Background

In a previous blog post, we discussed how particle size influences the time required for a particle to settle out of suspension within a vessel. The same post described that small particles will be especially susceptible to remaining airborne due to air currents within the vessel, leading to an environment that is enriched by small, reactive particles.

The threshold for what determines when a particle is "small" can be defined in terms of the dimensionless Stokes Number, which characterizes the relative characteristic times for the particle and fluid:

\[St=\frac{\text{Particle Characteristic Time}}{\text{Flow Characteristic Time}} = \frac{\tau_p}{\tau_f}\]

Particles with low Stokes numbers will be more susceptible to remaining airborne. Literature studies indicate that for particles with a Stokes Number of \(St \le 0.003\), particle velocities become entirely dependent on the flow velocity, following perfectly with the velocity of the local flow. Particles with a Stokes Number of \(St \ge 25 \) exhibit "ballistic" behavior, where their velocity is nearly-independent of the carrying gas phase.

An ability to estimate the airborne dust concentration and how it's distributed by diameter would be highly valuable for assessing the explosion risk in a vessel and for designing explosion protection systems. Currently, NFPA 652 requires that a "representative fine fraction" be tested for explosibility for systems or processes where the bulk material does not remain well mixed. Organizations rarely have data quantifying the representative fine fraction and simply follow the recommendation of ASTM E1226 (and others) to test the material with 95% of the sample having a diameter of 75 microns or less. This will be warranted in some cases, but it likely results in many occurrences of over-designed explosion protection systems. The conceptual framework described in this post is motivated toward building the analysis tools necessary for addressing this issue.

Model Development

The analysis here is organized around defining the relevant particle and flow characteristic times within a vessel so the distribution of Stokes Numbers can be estimated. For a particle-laden, turbulent jet, the flow characteristic time is generally calculated from the inlet diameter (\(D_i\)) and carrying gas velocity (\(v_f\)) as \(\tau_f=D_i/v_f\). However, our interest is knowing whether a particle will follow the bulk gas flow in the silo or if it will deviate on its own trajectory based on its momentum or inertia, making it more likely to settle into a pile at the bottom of the vessel. A more appropriate flow characteristic time is based on the time required for the incoming flow to decelerate from its inlet velocity (e.g., 20 m/s used for these calculations) to a magnitude which is on par with recirculation velocities in the vessel. As the flow moves further downstream from the inlet, the velocity will decay and the role of \(D_i\) in determining these recirculation velocities will be negligible.

To facilitate the calculation of the flow characteristic time, we need a model for the flow dynamics in the vessel. We will use an admittedly simple model (this is just a blog post -- not a rigorous engineering analysis). The purpose here is to illustrate a conceptual model that could be improved upon to yield a useful engineering tool.

The model is based on two distinct zones in the vessel: (1) an expanding turbulent jet which is actively swept and (2) an "unswept" region which is dominated by low velocity, recirculation effects. Each zone is illustrated in the 2D schematic below. The dynamics of the swept volume (i.e., the jet region) will be estimated using equations from Pope's turbulence textbook, which provide expressions for jet velocity profiles and jet spreading angles. The behavior of the zones will be linked through some simplifying assumptions:

• The volumetric flow rate through the swept volume is equal to the volumetric flow rate through the unswept volume.
• The swept volume terminates when its average velocity decays to be equivalent to the velocity in the unswept volume.
• No flow enters or leaves the jet except through the inlet at the top of the vessel and the terminal surface, as depicted in the schematic.

