In our discussion so far, we have assumed that the aerosol particles are suspended in a stagnant fluid. In most atmospheric applications, the air is in motion and one needs to describe simultaneously the air and particle motion. Equation (9.36) will be the starting point of our analysis.
Actually (9.36) is a simplified form of the full equation of motion, which is (Hinze 1959)
J 0 (t t I i where Vp is the particle volume. The second term on the RHS is due to the pressure gradient in the fluid surrounding the particle, caused by acceleration of the gas by the particle. The third term is the force required to accelerate the apparent mass of the particle relative to the fluid. Finally, the fourth term, the Basset history integral, accounts for the force arising as a result of the deviation of fluid velocity from steady state. In most situations of interest for aerosol particle in air, the second, third, and fourth terms on the RHS of (9.90) are neglected. Assuming that gravity is the only external force exerted on the particle, we again obtain (9.36).
Neglecting the gravitational force and particle inertia leads to the zero-order approximation that v ~ u; that is, the particle follows the streamlines of the airflow. This approximation is often sufficient for most atmospheric applications, such as turbulent
dispersion. However, it is often necessary to quantify the deviation of the particle trajectories from the fluid streamlines (Figure 9.12).
A detailed treatment of particle flow around objects, in channels of various geometries, and so on, is beyond the scope of this book. Treatments are provided by Fuchs (1964), Hinds (1999), and Flagan and Seinfeld (1988). We will focus our analysis on a few simple examples demonstrating the important concepts.
9.6.1 Motion of a Particle in an Idealized Flow (90° Corner)
Let us consider an idealized flow, shown in Figure 9.13, in which an airflow makes an abrupt 90° turn in a corner maintaining the same velocity (Crawford 1976). We would like to determine the trajectory of an aerosol particle originally on the streamline y = 0, which turns at the origin x = 0.
The trajectory of the particle is governed by (9.37). Neglecting gravity, the a and y components of the equation of motion are after the turning point
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