CONDENSATIONAL GROWTH OF
CLOUD DROPLETS
Equilibrium conditions
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Stable equilibrium:
• increase of the relative humidity (supersaturation) leads to a growth of cloud droplet until the new equilibrium is reached
• for the droplets to grow the
supersaturation has to rise until critical values (supersaturation, radius) are reached
Unstable equilibrium:
• increase of the realtive humidity
(supersaturation) above the critical value leads to a growth of the droplet and a decrease of the saturation equiibrium value
• water vapor diffuses toward the droplet causing its growth without an increase of suparsaturation
0,1 1 10 100 1000 µm Activation
Condensational growth
Condensational growth +
collision and coalescence Collision and coalescence
CN, CCN
cloud droplets
drizzle
rain S, CCN
P
Growth by diffusion of water vapor
• For a droplet to become a cloud droplet critical values of radius and supersaturation should be reached
• Before and after the critical values are reached a droplet grows by diffusion of water vapor towards the droplet surface
• We will describe a growth of a single droplet by diffusion of water
vapor
Condensational growth
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r R
𝜌!,#
𝜌!,$
The field of vapor molecules,
concentration n(R)
A droplet of radius r in a field of water vapor;
Concentration of vapor molecules, n(R) at distance R from the droplet center;
Vapor density or absolute humidity 𝜌% 𝑅 = 𝑛𝑚& .
The concentration of molecules is assumed to satisfy the diffusion equation:
D [m2s-1] is the molecular diffusion coefficient.
𝜕𝑛
𝜕𝑡 = 𝐷∇'𝑛
the ‘ambient’ or undisturbed value of vapor concentration The vapor concentration at the droplet’s surface
Boundary conditions
We assume the steady-state or ‘stationary’ conditions:
Vapor concentration around the droplet
r R
𝜌!,#
𝜌!,$
The field of vapor molecules,
concentration n(R)
𝜕𝑛
𝜕𝑡 = 0
𝜕𝑛
𝜕𝑡 = 𝐷∇'𝑛
𝐷∇' 𝑅 = 0 = 1 𝑅'
𝜕
𝜕𝑅 𝑅' 𝜕𝑛
𝜕𝑅
⟶ 𝐷∇'𝑛 = 0
𝑛 𝑅 = 𝐶( − 𝐶' 𝑅
𝑅 → ∞, 𝑛 → 𝑛$ 𝑅 → 𝑟, 𝑛 → 𝑛#
𝑛 𝑅 = 𝑛$ − 𝑟
𝑅 𝑛$ − 𝑛#
r R 𝜌!,#
𝜌!,$
The field of vapor molecules,
concentration n(R)
The flux of molecules onto the surface of the droplet:
the flux of mass over the surface 4pr2 the mass of one water molecule
the rate of mass increase
𝐷 𝜕𝑛
𝜕𝑅 )*#
𝑑𝑚
𝑑𝑡 = 4𝜋𝑟'𝐷 𝜕𝑛
𝜕𝑅 )*# 𝑚&
𝑛 𝑅 = 𝑛$ − 𝑟
𝑅 𝑛$ − 𝑛#
𝜕𝑛
𝜕𝑅 )*# = 𝑟
𝑅' 𝑛$ − 𝑛# 8
)*# = 1
𝑟 𝑛$ − 𝑛#
𝑑𝑚
𝑑𝑡 = 4𝜋𝑟𝐷 𝑛$ − 𝑛# 𝑚&
𝑑𝑚
𝑑𝑡 = 4𝜋𝑟𝐷 𝜌% − 𝜌%#
The diffusional growth equation for an isolated droplet at rest in a vapor field:
𝜌% = 𝑛𝑚&
the ambient vapor density the vapor density at the droplet’s surface
The droplet grows if 𝜌% > 𝜌%#
The droplet evaporates if 𝜌% < 𝜌%#
r R
•
The vapor density at the droplet’s surface ( r
vr) depends on:– the dropet size
– the chemical composition
– temperature (Tr), which is usually not the same as the ambient temperature (T), and must be determined by considering the heat transfer between the droplet and its surrounding.
