Attachment coefficient hybrid theory

The attachment coefficient, 3, corresponding to the hybrid theory can be shown to be (Fuchs, 1964)... 

This approximation may be considered to be an alternative to the hybrid theory. The value of di can be found by equating the attachment coefficients for the diffusion and kinetic theories (d x = 8D/v). 

The average attachment coefficient for the hybrid theory is of the form ... 

Figures 3 and 4 show the variation of the attachment coefficient with count median diameter for the diffusion, kinetic, hybrid and kinetic-diffusion theory for geometric standard deviations of 2 and 3 respectively.

Figure 5 illustrates the effect of the geometric standard deviation on the attachment coefficient using the hybrid theory. 

Figure 5. Attachment coefficient vs CMD using the hybrid theory for various particle distributions.

Figures 3 and 4 show the variation of the average attachment coefficient with CMD. It can be seen that for particles of CMD less than 0.06 ym and Og = 2 the kinetic theory predicts an attachment coefficient similar to the hybrid theory, whereas for CMD greater than about 1 ym the diffusion theory and the hybrid theory give approximately the same results. For a more polydisperse aerosol (Og = 3) the kinetic theory deviates from the hybrid theory even at a CMD = 0.01 ym. The diffusion theory is accurate for a CMD greater than about 0.6 ym. These results are easily explained since as the aerosol becomes more polydisperse, there are more large diameter particles (CMD >0.3 ym) which attach according to the diffusion theory. In contrast, the kinetic theory becomes more inaccurate as the aerosol becomes more polydisperse.

The kinetic-diffusion approximation predicts an attachment coefficient similar to the hybrid theory for all CMDs and for both Og m 2 and 3 (Figs. 3 and 4). The advantage of this theory is that the average attachment coefficient can be calculated from an analytical solution numerical techniques are not required. 

Figure 5 shows the variation of the hybrid theory with CMD for various Og. It is obvious that assuming an aerosol to be mono-disperse when it is in fact polydisperse leads to an underestimation of the attachment coefficient, leading in turn to large errors in calculation of theoretical unattached fraction. 

The variation of attachment coefficient with Og for CMD =0.2 ym and 0.3 ym is shown in Figures 6 and 7. Again it is apparent that the kinetic theory or diffusion theory are correct only at certain CMD and og. Neither is applicable under all circumstances. It is also evident that the kinetic-diffusion theory is a good approximation to the hybrid theory under all circumstances. 

Calculation of the attachment coefficient is required for theoretical prediction of the unattached fraction of radon progeny. The hybrid theory, which is a form of Fuchs theory with certain justifiable assumptions, can be used to describe attachment to aerosols under all conditions of Og and CMD. 

---