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vail during the lifetime of a fluid eddy in which the particle is traversing, are sampled assuming that these values possess a Gaussian probability distribution. They used the lifetime of the fluid eddy as a time interval over which the gas velocity remains constant. This assumption allowed the direct solution of the equations of motion to obtain local, closed-form solutions. They obtained the values of z, r, and 9 by a simple stepwise integration of the equation of the trajectory. Many random trajectories were evaluated, and a grade efficiency curve was constructed based on these results. The results agreed with the experimental results of Stairmand (1951).

Kessler and Leith (1991) approximated the drag force term using Stokes' law drag for spheres. They solved the equations of motion numerically using backward difference formulas to solve the stiff system of differential equations. They modeled the collection efficiency for a single, arbitrarily sized particle by running several simulations for that particle. For each particle size, the ratio of collected particles to total particles described the mean of a binomial distribution for the probability distribution. Figure 5.16.6 shows this curve, three other grade efficiency models (Barth [1956], Dietz [1981], and Iozia and Leith [1990]), and the direct measurements of Iozia and Leith.

Early collection efficiency treatments balanced the centrifugal and drag forces in the vortex to calculate a critical particle size that was collected with 50 or 100% efficiency. The tangential velocity of the gas and the particle were assumed to be equal, and the tangential velocity was given by the vortex exponent law of Equation 5.16(4). As previously shown, the maximum tangential velocity, Vmax occurs at the edge of the central core. The average inward radial velocity at the core edge is as follows:

For particles of the critical diameter, the centrifugal and drag forces balance, and these static particles remain suspended at the edge of the core. Larger particles move to the wall and are collected, while smaller particles flow into the core and out of the cyclone. Barth (1956) and Stairmand (1951) used different assumptions about Vmax to develop equations for the critical diameter. Collection efficiencies for other particle sizes were obtained from a separation curve—a plot that relates efficiency to the ratio of particle diameter to the critical diameter. However, theories of the critical particle type fail to account for turbulence in the cyclone (Leith, Dirgo, and Davis 1986); moreover, a single separation curve is not universally valid.

Later theories attempted to account for turbulence and predict the entire fractional efficiency curve (Dietz 1981; Leith and Licht 1972; Beeckmans 1973). Of these, the theory by Leith and Licht (1972) has been frequently used; however, some of the assumptions on which the theory is based have been invalidated. As with most efficiency theories, the interactions between particles is not taken into account. The assumption of complete turbulent mixing of aerosol particles in any lateral plane has been proven untrue.

Iozia and Leith (1989) used experimental data to develop an equation to predict the cut diameter d50, which is the particle size collected with 50% efficiency. Their empirical equations are as follows:

FIG. 5.16.6 Measured collection efficiency from studies by Barth, Dietz, and Iozia and Leith and modeled collection efficiency by Kessler and Leith.

dS0 l^ZcPpVimax j where:

where Vi is the inlet gas velocity, and zc is the core length which depends on core diameter dc as follows:

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