Settling velocity of tephra particles

Settling of volcanic particles depends on their Terminal Fall Velocity (TFV), which represents the dynamical balance between the gravity force, that accelerates the object downward, and the aerodynamic drag forces, that are opposed to the falling motion. The precise determination of the aerodynamic drag forces requires a detailed parameterization of particle morphology. A widely used shape parameter is particle sphericity (Ψ), defined as the ratio between the surface area of a sphere with the same volume as the particle (equivalent sphere) and the surface area of the particle. Sphericity can be mainly derived based on approximations to simple equivalent geometric shapes, from 2D image analysis of the projected surface of the particles or 3D scan analysis, and from direct measurements of surface area through gas adsorption.

An object which falls through the atmosphere accelerates until it reaches a maximum constant velocity, TFV. Particles with low TFV values can stay suspended in the atmosphere for longer times, and thus can travel greater distances, than particles with high TFV. TFV is defined by the so-called Impact Law:

TFV

where g is the gravitational acceleration, d is the diameter of the object, ρs and ρ are the density of the object and the density of the surrounding fluid, and CD is the drag coefficient. The drag coefficient is a dimensionless number, which depends on particle shape and on the flow regime of the fluid around the particle. The flow regime is related to the particle Reynolds number (Re), which is the ratio between inertial resistance of the particle and viscous resistance of the surrounding fluid. High particle Reynolds numbers (500) are associated with turbulent settling regime, whereas low particle Reynolds numbers (0.4) are associated with laminar settling regime. For intermediate values of particle Reynolds number, the flow is transitional and changes progressively from laminar to turbulent (intermediate settling regime).

The influence of particle morphology on TFV depends on particle size. In particular, the discrepancy between TFV calculated for irregular particles and TFV calculated for the equivalent sphere increases with grain size. The error associated with the assumption of a spherical shape is low (< 10 %) for small particles (φ ≥ 3), but it increases progressively up to 50 % for coarser grain size classes (φ < 3). In addition, the variation of Δ TFV % with grainsize is also related to flow-regime transition. In fact, the range of Reynolds number of the particles for the size and TFV range in which the trending points produce a break-in-slope are roughly coincident with the values of Reynolds number in which the flow regime pass from laminar, to intermediate to turbulent.

Difference ( %) between particle TFV, calculated using the model of Ganser (1993) and the sphericity of Riley et al. (2003), of the samples FL1, KMU and SHV and the TFV of the equivalent sphere plotted vs particle grain size showing the influence % of the shape on the calculated TFV. Red areas A and B indicate the break-in-slope of the trend of the points.
Difference (Δ %) between particle TFV and the TFV of the equivalent sphere plotted vs particle grain size showing the influence % of the shape on the calculated TFV. Red areas A and B indicate the break-in-slope of the trend of the points.

Further readings:

ALFANO F., Bonadonna C., Delmelle P., Costantini L. (2011) Insights on settling velocity from morphological observations. Journal of Volcanology and Geothermal Research, 208(3-4), 86-98. doi:10.1016/j.jvolgeores.2011.09.013. [link]

 

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