droplet growth

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Kelvin Curve
Köhler Curve
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Growth of Individual Cloud Droplet

• Depends upon
• Type and mass of hygroscopic nuclei
• surface tension
• humidity of the surrounding air
• rate of transfer of water vapor to the droplet
• rate of transfer of latent heat of condensation
  away from the droplet

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Assumptions

• Isolated, spherical water droplet of mass M,
  radius r and density ?w
• Droplet is growing by the diffusion of water
  vapor to the surface
• The temperature T? and water vapor density ?v? of
  the remote environment remain constant
• A steady state diffusion field is established
  around the droplet so that the mass of water
  vapor diffusing across any spherical surface of
  radius R centered on the droplet will be
  independent of R and time t

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Ficks Law of Diffusion

• Flux of water vapor toward the droplet through
  any spherical surface is given as

where D - diffusion coefficient of water vapor in
air ?v - density of water vapor Note that Fw has
units of mass/(unit areaunit time)
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Mass Transport

• Rate of mass transfer of water vapor toward the
  drop through any radius R is denoted Tw and

Note that Tw A1(a constant) because we
assumed a steady state mass transfer
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Mass Transport - continued

• Integrate the equation from the surface of the
  droplet where the vapor density is ?vr to ? where
  it is ?v?

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Conduction of Latent Heat

• Assume that the latent heat released is
  dissipated primarily by conduction to the
  surrounding air. Since we assume that the mass
  growth is constant (A1), then the latent heat
  transport is a constant (A2).
• The equation for conduction of heat away from the
  droplet may be written as

K is the thermal conductivity of air
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Conduction of Latent Heat - continued

• Integrate the equation from the droplet surface
  to several radii away which is effective ?

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Radial Growth Equations
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Radial Growth - continued

• Note that, the radius of a smaller droplet will
  increase faster than a larger droplet..

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• Important Variables
• e? Ambient water vapor pressure
• es? Equilibrium water vapor pressure at
  ambient temperature
• es? CC(T?)
• er Equilibrium water vapor pressure for a
  droplet
• er ehrCC(Tr) f(r)
• f(r)
• esr Equilibrium water vapor pressure for
  plane water at the same temperature as the
  droplet
• esr CC(Tr)

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Additional Equations

• Clausius-Clapeyron equation
• Combined curvature and solute effects
• Integrate the CC equation from the saturation
  vapor pressure at the temperature of the
  environment es(T?), denoted as es? , to the
  saturation vapor pressure at the droplet surface
  es(Tr), denoted esr to obtain

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Final Set of Growth Equations

• Mass diffusion to the droplet

• Conduction of latent heat away

• Combined curvature and solute effects

• Clausius-Clapeyron equation

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Summary

• The four equations are a set of simultaneous
  equations for er, esr , Tr , and r.
• If we know the vapor pressure and temperature of
  the environment and the mass of solute, the four
  unknowns may be calculated for any value of r.
  Then, r may be calculated by numerical
  integration.

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Derivation of Droplet Radius Dependence on Time

• Steps to solve the Problem
• Expand Clausius-Clapeyron Equation
• Substitute for Tr - T? in (2) using the expansion
• Express the ratio (esr/es????in terms of radial
  growth rate from (1)
• Solve resulting equation for r (dr/dt)

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Derivation
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Derivation - continued
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Derivation - continued

• But, from Eq. (2) we can write

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Derivation - continued

• Note that some quantities always appear together.
  Lets define

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Derivation - continued
or
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Derivation - continued

• where

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Radius as a Function of Time
Note that, in general, this requires a numerical
integration
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Analytic Approximation

• Since (er /esr )???1 after nucleation

Consider the case where S, C1, and C2 are
constant. Integrate as
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