•
The ambient vapor density ( r
v) may bedetermined from given environmental conditions
𝑇 𝜌% 𝑇# 𝜌%#
𝑑𝑚
𝑑𝑡 = 4𝜋𝑟𝐷 𝜌% − 𝜌%#
Associated with condensation is the release of latent heat, which tends to raise the droplet temperature above the ambient value.
The diffusion of heat away from the droplet (Q) is given by an equation analogous to the mass diffusion:
ambient temparature temperature at the
surface of the droplet coefficient of thermal
conductivity of air [J m-1s-1 K-1]
On the right hand side the signs are reversed comparing to the mass diffusion equation beacuse the positive heat flux is ‘outside’ the droplet, and the positive mass flux is ‘towards’
the droplet.
Equations of mass and heat fluxes were first derived by James Clerc Maxwell in 1877, in slightly different form, in an article in Encyclopaedia Britannica on the theory of the wet- bulb thermometer (Maxwell, 1890).
The theory of the steady-state growth of a spherical drop in rest in a vapor field regarded as a continuum is therefore often called the Maxwell theory.
𝑑𝑄
𝑑𝑡 = 4𝜋𝑟𝐾 𝑇# − 𝑇
density of water specific heat capacity of water
Weak dependence on temperature and pressure; Tabele 7.1 R&Y
The change of cloud droplet’s temperature (Tr) in the condensation process is due to:
• the release of the latent heat of condensation
• the heat release to the environment:
𝑚𝑐𝑑𝑇# = 𝐿𝑑𝑚 − 𝑑𝑄 4
3𝜋𝑟+𝜌,𝑐 𝑑𝑇#
𝑑𝑡 = 𝐿 𝑑𝑚
𝑑𝑡 − 𝑑𝑄 𝑑𝑡
For stationary state: 𝑑𝑇#
𝑑𝑡 = 0 𝐿𝑑𝑚
𝑑𝑡 = 𝑑𝑄
𝑑𝑡 ⟶ 4𝜋𝑟𝐷𝐿 𝜌% − 𝜌%# = 4𝜋𝑟𝐾 𝑇# − 𝑇 𝜌% − 𝜌%#
𝑇# − 𝑇 = 𝐾 𝐿𝐷
𝑑𝑚
𝑑𝑡 = 4𝜋𝑟𝐷 𝜌% − 𝜌%#
𝑑𝑄
𝑑𝑡 = 4𝜋𝑟𝐾 𝑇# − 𝑇
Ordinarily the drop temperature (Tr) and the water vapor density at its surface (rvr) are unknown.
However we know the relation between Tr and rvr :
the equilibrium vapor pressure over a plane surface at temperature Tris given by the Clausius-Clapeyron equation
𝜌%# = 𝑒! 𝑟
𝑅%𝑇# = 1 + 𝐴
𝑟 −𝐵!
𝑟+ 𝑒! 𝑇# 𝑅%𝑇#
Equations:
comprise a simultaneous systems which can be solved numerically for rvr and Tr to permit evaluation of the rate of drop growth by condensation
𝜌%# = 1 + 𝐴
𝑟 − 𝐵!
𝑟+ 𝑒! 𝑇# 𝑅%𝑇#
𝜌% − 𝜌%#
𝑇# − 𝑇 = 𝐾 𝐿𝐷
𝑑𝑚
𝑑𝑡 = 4𝜋𝑟𝐷 𝜌% − 𝜌%#
Condensational growth
Mason’s solution
Analytical approximation (Mason, 1971)
As an alternative to the numerical method of solution, Masson (1971) introduced a useful analytical approximation for calculating the rate of growth of a drop by condensation.
