Coagulation
rain drop, ice crystal grow up to few mm, snowflakes, hailstones –up to few cm
Rate of droplet enlargement due to coagulation ~ r
Rate of droplet enlargement due to condensation ~ 1/r
Precipitation from stratiform clouds.
Scheme of formation
Water droplet clouds
Air cooling down
to supersaturated state
Formation and growth of germ droplets
Droplet growth due to condensation
Formation of
Precipitation germs
Growth of the drops due to coagulation and their falling
Partial evaporation under clouds
Precipitating drizzle
Mixed clouds
Air cooling down
to supersaturated state
Formation and growth of germ droplets
Droplet growth due to condensation
Some ice particles appearance
Growth of the ice particles due to sublimation
Growth of the ice particles due to coagulation
Winter |
Summer |
Precipitating snow, |
Melting of precipitating |
snow tapioca (soft hail) |
ice particles |
rainfall
The rate of fall for solid and liquid particles in the atmosphere
Atm. Particle is affected by: |
buoyancy |
• external forces |
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• air resistance force |
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• force interaction between particles |
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According to the 2 nd Newton’s law
m dV |
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(26.1) |
F |
G |
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dt |
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V is a particle velocity |
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F is a resis tan ce force, ~ V , r, ...
G is an external force (ex., gravity)
According to Stocks law for ball like particles
F0 6 rV |
(26.2) |
is a molecular vis cos ity coefficient
(26.2) is valid for lowr Reynoldsis a particlenumber Re*<3radius
According to experiments"0" indicates ball like particle
(26.3) is correct for 3 < Re < 400
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(1 |
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(26.3) |
F |
F |
1 Re |
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0 |
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6 |
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dimensionless number, gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.
Re |
VL |
VL |
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• V is the mean velocity of the object relative to the fluid (SI units: |
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m/s) |
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•L is a characteristic linear dimension, (travelled length of the fluid; |
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hydraulic diameter when dealing with river systems) (m) |
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μ is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s)) |
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ν is the kinematic viscosity (ν = μ / |
ρ |
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is the density of the fluid (kg/m³) |
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laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion; while turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.
Ball-like particle fall under influence of gravity, small Re
External force= gravity + buoyancy forces
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G 4 |
r3 d g |
4 r3 g |
(26.4) |
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3 |
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Acting downward |
acting upward |
air |
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density |
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G |
r3 g( d ) |
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G |
4 |
r3 d g |
due to |
d |
(26.5) |
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3 |
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ice particle 960 kg / m3 , |
d 1000 kg / m3 , 1.29 kg / m3 |
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Owing to the fact that F and G are directed along vertical line,
Using (26.2) and (26.5) :
m dV |
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4 r3 |
d g 6 rV |
(26.6) |
dt |
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when it is directed downward . |
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V is positive |
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(26.6) can be presented in |
form |
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dV V |
g 0 |
(26.7) |
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dt |
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is time of relaxation
6 m r 2r2 d / 9
Steady state motion dV/dt=0
Vs is a speed of steady state motion
V g or |
V 2g d r2 |
(26.8) |
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9 |
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it is Stokes' |
formula, works |
when r 10 5...5*10 3 cm |
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So, Vs ~ r2 at small Re (or small r)
Unsteady motion dV/dt ≠ 0
let’s use an eq-on of the particle motion
dV V |
g 0 |
(26.9) |
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dt |
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initial |
moment of |
a particle falling |
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t0 ,V0 |
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solution |
of (26.9) V V0 exp( t / ) |
(26.10) |
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Since V=dx/dt, after integrating (26.10)
V V0 (1 exp( t / )) |
(26.11) |
At t=τ V ↓ in e times as compared with V0 physical meaning of relaxation time
Assuming max imal dis tan ce Li particle passes through in a resis tan t environment
having initial speed V0
L |
V |
2V r2 |
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(26.12) |
0 |
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d |
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0 |
9 |
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inertial path of the particle run L |
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All above is true for ball-like particle |
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As droplet reaches r=> 0.5 mm deformations
Non ball-like particle speed is determined |
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accuracy about 3.7% |
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empirically:3 |
at 200 re 500 |
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Vs |
8.03*10 re 0.013 |
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V |
3.67 *10 10 r3e 3.27 *10 6 r3e 9.57 *10 3 r 0.121 |
at r 500 |
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e |
re |
is equivalent drop radius |
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( the drop would have if |
it were ball like at its real volume) |
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Vs |
in m / s |
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