Use of these assumptions permits calculation of the jet length as: \[x = \frac{5}{\sqrt{2}} \Bigg(\frac{4V_{\text{silo}}}{\pi \frac{L}{D}} \Bigg)^{1/3} \]

This jet length represents the distance over which the inlet jet decays to the residual velocity in the vessel. By using the expression for the average jet velocity, \(\bar{u}_j = \frac{5D_i}{2x}U\), we may integrate along the path of the decelerating jet to obtain the characteristic flow time as:

\[\tau_f = \frac{2x}{5 D_i U} \int_0^x{xdx} = \frac{x^2}{5 D_i U} \]

Arguments for the form of the particle characteristic time are more complicated. We'll simply use the final result of \(\tau_p = \frac{\rho_p d_p^2}{18\mu_f}\) and forego the derivation. By combining the definitions for the characteristic times, we obtain:

\[St = \frac{\tau_p}{\tau_f} = \frac{\rho_p d_p^2 / 18 \mu_f}{x^2 / 5 D_i U} \]

From our equations, it's evident that once we specify the volume of the silo and its \(L/D\) ratio, we may calculate the jet length, \(x\). Subsequent use of the particle characteristics of density and particle size will enable calculation of the Stokes Number.

Example Analysis

For illustrative analysis, let's consider pneumatically conveying powder into a silo. The particle size distribution is given in the plot below and the particle density is 1500 kg/m³ (similar to wheat flour). The conveying medium is air for which we'll use an absolute viscosity of \(\mu_f = 1.81 \times 10^{-5} \frac{\text{kg}}{\text{m} \cdot \text{s}}\). The analysis considers silo volumes from 10 - 100 m³, with a fixed \(L/D\) of 3.8 and an inlet diameter of \(D_i = 6 \text{ in.}\). The silos are assumed to be perfectly cylindrical.

The results of the Stokes Number calculations are shown in the following contour plot, organized by particle diameter and silo volume. It's noteworthy that while none of the Stokes Numbers approach the ballistic range, they do vary over two orders of magnitude even for a given silo volume. For example, considering the 50 m³ silo case, the Stokes Number increases from approximately 0.001 to 0.21 as the particle diameter increases from 25 microns to 500 microns.

If we assume that the airborne concentration for a given particle diameter is inversely proportional to its Stokes Number, we can use the results to estimate the composition of the dust that is present in the airborne phase. The results of the calculation are shown in the following figures, where the PSDs and cumulative distributions for the raw and airborne samples are reported. For a given diameter (x-axis), the cumulative distributions characterize the percentage of the sample having a size that is less than or equal to the given diameter.

The results indicate that the variation in the Stokes Numbers leads to a substantial difference in the particle size distribution between the raw material and what is retained in the airborne phase. As a point of comparison, we may consider the fraction of "fines" in the sample, corresponding to a particle size less than 75 microns. Only 0.4% of the raw sample is made up of fines, while 6.1% of the airborne phase is estimated to be fines. This is a substantial, 15-fold increase, but it's still far less than the 95% recommended by the ASTM standards for testing the sample. This preliminary analysis is not meant to contradict the ASTM recommendation, especially because we're not accounting for many of the complex dynamics which can lead to further concentration of small particles. However, it is meant to justify further analysis on the topic so more accurate and customized solutions could be provided.

The plot of the PSDs also includes vertical lines marking the Sauter mean diameter (SMD, D_3,2). There are many different "mean" diameters we could use to characterize the PSD, but a 2014 study published by Castellanos et al. indicated that the SMD yields the best correlation with the dust deflagration index (K_St). The SMD corresponds to the diameter of a particle that has the same surface area-to-volume ratio as the overall particle sample.

Wrap Up

The analysis here provides a cursory look at how the silo filling process yields an airborne PSD which is distinct from the raw sample PSD. With better analysis tools, organizations can be equipped to estimate the nature of the reactive mixture in an enclosed vessel, yielding an improved basis for designing explosion protection and isolation systems.

A reliable and complete analysis would need to address the complex, transient flow dynamics in the vessel and the influence of preferential concentration, which will lead to spatially inhomogeneous particle size distributions due to interactions between particle size and varying turbulence length scales. Experimental validation will also be an important step toward applying the models.

If your organization is interested in applying the type of analysis described here or in helping to grow our understanding of silo filling dynamics, please contact us!