In a field of saturated vapor, changes in vapor density are related to changes in tempearture by:
We will integrate this equation from temperature Tr to temperature T, assuming T/Tr~1
Clausius-Clapeyron equation
𝜌%! = 𝑒! 𝑅%𝑇
𝑑𝑒!
𝑒! = 𝐿𝑑𝑇 𝑅%𝑇'
⟶ 𝑑𝜌%!
𝜌%! = 𝑑𝑒!
𝑒! − 𝑑𝑇 𝑇
𝑑𝜌%!
𝜌%! = 𝐿 𝑅%
𝑑𝑇
𝑇' − 𝑑𝑇 𝑇
E
-!
- 𝑑𝜌%!
𝜌%! = E
-!
- 𝐿
𝑅% 𝑑𝑇
𝑇' − 𝑑𝑇 𝑇
ln 𝜌%!
𝜌%!# = − 𝐿 𝑅% 1H
𝑇 -
!
-
− ln 𝑇8
-! -
ln 𝜌%!
𝜌%!# = − 𝐿 𝑅%
1
𝑇 − 1
𝑇# − ln 𝑇 𝑇#
ln 𝜌%!
𝜌%!# = − 𝐿 𝑅%
𝑇# − 𝑇
𝑇𝑇# − 𝑇
𝑇# − 1 = 𝑇# − 𝑇 𝐿
𝑅%𝑇𝑇# − 𝑇# − 𝑇 𝑇# 𝑇# − 𝑇
ln 𝜌%!
𝜌%!# = 𝑇# − 𝑇 𝐿
𝑅%𝑇𝑇# − 1 𝑇#
𝑇
𝑇# ≈ 1 ; 𝑇
𝑇# = 1 + 𝜀 ln 1 + 𝜀 ≈ 𝜀 ; ln 𝑇
𝑇# = 𝑇
𝑇# − 1
T-Trcan be replaced from the equation of the heat diffusion
for a steady-state
𝑑𝑄
𝑑𝑡 = 4𝜋𝑟𝐾 𝑇# − 𝑇 𝑇# − 𝑇 = 1 4𝜋𝑟𝐾
𝑑𝑄 𝑑𝑡
𝑇# − 𝑇 = 𝐿 4𝜋𝑟𝐾
𝑑𝑚 𝑑𝑡 𝑑𝑄
𝑑𝑡 = 𝐿𝑑𝑚 𝑑𝑡
𝜌%! − 𝜌%!#
𝜌%!# = 1 − 𝐿 𝑅%𝑇
𝐿 4𝜋𝑟𝐾𝑇
𝑑𝑚 𝑑𝑡 𝜌%!
𝜌%!# ≈ 1 ; ln 𝜌%!
𝜌%!# ≈ 𝜌%!
𝜌%!# − 1 = 𝜌%! − 𝜌%!#
𝜌%!#
ln 𝜌%!
𝜌%!# = 𝑇# − 𝑇 𝐿
𝑅%𝑇𝑇# − 1 𝑇#
𝑇 L 𝑇# ≈ 𝑇'
𝜌%! − 𝜌%!#
𝜌%!# = 𝑇 − 𝑇# 𝑇
𝐿
𝑅%𝑇 − 1
Subtracting (2) from (1), assuming rvr=rvsrand neglecting the effect of curvature
(1)-(2)
saturation over plane water surface
𝑑𝑚
𝑑𝑡 = 4𝜋𝑟𝐷 𝜌% − 𝜌%# 1 𝜌% − 𝜌%#
𝜌%# = 4𝜋𝑟𝐷𝜌%# .( 𝑑𝑚 𝑑𝑡 2 𝜌%! − 𝜌%!#
𝜌%!# = 1 − 𝐿 𝑅%𝑇
𝐿 4𝜋𝑟𝐾𝑇
𝑑𝑚 𝑑𝑡
𝜌% − 𝜌%#
𝜌%# − 𝜌%! − 𝜌%!#
𝜌%!# ≈ 𝜌% − 𝜌%!
