Collision-Coalescence r1 r2 y
Collision-Coalescence Reading Wallace & Hobbs pp 224 – 232
Collision-Coalescence Objectives Be able to recall that condensational growth produces a monodisperse spectra Be able to define collision efficiency Be able to define terminal fall speed Be able to draw a force diagram for a fall cloud droplet
Collision-Coalescence Objectives Be able to compare the collection efficiencies of various sized collector drops Be able to explain the variations in collection efficiencies of collector drops Be able to define coalescence efficiency Be able to list the factors that determine coalescence efficiency
Collision-Coalescence Objectives Be able to explain the variations in coalescence efficiencies of collector drops Be able to define collection efficiency Be able to list the assumptions of the continuous collection model Be able to state the relationship between rate of growth and collector radius
Collision-Coalescence Objectives Be able to describe the change in collector drop size in relation to height above cloud base Be able to recall the size at which cloud drops break up Be able to describe the stochastic collision model
Collision-Coalescence Overview Collision Efficiency Droplet Terminal Fall Speed Coalescence Efficiency Collection Efficiency Continuous Collection Model Stochastic Collision Model
Overview Observations 20 min. from cloud formation to precipitation
Overview Theory 10 min. to grow to 20 mm at SS = 0.5% Cloud Droplet
Overview Spectrum tends to be monodisperse 10 1 Supersaturation (%) Time (s) Droplet Radius (mm) Supersaturation (%) .01 .1 1 10 100 Supersaturation
Overview Droplets have same settling speed No collisions No broadening of spectrum
Overview How does spectrum broaden? Giant Sea Salt Particles? Bigger is better! Mixing & Turbulence?
Overview Broadening of droplet spectrum accounts for growth from cloud droplet to precipitation sized particles Cloud Droplet (10 mm) Typical Raindrop (1000 mm)
Collision-Coalescence Overview Collision Efficiency Droplet Terminal Fall Speed Coalescence Efficiency Collection Efficiency Continuous Collection Model Stochastic Collision Model
Collision Efficiency Ratio of the actual number of collisions to the number of collisions for complete geometric sweepout
Pay up, or I has ta break youse kneecaps. Collision Efficiency Terminology Collector Drop the drop doing the collecting Pay up, or I has ta break youse kneecaps. x x Collector Drop
Collision Efficiency Terminology Droplet Terminal Fall Velocity Vt1 Speed at which a droplet falls Vt1 Vt2
Droplet Terminal Fall Speed Balance between drag force and gravitational force Drag Force Gravitational Force
Droplet Terminal Fall Speed Function of Droplet Radius Reynolds Number Drag Coefficient Viscosity of Air Drag Force Gravitational Force for droplets < 20 mm
Collision Efficiency Droplet Terminal Fall Velocity Large droplets fall faster than small ones Vt1 Vt1 > Vt2 Vt2
Collision Efficiency
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Collision Efficiency Small drops have a low collection efficiency Tend to follow streamlines Low inertia
Collision Efficiency Large Droplet
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Collision Efficiency Large droplets have a large collection efficiency Cross streamlines Large inertia
# of collisions for a complete geometric sweepout Collision Efficiency # of actual collisons # of collisions for a complete geometric sweepout E =
Collision Efficiency Effective Collision Cross Section (y) Critical distance between Centerline of collector drop Center of the droplet r1 r2 y
Collision Efficiency Effective Collision Cross Section Area Droplets whose center within this area are collected r1 r2 y
Collision Efficiency Geometric Collision Cross Section (r1+r2) r1 r2 y
Collision Efficiency Geometric Collision Cross Section Area Area swept out by collector drop r1 r2 y
# of collisions for a complete geometric sweepout Collision Efficiency # of actual collisons # of collisions for a complete geometric sweepout E = r1 r2 y
Collision Efficiency Problems Not so simple Droplets influence each other’s motion r1 r2 y
Collision Efficiency (E) Collector Drops 10 70 mm 60 50 40 1 30 Collision Efficiency (E) 20 .1 10 .01 .2 .4 .6 .8 1.0 r2/r1
Collision Efficiency (E) Collision efficiency increases as collector drop size increases Collision efficiency low for drops < 20 mm 10 20 30 40 50 60 70 mm .01 .1 1 .2 .4 .6 .8 1.0 r2/r1 Collision Efficiency (E) Collector Drops
