$1\mathrm{.}3.$ Dependence of gravitational settling velocity on $D_p$

Refer to the sketch above, which shows the settling velocity of particles $(v_s)$ in air as a

function of the particle diameter $(D_p)$ in $log-$log space. The figure spans six orders of

magnitude. The particle size range is divided into three regions. In the first region, the settling

velocity increases in proportion to the first power of particle diameter. In the second region,

the settling velocity increases in proportion to the square of the particle diameter. In the third

region, the settling velocity roughly increases in proportion to the first power of particle

diameter. This figure is for unit particle density in standard conditions, but this relationship

holds for a broad range of conditions.

a$.$ Approximately what particle diameter marks the transition point between Region $1$ and $2?$

b$.$ Explain both the underlying physical reason and mathematical basis for the change in

dependence of settling velocity on particle diameter between Regions $1$ and $2.$

c$.$ Explain the physical phenomenon that accounts for the change in how settling velocity

depends on $D_p$ between Regions $2$ and Regions $3.$

$1\mathrm{.}4.$ Particle transport in the stratosphere

To begin, confirm that you can correctly calculate the slip correction factor, settling velocity,

and diffusion coefficient for a $0\mathrm{.}1m$ particle at standard conditions. See attached table.

Now, consider a position in the stratosphere, $50km$ above the earth's surface. The air

temperature is $271K$ and the pressure is $0\mathrm{.}8\frac{m}{b}ar=80Pa.$ Compute the particle transport

properties indicated below. In carrying out your analysis, consider the possible effects of

altitude, temperature and pressure on gravitational acceleration, air viscosity, mean$-$free$-$path

of gas molecules, and air density.

a$.$ What is the Brownian diffusivity of a $0\mathrm{.}3m$ particle?

b$.$ What is the gravitational settling velocity of a $0\mathrm{.}3m$ particle? Assume unit density.