𝜌%!# ≈ 𝜌%
𝜌%! − 1 ≈ 𝑆 − 1
𝑆 − 1 = 𝑑𝑚 𝑑𝑡
1
4𝜋𝑟𝐷𝜌%# + 𝐿
𝑅%𝑇 − 1 𝐿 4𝜋𝑟𝐾𝑇
𝑑𝑚
𝑑𝑡 = 𝑆 − 1
4𝜋𝑟𝐷𝜌1 %# + 𝐿
𝑅%𝑇 − 1 𝐿 4𝜋𝑟𝐾𝑇
the term associated
with vapor diffusion the term associated with heat conduction
Fd+Fkdepends on temperature and pressure (the dependence on temperature is stronger)
𝑑𝑚
𝑑𝑡 = 𝑆 − 1
4𝜋𝑟𝐷𝜌1 %# + 𝐿
𝑅%𝑇 − 1 𝐿 4𝜋𝑟𝐾𝑇
𝑑𝑚
𝑑𝑡 = 𝑑 4
3 𝜋𝜌,𝑟+ 𝑑𝑡
= 4𝜋𝜌,𝑟' /#/0 = 4𝜋𝜌,𝑟 𝑟/#/0
𝑟𝑑𝑟
𝑑𝑡 = 𝑆 − 1
4𝜋𝜌,𝑟
4𝜋𝑟𝐷𝜌%# + 𝐿
𝑅%𝑇 − 1 4𝜋𝜌,𝑟𝐿
4𝜋𝑟𝐾𝑇 𝜌%# ≈ 𝑒!(𝑇)
𝑅%𝑇
𝑟𝑑𝑟
𝑑𝑡 = 𝑆 − 1
𝜌,𝑅%𝑟
𝐷𝑒! 𝑇 + 𝐿
𝑅%𝑇 − 1 𝜌,𝐿 𝐾𝑇
𝑟𝑑𝑟
𝑑𝑡 = 𝑆 − 1
𝐹/ + 𝐹1 𝐹/ = 𝜌,𝑅%𝑟
𝐷𝑒! 𝑇 𝐹1 = 𝐿
𝑅%𝑇 − 1 𝜌,𝐿 𝐾𝑇
The equation is an excellent approximation to that obtained by simultaneous solution of equations for rvr i Tr.
In the term represented by Fk, it may be shown that L/RvT>>1, and we can neglect 1 in brackets
𝑟𝑑𝑟
𝑑𝑡 = 𝑆 − 1 − 𝐴
𝑟 +𝐵! 𝑟+ 𝐹/ + 𝐹1
𝑆23 = 1 + 𝐴
𝑟 − 𝐵! 𝑟+ 𝑟𝑑𝑟
𝑑𝑡 = 𝑆 − 𝑆23 𝐹/ + 𝐹1 𝑟𝑑𝑟
𝑑𝑡 = 𝑆 − 1
𝐹/ + 𝐹1 saturation over plane water surface
Saturation over plane water surface will be replaced by the equilibrium saturation over droplet solution
𝐹/ = 𝜌,𝑅%𝑇
𝐷𝑒! 𝑇 𝐹1 = 𝐿
𝑅%𝑇 − 1 𝜌,𝐿
𝐾𝑇 ≈ 𝜌,𝐿' 𝐾𝑅%𝑇'
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K depends on T, D depends on T and p
Dynamic viscosity of the air Kinematic viscosity of the air
In the term represented by Fk, it may be shown that L/RvT>>1, and we can neglect 1 in brackets
𝐹/ = 𝜌,𝑅%𝑇
𝐷𝑒! 𝑇 𝐹1 = 𝐿
𝑅%𝑇 − 1 𝜌,𝐿
𝐾𝑇 ≈ 𝜌,𝐿' 𝐾𝑅%𝑇'
𝐾 ∝ 𝜇 , 𝜇
𝐷 ∝ T𝜇
𝜌 , 𝜇T
𝜌 = 𝜈
𝜇 𝑇 = 1.72 L 10.4 393 𝑇 + 120
𝑇 273
+5 '
𝐾 = 4.1868 L 10.+ 5.69 + 0.017 𝑇 − 𝑇& 𝑊𝑚.(𝐾.(
𝐷 = 2.11 L 10.4 𝑇 𝑇&
(.78 𝑝&
𝑝 𝑚'𝑠.(
𝑇& = 273.15 𝐾; 𝑝& = 1013.25 ℎ𝑃𝑎
For very small drops, for which the Kelvin and Raoult effects are important the equation of condensational growth cannot be solved analiticaly.