Collision Efficiency (E) Collision efficiency low if collector drop much bigger than droplets (low inertia) 10 20 30 40 50 60 70 mm .01 .1 1 .2 .4 .6 .8 1.0 r2/r1 Collision Efficiency (E) Collector Drops
Collision Efficiency (E) Collision efficiency falls off for 20 & 30 mm droplets as size approaches collector (0.8-0.9) Due to comparative terminal velocity 10 20 30 40 50 60 70 mm .01 .1 1 .2 .4 .6 .8 1.0 r2/r1 Collision Efficiency (E) Collector Drops
Collision Efficiency (E) Collection efficiency exceed 1 due to wake capture 10 20 30 40 50 60 70 mm .01 .1 1 .2 .4 .6 .8 1.0 r2/r1 Collision Efficiency (E) Collector Drops
Collision Efficiency Wake Capture
Collision-Coalescence Overview Collision Efficiency Droplet Terminal Fall Speed Coalescence Efficiency Collection Efficiency Continuous Collection Model Stochastic Collision Model
Coalescence Efficiency The fraction of collisions that result in coalescence
Coalescence Efficiency Possible Scenarios Droplets Bounce Apart Droplets Coalescence & Remain United Droplets Coalesce Temporarily & Separate Into Original Identities Droplets Coalesce Temporarily & Separate Into a Number of Small Drops
Coalescence Efficiency Depends on Droplet Size & Terminal Speed
Coalescence Efficiency Depends on Droplet Size & Terminal Speed Droplet Trajectories
Coalescence Efficiency + + + + + + + + + + + + + + + Depends on Droplet Size & Terminal Speed Droplet Trajectories Electrical Forces + - - - - - - - - - - - - - - - - - - - - -
Coalescence Efficiency 1.0 Collector Drops 400 – 2000 mm .8 .6 Collector Drops 50 – 100 mm Coalescence Efficiency .4 .2 10-3 10-2 10-1 100 r2/r1
Fig. 6.22 (W&H) Coalescence efficiencies
Coalescence Efficiency Coalescence Efficiency for Large Collector and Small Droplets is Good Collector Drops 400 – 2000 mm 50 – 100 mm r2/r1 10-3 10-2 10-1 100 .2 .4 .6 .8 1.0 Coalescence Efficiency
Collision-Coalescence Overview Collision Efficiency Droplet Terminal Fall Speed Coalescence Efficiency Collection Efficiency Continuous Collection Model Stochastic Collision Model
Collection Efficiency The Product of Collision Efficiency & Coalescence Efficiency Collection Efficiency Collision Efficiency Coalescence Efficiency = x
Collision-Coalescence Overview Collision Efficiency Droplet Terminal Fall Speed Coalescence Efficiency Collection Efficiency Continuous Collection Model Stochastic Collision Model
Continuous Collection Model Assumptions Still Air Droplets Uniformly Distributed Monodisperse Droplet Spectrum Droplets Are Collected Uniformly r1 Vt1 Vt2
Continuous Collection Model Rate Increase of Mass of Collector Drop r1 wl = Liquid Water Content Ec = Collection Efficiency V1 V2
Continuous Collection Model r1 Mass of Collector rl = Density of Liquid Water V1 V2
Continuous Collection Model r1 V1 V2
Continuous Collection Model r1 Assuming v1>>v2 Collection Efficiency = Collision Efficiency V1 V2
Continuous Collection Model r1 Rate of Growth Increases with Collector Radius Accelerating Process As r1 Increases E Increases v1 Increases V1 V2
Continuous Collection Model W r1 Let’s Add An Updraft! velocity of collector V1 velocity of droplets V2
Continuous Collection Model W r1 Velocity of Collector h = height above cloud base V1 V2
Continuous Collection Model W r1 Change in Collector Size Above Cloud Base V1 V2
Continuous Collection Model Integrate From Cloud Base (0) to Some Height (H) H
Continuous Collection Model Assuming Liquid Water Content is Constant with Height H
Continuous Collection Model First Integral Dominates When w>v1 Small Droplet Sizes Droplet Rises in Cloud H
Continuous Collection Model Second Integral Dominates When w<v1 Large Droplet Sizes Droplet Descends Through Cloud H
Continuous Collection Model Droplet Breakup r > 1 mm Produces More Collector Drops
Continuous Collection Model Model Predictions Warm Clouds with Strong Updrafts Produce Rain in Short Time Must Have Vertical Development (Deep)
Continuous Collection Model Model Predicition Collector Drops Grow to Same Size Not Realistic
Collision-Coalescence Overview Collision Efficiency Droplet Terminal Fall Speed Coalescence Efficiency Collection Efficiency Continuous Collection Model Stochastic Collision Model
Stochastic Collision Model Collisions Are Individual Events Statistically Distributed in Time and Space
Stochastic Collision Model 100 Higher Collection Efficiency 10 90 9 9 1 18 81
Stochastic Collision Model Importance Spectrum Broadening Fast Growth of Small Number of Large Droplets 100 90 81 10 18 1 9