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Chapter_08_equations_of_particle_motion

Chapter_08_equations_of_particle_motion - 8.4 Equations of...

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8.4 Equations of Particle Motion To design particle collection devices and to predict their performance, engineers must be able to predict the trajectories of particles passing through the devices. The objective of this section is to establish equations that predict the trajectories of particles. As an aerosol particle moves through air, it disturbs the air, changing the velocity and pressure fields of the air flow. In addition, particles collide with each other and are influenced by each others’ wakes. Exact analysis of particles moving in air is therefore nearly impossible, even for simple air flows, since the equations for the air flow and for each particle’s trajectory are coupled. Fortunately, some simplifications are possible, which make the problem more tractable. Two major assumptions are made here: 1. Particles move independently of each other. 2. Particles do not influence the flow field of the carrier gas. The first assumption is valid if two conditions are met: (a) particle collisions are infrequent and inconsequential, and (b) particles are not significantly affected when they pass through each others’ wakes. A useful rule of thumb that quantifies these conditions for a monodisperse aerosol is that the average distance between particles is at least 10 times the particle diameter. Assuming that 8 particles are located at the corners of a cube of dimension L, one finds that L/D p > 10 when () p 3 number p 4 8 1000 3c c D πρ => (8-51) where c is the mass concentration of particles, ρ p is the particle density, and c number = n t /V is the particle number concentration , defined as the total number of particles per unit volume of gas. Table 8.3 illustrates these upper limits and indicates that the particle number concentration has to be exceedingly large for the particles to influence each other. For water droplets ( ρ p = 1,000 kg/m 3 ), application of Eq. (8- 51) leads to a particle mass concentration c = 4.2 kg/m 3 , corresponding to the upper limit of 1,000 in Eq. (8-51). For most problems in indoor air pollution, particle concentrations of water and particulate pollutants are hundreds of times smaller than 4.2 kg/m 3 . Consequently the first assumption above is clearly valid, and one can calculate the trajectory of one individual particle at a time. The second assumption is more difficult to quantify. Large objects moving through air generate air flows due to displacement effects and the effects of aerodynamic wakes. For example, a pedestrian walking along the side of a road feels the “wind” created by a passing vehicle. The same effect can be experienced when a person walks past another person in otherwise quiescent air. Automobiles and people, however, are very large objects; particles usually associated with indoor air pollution are many orders of magnitude smaller. As shown below, most particles of interest to indoor Table 8.3 Particle number concentrations beyond which particles influence each other.

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Chapter_08_equations_of_particle_motion - 8.4 Equations of...

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