Slip factor correction

The slip correction factors are important for particles smaller than 1 pm in diameter, which is rarely the case for pharmaceutical aerosols. Slip correction is required for the Stokes equation to remain predictive of particle behavior for these small particles. Therefore, assuming the absence of shape effects for particles in the Stokes regime of flow, Eq. (1) collapses into the following expression ... 

Drag on a particle at nonnegligible Kn, but low Ma and Re, is conveniently expressed by the slip correction factor ... 

For nonspherical particles, values for the slip correction factor are available in slip flow (MU) and free-molecule flow (Dl). To cover the whole range of Kn and arbitrary body shapes, it is common practice to apply Eq. (10-58) for nonspherical particles. The familiar problem then arises of selecting a dimension to characterize the particle. Some workers [e.g. (H2, P14)] have used the diameter of the volume-equivalent sphere this procedure may give reasonable estimates for particles only slightly removed from spherical, or in near-con-tinuum flow, but gives the wrong limit at high Kn. An alternative approach... 

Figure 9.15 shows the Cunningham slip correction factors for air at 1 atm pressure and 20°C for the smallest particles, there is a significant correction to the speeds calculated using Eq. (O). 

Smoluchowski s equation was extended up to Kn < 0.25, by introducing a slip correction factor C(Kri) in the expression for the diffusion coefficient. The coagulation coefficient is given for Kn < 0.25 by the expression... 

The lower bound for the coagulation coefficient, ft, is calculated from Eq. [81] using the values of the constant A determined from the numerical solution of Eqs. [74], [77], and [78]. The upper bound for the coagulation coefficient, ft, is obtained from Eq. [97]. In the above calculations, the Philips slip correction factor (Eq. [6]), is used for the calculation of the diffusion coefficient of particles. The results are expressed in terms of the dimensionless coagulation coefficient y defined as the ratio between the coagulation coefficient and the Smoluchowski coagulation coefficient... 

Table 1 compares the dimensionless coagulation coefficient predicted by the present model with other models. Since the Hamaker constant for most of the aerosol systems is of the order of 10"12 eig, this value is used in the calculation of the lower bound. Particle diffusion coefficients based on Philips slip correction factor for an accommodation coefficient of unity are used for the calculation of the coagulation coefficients ft (the Fuchs interpolation formula) and fts (the Sitarski... 

It was assumed that the motion of the fictitious particle within the above time steps is rectilinear. This simplification, which accelerates the computer calculation of the trajectories of the fictitious particle, has been show n to be justified (9). The Philips slip correction factor for an accommodation coefficient of unity (Eq. [4]) was used in the calculation of the diffusion coefficients of particles. The values of the dimensionless coagulation coefficients % obtained by the computer simulation for different particle sizes, are given in Table I. The statistical errors of the Monte Carlo simulation were estimated by the standard 3 a method (corresponding to a probability of 0.997) (13). The number of particle pairs that must be generated in order to lower the error to a reasonable level depends both on the initial distance of separation between... 

Show, using Stokes law and the slip correction factor, that for very small particles the resisting force is proportional to d2. 

To determine the thermophoretic velocity, Stokes law can be utilized by assuming that the Cunningham or slip correction factor (Eq. 5.3) is applicable for cases where Kn > 1. Thermophoretic velocity will be independent of particle diameter since Cc Kn(A + Q) when Kn > 1. Then, equating the thermal force (Eq. 11.8) with the resisting force (Stokes law) and solving for the thermophoretic velocity vT give (Talbot et al., 1980)... 

Figure 18. Cunningham slip correction factors (Q) for spherical particles in air as calculated from Equation (19) at 298 K and 1 atm.

When the particles become very small so that using the Stokes picture of an object moving in a continuous medium is no longer permissible, the so-called slip correction factor, Cc, must be applied [22], The particle can slip through the medium, and, consequently, its velocity is greater than that predicted by the previous equation. 

Here cc is the Cunningham slip correction factor given in Table 2.2. It should be included when rp is comparable with the mean free path of the gas molecules. 

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