N(z, t)=N0 z oo where the z coordinate is taken as vertically upward.

The solution of (9.80) and (9.81) for the vertical profile of the number distribution

We can calculate the deposition rate of particles on the z = 0 surface from the expression for the flux of particles at z = 0,

Recall that N(0,t) = 0 in (9.83). Combining (9.82) and (9,83) we obtain

According to (9.84), there is an infinite removal flux at r = 0, bccause of our artificial specification of an infinite concentration gradient at z = t = 0. We can identify a characteristic time zd, for the system xds —

and observe the following limiting behavior for the particle flux at short and long times:

Thus, at very short times, the deposition flux is that resulting from diffusion plus one-half that due to settling, whereas for long times the deposition flux becomes solely the settling flux. For particles of radii 0.1 pm and 1 pm in air (at 1 atm, 298 K), is about 80s and 0.008 s, respectively, assuming a density of 1 gem-3. For times longer than thai, Brownian motion does not have any effect on the particle motion. The aerosol number concentration and removal flux are shown in Figures 9,9 and 9.10. The system reaches a steady state after roughly 100 s and at this state 0.23 particles are deposited per second on each cm" of the surface (Figure 9.10). Note that the concentration profile changes only over a shallow layer of approximately 1 mm above the surface (Figure 9.9). The depth of this layer is proportional to D/v,. Note that the example above is not representative of the ambient atmosphere, where there is turbulence and possibly also sources of particles at the ground.

FIGURE 9.9 Evolution of ihe concentration profile (times 1, 10T 100T and 400s) of an aerosol population settling and diffusing in stationary air over a flat perfectly absorbing surface. The particles are assumed to be monodisperse with Dp = 0.2 j.im and have an initial concentration of lOOOcm-3.
FIGURE 9.10 Removal rate of particles as a function of time for the conditions of Figure 9.9. 9.5.3 Mean Free Path of an Aerosol Particle

The concept of mean free path is an obvious one for gas molecules. In the Brownian motion of an aerosol particle there is not an obvious length that can be identified as a mean free path. This is depicted in Figure 9. i 1 showing plane projections of the paths followed by an air molecule and an aerosol particle of radius roughly equal to 1 pm. The trajectories

FIGURE 9.11 A two-dimensional projection of the path of (a) an air molecule and (b) the center of a 1-pm particle. Also shown is the apparent mean free path of the particle.

FIGURE 9.11 A two-dimensional projection of the path of (a) an air molecule and (b) the center of a 1-pm particle. Also shown is the apparent mean free path of the particle.

of the gas molecules consist of straight segments, each of which represents the path of the molecule between collisions. At each collision the direction and speed of the molecule are changed abruptly. With aerosol particles the mass of the particle is so much greater than that of the gas molecules with which it collides that the velocity of the particle changes negligibly in a single collision. Appreciable changes in speed and direction occur only after a large number of collisions with molecules, resulting in an almost smooth particle trajectory.

The particle motion can be characterized by a mean thermal speed cp:

To obtain the mean free path Xp, we recall that in Section 9.1, using kinetic theory, we connected the mean free path of a gas to measured macroscopic transport properties of the gas such as its binary diffusivity. A similar procedure can be used to obtain a particle mean free path hp from the Brownian diffusion coefficient and an appropriate kinetic theory expression for the diffusion flux. Following an argument identical to that in Section 9.1, diffusion of aerosol particles can be viewed as a mean free path phenomenon so that

and the mean free path Xp combining (9.73), (9.87), and (9.88) is then

TABLE 9.5 Characteristic Quantities in Aerosol Brownian Motion

Dp, (im

D, cm2 s 1

cp, cms 1


Xp (pm)


1.28 x 10"2


1.33 x 10-9

6.59 x 10"2


3.23 x 10~3


2.67 x 10"9

4.68 x 10"2


5.24 x 10~4


6.76 x 10"9

3.00 x 10"2


1.30 x 10"4


1.40 x 10~8

2.20 x 10"2


3.59 x HT5


2.98 x 10~8

1.64 x 10~2


6.82 x 10~6


9.20 x 10"8

1.24 x 10"2


2.21 x 10~6


2.28 x 10~7

1.13 x 10"2


8.32 x 10"7


6.87 x 10~7

1.21 x 10-2


2.74 x 10"7


3.60 x 10"6

1.53 x 10-2


1.27 x 10"7


1.31 x 10-5

2.06 x 10-2


6.1 x 10-8

5.55 x 10"2

5.03 x 10"5

2.8 x 10~2


2.38 x 10~8

1.40 x 10~2

3.14 x 10"4

4.32 x 10"2


1.38 x 10~8

4.96 x 10~3

1.23 x 10~3

6.08 x 10"2

Certain quantities associated with the Brownian motion and the dynamics of single aerosol particles are shown as a function of particle size in Table 9.5. All tabulated quantities in Table 9.5 depend strongly on particle size with the exception of the apparent mean free path Xp, which is of the same order of magnitude right down to molecular sizes, with atmospheric values Xp ~ 10-60nm.

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