Aerosol Transport – Inertia

:: Section 7

Straight-Line Particle Acceleration

II. Acceleration Process

In the previous sections, we discussed the terminal settling velocity under the equilibrium condition (i.e. the net force acting on the particle is zero and the velocity of the particle is constant). Now, we are going to consider the particle during the acceleration process. If a particle is released with zero initial velocity in still air. How long will it take for the particle to reach its terminal settling velocity?

When the particle is released, there are two forces acting on it: the gravitational force pulling it down and the drag force to slow it. Therefore, the net force of these two results in the acceleration of the particle.

According to the Newton's law,

If we multiple particle mobility (B) to each term of the above equation and knowing that the particle's relaxation time (τ) equals to mB, we can get:

Thus, knowing that τg equals to the settling velocity (V_TS), the velocity of a particle at any time after it is released in still air in a gravitational field can be calculated by the following equation via integration of the above one:

where V(t) is the particle's velocity after it is released for a period of time, t.

A particle will reach 63% of its terminal settling velocity after an elapsed time of τ, and reach 95% of the terminal settling velocity after 3τ.

1. Does a smaller or a larger particle have a smaller relaxation time? Why? 2. Does a smaller or a larger particle reach its terminal velocity faster? Why?