Numerical solution
The table gives the time necessary for a droplet to grow from the initial radius 0,75 µm, for different initial mass of NaCl in the condensation nucleus.
Supersaturation 0,05%, p=900hPa, T=273 K Droplets formed on big CCN grow initially faster (shorter time) than droplets formed an small CCN.
From a certain drop’s diameter (~10 µm) the rate of growth is similar.
𝑟𝑑𝑟
𝑑𝑡 = 𝑆 − 1 − 𝐴
𝑟 + 𝐵! 𝑟+ 𝐹/ + 𝐹1
When a droplet becomes larger (approx. 10 µm) the curvature and solution effects (A/r i Bs/r3) become negligible compared to (S-1).
When a droplet becomes larger (approx. 10 µm) the curvature and solution effects (A/r i Bs/r3) become negligible compared to (S-1).
When a droplet becomes larger (approx. 10 µm) the curvature and solution effects (A/r i Bs/r3) become negligible compared to (S-1).
The equation for condensational growth can be solved analiticaly.
Smaller droplets grow faster. The droplet spectrum becomes narrower due to the condensational growth.
The rate of growth of a droplet’s surface does not depend on radius
𝑟𝑑𝑟
𝑑𝑡 = 𝑆 − 1 𝐹/ + 𝐹1 𝑑𝑟'
𝑑𝑡 = 2𝜉 𝑆 − 1 , 𝜉 = 1 𝐹/ + 𝐹1 𝑟' 𝑡 − 𝑟' 𝑡& = 2𝜉 𝑆 − 1 L 𝑡 − 𝑡&
𝑟 𝑡 = 𝑟' 𝑡& + 2𝜉 𝑆 − 1 L 𝑡 − 𝑡&
x is a function of temperature and pressure Solid lines representlog10 x
𝜉 = 1 𝐹/ + 𝐹1
𝜉 = 10(.9+8 = 68.2 𝜇𝑚'𝑠.(
x is a function of temperature and pressure Solid lines representlog10 x
Dashed lines represent pseudoadiabats corresponding T= 0°C and 20°C 𝜉 = 1
𝐹/ + 𝐹1
𝜉 = 10(.9+8 = 68.2 𝜇𝑚'𝑠.(
𝑝 = 800 ℎ𝑃𝑎 , 𝑇 = 0℃
𝜉 = 1 𝐹/ + 𝐹1
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For the same supersaturation values smaller droplets grow faster (faster increase of their radius) than bigger droplets.
Condensational growth leads to a narrowing of the drop-size distribution.
For a given thermodynamic conditions the rate of growth of droplet surface (~r2) is constant and depends only on supersaturation .
𝑑𝑟
𝑑𝑡 = 𝜉𝑆 − 1
𝑟 , 𝜉 = 1
𝐹/ + 𝐹1 𝑑𝑟'
𝑑𝑡 = 2𝜉 𝑆 − 1
T=10oC, p=1000 hPa, S=0.5%
In realistic cloud conditions, growth by water-vapor diffusion seldom produces droplets with radii close to 20 µm because of the low magnitude of the supersaturation field and
/30 30 S=0.05%
S=0.5%
