1. Collision-Coalescencer1 r2 y

2. Collision-Coalescence
Reading
Wallace & Hobbs
pp 224 – 232

3. 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

4. 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

5. 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

6. 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

7. Collision-Coalescence
Overview
Collision Efficiency
Droplet Terminal Fall Speed
Coalescence Efficiency
Collection Efficiency
Continuous Collection Model
Stochastic Collision Model

8. Overview Observations 20 min. from cloud formation to precipitation

9. Overview Theory 10 min. to grow to 20 mm at SS = 0.5% Cloud Droplet

10. Overview Spectrum tends to be monodisperse 10 1 Supersaturation (%)
Time (s)
Droplet Radius (mm)
Supersaturation (%)
.01 .1 1 10
100
Supersaturation

11. Overview Droplets have same settling speed No collisions
No broadening of spectrum

12. Overview How does spectrum broaden? Giant Sea Salt Particles?
Bigger is better!
Mixing & Turbulence?

13. Overview
Broadening of droplet spectrum accounts for growth from cloud droplet to precipitation sized particles
Cloud Droplet
(10 mm)
Typical Raindrop
(1000 mm)

14. Collision-Coalescence
Overview
Collision Efficiency
Droplet Terminal Fall Speed
Coalescence Efficiency
Collection Efficiency
Continuous Collection Model
Stochastic Collision Model

15. Collision Efficiency
Ratio of the actual number of collisions to the number of collisions for complete geometric sweepout

16. 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

17. Collision Efficiency Terminology Droplet Terminal Fall Velocity Vt1
Speed at which a droplet falls
Vt1
Vt2

18. Droplet Terminal Fall Speed
Balance between drag force and gravitational force
Drag Force
Gravitational Force

19. Droplet Terminal Fall Speed
Function of
Droplet Radius
Reynolds Number
Drag Coefficient
Viscosity of Air
Drag Force
Gravitational Force
for droplets < 20 mm

20. Collision Efficiency Droplet Terminal Fall Velocity
Large droplets fall faster than small ones
Vt1
Vt1 > Vt2
Vt2

21. Collision Efficiency

22. Collision Efficiency

23. Collision Efficiency

24. Collision Efficiency

25. Collision Efficiency

26. Collision Efficiency

27. Collision Efficiency Small drops have a low collection efficiency
Tend to follow streamlines
Low inertia

28. Collision Efficiency
Large Droplet

29. Collision Efficiency

30. Collision Efficiency

31. Collision Efficiency

32. Collision Efficiency

33. Collision Efficiency

34. Collision Efficiency Large droplets have a large collection efficiency
Cross streamlines
Large inertia

35. # of collisions for a complete geometric sweepout
Collision Efficiency
# of actual collisons
# of collisions for a complete geometric sweepout
E =

36. Collision Efficiency Effective Collision Cross Section (y)
Critical distance between
Centerline of collector drop
Center of the droplet r1 r2 y

37. Collision Efficiency Effective Collision Cross Section Area
Droplets whose center within this area are collected r1 r2 y

38. Collision Efficiency
Geometric Collision Cross Section (r1+r2) r1 r2 y

39. Collision Efficiency Geometric Collision Cross Section Area
Area swept out by collector drop r1 r2 y

40. # of collisions for a complete geometric sweepout
Collision Efficiency
# of actual collisons
# of collisions for a complete geometric sweepout
E = r1 r2 y

41. Collision Efficiency Problems Not so simple
Droplets influence each other’s motion r1 r2 y

42. 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

43. 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

44. 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

45. Collision Efficiency (E)
Collision efficiency falls off for 20 & 30 mm droplets as size approaches collector ( )
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

46. 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

47. Collision Efficiency
Wake Capture

48. Collision-Coalescence
Overview
Collision Efficiency
Droplet Terminal Fall Speed
Coalescence Efficiency
Collection Efficiency
Continuous Collection Model
Stochastic Collision Model

49. Coalescence Efficiency
The fraction of collisions that result in coalescence

50. 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

51. Coalescence Efficiency
Depends on
Droplet Size & Terminal Speed

52. Coalescence Efficiency
Depends on
Droplet Size & Terminal Speed
Droplet Trajectories

53. Coalescence Efficiency
Depends on
Droplet Size & Terminal Speed
Droplet Trajectories
Electrical Forces + -

54. 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

55. Fig. 6.22 (W&H) Coalescence efficiencies

56. 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

57. Collision-Coalescence
Overview
Collision Efficiency
Droplet Terminal Fall Speed
Coalescence Efficiency
Collection Efficiency
Continuous Collection Model
Stochastic Collision Model

58. Collection Efficiency
The Product of Collision Efficiency & Coalescence Efficiency
Collection Efficiency
Collision Efficiency
Coalescence Efficiency = x

59. Collision-Coalescence
Overview
Collision Efficiency
Droplet Terminal Fall Speed
Coalescence Efficiency
Collection Efficiency
Continuous Collection Model
Stochastic Collision Model

60. Continuous Collection Model
Assumptions
Still Air
Droplets Uniformly Distributed
Monodisperse Droplet Spectrum
Droplets Are Collected Uniformly r1
Vt1
Vt2

61. Continuous Collection Model
Rate Increase of Mass of Collector Drop r1
wl = Liquid Water Content
Ec = Collection Efficiency V1 V2

62. Continuous Collection Modelr1
Mass of Collector
rl = Density of Liquid Water V1 V2

63. Continuous Collection Modelr1 V1 V2

64. Continuous Collection Modelr1
Assuming
v1>>v2
Collection Efficiency = Collision Efficiency V1 V2

65. Continuous Collection Modelr1
Rate of Growth Increases with Collector Radius
Accelerating Process
As r1 Increases
E Increases
v1 Increases V1 V2

66. Continuous Collection ModelW r1
Let’s Add An Updraft!
velocity of collector V1
velocity of droplets V2

67. Continuous Collection ModelW r1
Velocity of Collector
h = height above cloud base V1 V2

68. Continuous Collection ModelW r1
Change in Collector Size Above Cloud Base V1 V2

69. Continuous Collection Model
Integrate From Cloud Base (0) to Some Height (H) H

70. Continuous Collection Model
Assuming Liquid Water Content is Constant with Height H

71. Continuous Collection Model
First Integral Dominates When w>v1
Small Droplet Sizes
Droplet Rises in Cloud H

72. Continuous Collection Model
Second Integral Dominates When w<v1
Large Droplet Sizes
Droplet Descends Through Cloud H

73. Continuous Collection Model
Droplet Breakup
r > 1 mm
Produces More Collector Drops

74. Continuous Collection Model
Model Predictions
Warm Clouds with Strong Updrafts Produce Rain in Short Time
Must Have Vertical Development (Deep)

75. Continuous Collection Model
Model Predicition
Collector Drops Grow to Same Size
Not Realistic

76. Collision-Coalescence
Overview
Collision Efficiency
Droplet Terminal Fall Speed
Coalescence Efficiency
Collection Efficiency
Continuous Collection Model
Stochastic Collision Model

77. Stochastic Collision Model
Collisions Are Individual Events
Statistically Distributed in Time and Space

78. Stochastic Collision Model
100
Higher Collection Efficiency 10 90 9 9 1 18 81

79. Stochastic Collision Model
Importance
Spectrum Broadening
Fast Growth of Small Number of Large Droplets
100 90 81 10 18 1 9