Stokes regime

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 ... 

Sedimentation of particles follows the principle outlined above [Eq. (1)] in which particles in the Stokes regime of flow have attained terminal settling velocity. In the airways this phenomenon occurs under the influence of gravity. The angle of inclination, t /, of the tube of radius R, on which particles might impact, must be considered in any theoretical assessment of sedimentation [14,19]. Landahl s expression for the probability, S, of deposition by sedimentation took the form ... 

The foregoing expressions give the suspension velocity (Fs) relative to the single particle free settling velocity, V0, i.e., the Stokes velocity. However, it is not necessary that the particle settling conditions correspond to the Stokes regime to use these equations. As shown in Chapter 11, the Dallavalle equation can be used to calculate the single particle terminal velocity V0... 

This is particularly useful because equation 9.7 for the Stokes regime can be extended to Re - 2 with negligible error. 

Time for a solid spherical particle to reach 99 per cent of its terminal velocity when falling from rest in the Stokes regime... 

Consider a spherical particle of diameter dp and density pp falling from rest in a stationary fluid of density p and dynamic viscosity p.. The particle will accelerate until it reaches its terminal velocity a,. At any time t, let a be the particle s velocity. Recalling that the drag force acting on a sphere in the Stokes regime is of magnitude iirdppu, application of Newton s second law of motion can be written as... 

Stokes Regime (Small Particles). At the time when a particle, originally of size Rq, has shrunk to size R, we may write... 

This relationship of size versus time for shrinking particles in the Stokes regime is shown in Figs. 25.9 and 25.10, pp. 582 and 583, and it well represents small burning solid particles and small burning liquid droplets. 

Particles of constant size Gas film diffusion controls, Eq. 11 Chemical reaction controls, Eq. 23 Ash layer diffusion controls, Eq. 18 Shrinking particles Stokes regime, Eq. 30 Large, turbulent regime, Eq. 31 Reaction controls, Eq. 23... 

Substituting, respectively, the expressions for the drag coefficient, CD, yields the relationships for the corresponding flow regimes. For example, for the Stokes regime, substituting Eq. (2.6) into Eq. (2.14) results in... 

As an example, let us now examine the specific case of particle motion at 25°C and atmosphere pressure. The related physical properties are dp = 0.001 m, pp = 1000 kg-nf p, = 1.145 kg-in. and pg = 1.798 Pa s. The calculated values for the terminal velocity and the operational condition ranges are given in Table 2.2. An important conclusion that can be drawn from the data listed in the fourth column of Table 2.2 is that the Stokes regime cannot exist in co-axial horizontal impinging streams while other regimes are applicable in this kind of impinging stream. 

In gas-solid flows well beyond the Stokes regime, the effect of convective acceleration of the gas surrounding the particle is important. To incorporate this effect into the preceding formulation, modifications of the expressions for the Stokes drag, carried mass, and Basset force in the BBO equation are necessary [Odar and Hamilton, 1964]. The modified BBO equation takes the form [Hansell et al., 1992]... 

In actual applications, the gas flow in a gravity settler is often nonuniform and turbulent the particles are polydispersed and the flow is beyond the Stokes regime. In this case, the particle settling behavior and hence the collection efficiency can be described by using the basic equations introduced in Chapter 5, which need to be solved numerically. One common approach is to use the Eulerian method to represent the gas flow and the Lagrangian method to characterize the particle trajectories. The random variations in the gas velocity due to turbulent fluctuations and the initial entering locations and sizes of the particles can be accounted for by using the Monte Carlo simulation. Examples of this approach were provided by Theodore and Buonicore (1976). 

Pettyjohn and Christiansen [24] restricted their calculations to the Stokes regime (Re < 0.25) and defined a Stokes shape factor (/(,) from modification to equations (6.4) and (6.5) so that ... 

At the lower temperature (783 K open symbols in Fig. 70) a substantially different behavior is observed. The imide band (A in Fig. 69 bottom) decreases quasi-linearly with the elapsed time (see Eq. 24). The aromatic band (V in Fig. 70 top) is complex, revealing two distinct decomposition patterns. At the beginning (first half) of the normalized time a slow linear decrease is observed, followed by a fast decrease. The decrease of the imide band and the change of the aromatic band in the second part of the curve are typical for a film diffusion-controlled reaction of shrinking particles in a gas flow in the Stokes regime. To confirm this observation a new mathematical model is used to fit the curves [321]. Starting from Eq. 20, the reaction velocity ks is substituted with kg=D Rf1 [321]. D is the diffusion velocity and kg the mass transfer coefficient between fluid and particle. The differential equation is solved and the time necessary to reduce a particle from a starting radius R0 to Rt is obtained [see Eq. (22)] [321],... 

The data points (open symbols) in Fig. 70 were fitted using Eq. 24 with the exponent as variable. The obtained values, around 1.5 0.1, confirm the hypothesis of the film diffusion-controlled reaction of shrinking particles in the Stokes regime. 

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