplets in radius, the curvature and solute effects must be considered,
We could also write the numerator as S∞ – awexp(Kelvin)
mass of salt
molecular weight of salt
Environmental saturation ratio
saturation ratio over the droplet
Van’t Hoff factor
molecular weight of water
Condensational growth implies,
which means that r2
Condensational growth alone would tend to
produce a monodisperse cloud droplet size
distribution r1 r0
10 hours
time

14. growth rate
small
dm/dt is
proportional
to r
growth rate
large
Houghton, 1985
~20s
~40s

15. Evaporation is described by the same equation as condensation (but opposite sign for the driving force, of course), so we should also expect smaller drops to evaporate faster than larger ones:
Distance a drop falls before evaporating, assuming isothermal atmosphere and constant S (Table 7.3, Rogers and Yau)
Initial radius (microns)
Distance fallen 1
2 µm 3
0.17 mm 10
2.1 cm 30
1.69 m
0.1 mm
208 m
0.15 mm
1.05 km

16. Rate of growth droplets by condensation on salt nuclei
(after Best 1951b)
Temperature T=273K
Pressure=900mb
Nuclear mass m
Supersaturation=100(S-1)%
10-14 g
0.05
10-13 g g
Radius (µm)
Time (s) to grow from initial radius of 0.75µm 1
2.4
0.15
0.013 2
130
7.0
0.61 5
1000
320 62 10
2700
1800
870 15
5200
4200
2900 20
8500
7400
5900 25
12500
11500
9700 30
17500
16000
14500 35
23000
22000
20000 50
44500
43500
41500
Growth rate (initially) faster for larger initial salt mass (solute effect)
For a given salt mass, the growth rate decreases with increasing size.
Conclude that diffusional growth alone cannot produce precipitation sized drops within reasonable cloud lifetimes. r0 r1
Condensational growth tends to produce
uniform size distribution r2 r0 r1 r2

17. Consider the growth of a population of droplets
In clouds many droplets grow at the same time (on activated CCN). Typical concentrations may be a few hundred per cm3 in maritime clouds and ~ cm-3 for continental clouds. These droplets compete for available H2O vapor made available by condensation associated with the rising air parcels. This of course creates a supersaturation which is reduced by condensational growth on the cloud droplets.
Time rate of change of supersaturation,
supersaturation
Net rate of change = rate of production of supersaturation – rate of consumption (by condensation)
condensational growth rate
saturation mixing ratio; updraft velocity
where is the droplet concentration, assumed to be

18. INITIAL RADII α salt mass radii Activated droplets6 60
MONODISPERSE
INITIAL RADII α salt mass radii
Activated droplets
Increasing
salt mass
non-activated droplets
1) Max supersaturation achieved few tens of meters above cloud base
cloud droplet concentration determined
2) drop growth maximum around Smax
3) smallest particles become haze droplets
4) Activated droplets become monodisperse in size