# Dust Explosions in Spherical Vessels: The Role of Flame Thickness in the Validity of the ‘Cube-Root Law’ – Summary

1-Sentence-Summary: A three stage model for propagation of thick dust flames is developed and illustrates several important aspects of explosion scaling from small experimental chambers.

Authors: A. Dahoe, J. Zevenbergen, S. Lemkowits, and B. Scarlett

Favorite quote from the paper:

The current authors develop a mathematical model for thick flame propagation in spherical chambers. This is of particular importance for dust explosion where the flame thicknesses approaches the vessel radius, invalidating the thin flame models typically used to scale experimental data.

The typical thickness of a gas flame is on the order of millimeters, while dust flames can range from 15 to 80 centimeters. This causes the flame front to reach the wall of the vessel before the fuel is fully consumed, and changes the shape of the pressure-time profile. This also typically lowers the maximum rate of pressure rise.

The model developed in this work takes flame thickness into account by reconfiguring the traditional modeling approach used for gas flames (e.g., See the textbook Combustion, Flames, and Explosion of Gases by Lewis and von Elbe). In the current model the flame is broken into three zones: unburnt fuel ahead of the flame, a flame of finite thickness where the fuel is consumed linearly to completion, and burnt fuel behind the flame. This is in contrast to the traditional approach for gases that assumes the flame is infinity thin.

The current authors propose that the thin flame approach is only applicable when the flame thickness is below 1% of the vessel radius, which is typically not the case for dust explosion. The assumptions used to develop the three zone model include: spherical flame surface with center ignition, constant specific heat, constant turbulence level, and idealized compression ahead of the flame.

The results produced by the model can broadly be broken into two groups: the case where the flame thickness is less than the vessel radius, and the case where the flame thickness is larger than the vessel radius. In the paper pressure-time profiles are simulated in spherical vessels of the sizes typically used in industry. The effect of the flame thickness on explosion scaling is also investigated.

Three of the main findings from using the model are:

1. The deflagration index (KSt, bar-m/s) is increasingly underpredicted as the flame thickness increases with respect to the vessel radius.
2. A unique curve can be identified for deflagration index verses normalized flame thickness for vessels ranging from 20-L to 10 m3.
3. The current model can be used to predict flame thickness and nominal flame propagation rates for dust explosions, as well as pressure-time traces in any size vessel from experimental data at one size.

The following sections outline the main findings in more detail. The interested reader is encouraged to view the complete article at the link provided below.

## Finding #1: Deflagration index is underpredicted for thick dust flames

Results from the mathematical model demonstrate that as the flame thickness increases with respect to the vessel radius, the maximum rate of pressure rise occurs earlier in the pressure-time profile. In addition, the deflagration index decreases systematically with an increase in the normalized flame thickness (flame thickness divided by vessel radius). The decrease in deflagration index is largest for the 20-L chamber, which was the smallest vessel simulated. For a flame thickness of 10 cm the authors propose that the deflagration index is half of that produced in the 1 m3 chamber, and a third of that produced in a 10 m3 chamber at the same turbulence level. Note that the turbulence levels under standard testing conditions may not be the same in each chamber, and this may partially adjust the deflagration indices.

## Finding #2: Deflagration index between vessels can be scaled by normalized flame radius

The authors demonstrate that the decreasing deflagration indices produced in different vessels collapse onto a single unique curve, when they are plotted as a function of the normalized flame radius (the actual flame radius divided by the vessel radius). The dependence between deflagration index and normalized flame thickness also appears to be log-linear in a certain operational range. These findings may be useful to develop alternative scaling laws for laboratory dust explosion testing to industrial geometries.

## Finding #3: Flame thickness and nominal flame propagation rate can be determined by fitting the model to experimental tests

The difficulty with using the scaling concepts developed in this paper is that the flame propagation rate and therefore flame thickness for dusts is not typically known. The authors give an example of how the model can effectively be used in reverse to predict these values from a given set of experimental data. They use data from cornstarch explosion experiments to compute a flame thickness of 4.1 cm and nominal flame propagation rate of 3.06 m/s, which compare well with experimental data published in the literature. These values can then be used to predict pressure-time traces in different size vessels.

## My Personal Take-Aways From “Dust Explosions in Spherical Vessels: The Role of Flame Thickness in the Validity of the ‘Cube-Root Law’”

This paper gives an excellent review of the assumptions in the typical ‘cubed-root law’ and demonstrates the issues that may occur for dust explosion scaling. The model developed is useful to both illustrate features of thick flames in small vessels, and as a starting point to develop better scaling laws. A difficulty in changing the scaling procedure is that the turbulence level in the standard chambers (20-L and 1m3) has historically been adjusted in-part to account for issues such as flame thickness. Providing an alternative scaling approach also requires understanding how the historic data can be used moving forward.

This article builds on previous work such as Bradley and Mitcheson, 1976 and Nomura and Tanaka, 1992 to develop a theoretical model for dust explosion. It would also be interesting to compare the current model to the work of Singh, 1988, to see how the explosion process for thick flames may change in different geometry vessels and ignition point locations.

Full Citation:

• A. E. Dahoe, J. F. Zavenbergen, S. M. Lemkowitz, and B. Scarlett, “Dust explosion in spherical vessels: the role of flame thickness in the validity of the `cube-root law’.,” Journal of loss prevention in the process industries, vol. 9, pp. 33-44, 1996.
[Bibtex]
``````@ARTICLE{Dahoe1996,
title={Dust explosion in spherical vessels: the role of flame thickness in the validity of the `cube-root law'.},
author={Dahoe, A.E. and Zavenbergen, J.F. and Lemkowitz, S.M. and Scarlett, B.},
journal={Journal of Loss Prevention in the Process Industries},
volume={9},
pages={33--44},
year={1996},