Collisional ProcessesIn chemistry, the rate of 2-body reactions is based on the rate of collisions between

the two species times the probability that a collision leads to a reaction. This is expressed by the reactantconcentrations and a rate coefficient, k:

X + Y W + ZR = k[X][Y]

RdtZd

dtYd

dtXd

][][][

We can think of various collisional processes in the atmosphere the same way, where instead of species X and Y, we have particles of size Dp1 and Dp2

combining to form a particle of size Dp3, where

Likewise, instead of concentrations [X] and [Y], we have number concentrations N1, N2.

R = K12N1N2

RdtdN

dtdN

dtdN

321

3 32

313 ppp DDD

Review of CoagulationCoagulation by diffusion:

• Collision rate depends on the particles’ diffusivities, D1 and D2.• Important for aerosol smaller than 1 m.• Main sink for the Aitken/nucleation modes of aerosols• Main source for the Accumulation mode

))((2 212112 DDDDK pp

• Di usually includes the slip correction factor, since Dpi is generally small• is a kinetic correction that is important for Dpi < 100 nm.• You usually use tables (13.2, 13.3, Fig 13.5) for K12.

Useful formulae:

pi

pic

i

DDkTC

kTBD

3)(

38kTKii

For Kn << 1(i.e. Dpi >> 100 nm)

“self-coagulation”Some particles bounce instead of sticking on collision.

If the probability of the particles “sticking” on collision < 1, use S+P Equation (13.56) to calculate K and .

Shear CoagulationCoagulation by laminar shear:

• Collision rate depends on airflow shear• Important for aerosol and cloud drops larger than 100 nm.• Usually relevant in laboratory conditions• The basic concept between turbulent coagulation, relevant in

atmosphere (discussed later)• Ignores kinetic effects + sticking probability – basically an overestimate

32112 61

ppLS DD

dyduK

Shear CoagulationCoagulation by turbulent shear:

• Collision rate depends on turbulent kinetic energy density, k.• Important for cloud drops (generally > 5 m).• Same concept as laminar shear – uses turbulent theory to estimate

typical shear near a droplet in a turbulent field• Purely theoretical – ignores a number of effects, including inertial motion

of particles, flow-following behavior of smaller drops, sticking probability, etc.

321

2/1

12 120 ppkTS DDK

Gravitational SettlingCoagulation by gravitational settling:

• Collision rate depends on differential terminal velocities, vt,i.• Important for drizzle and rain drops (generally > 100 m).• Depends on collision volume and a collision efficiency, E• Usually, Dp1 = drizzle drop (> 100 m), Dp2 = cloud drop (< 100 m)

21212

2112 ,4 ppttpp

GS DDEvvDDK

• For drizzle drops, find terminal velocity from • S+P Section 9.3 using Figure 9.7 –OR-• S+P Eqs. 9.32 and 9.44 (less accurate)

• E is the combination of • The probability of a collision due to bending of the trajectories• The probability of coalescence due to bounceoff

Collision Efficiency

Et=total collection efficiencyE = Collision efficiencyEc = Coalescence probability

Ct EEE

21

2

21

2

21

1

pp

pp

pp

pC

DD

DDyE

DDD

E

For a very detailed empirical formulation of E (hence y), see the discussion surrounding S+P Eq. 20.53.

Otherwise, simply use S+P figure 17.29

The Big PictureConvolve this picture (for 1 m particles only) with the concentration of the other particle sizes to estimate the collisional lifetime of the drop.

Note that collisions with smaller particles will only incrementally affect the size of the 1 mm target.

Note that shear never dominates – it can be a correction, though.

Why does Brownian motion have have that dip in the middle?

Why does sedimentation have a much stronger dip in the middle?