DOI: 10.3303/CET2290082 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 13 December 2021; Revised: 4 March 2022; Accepted: 30 April 2022 
Please cite this article as: Scotton M.S., Barozzi M., Derudi M., Copelli S., 2022, A Mathematical Model for the Prediction of the Kst for Metallic 
Dusts as a Function of the Particle Size Distribution, Chemical Engineering Transactions, 90, 487-492  DOI:10.3303/CET2290082 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 90, 2022 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Aleš Bernatík, Bruno Fabiano 
Copyright © 2022, AIDIC Servizi S.r.l. 
ISBN 978-88-95608-88-4; ISSN 2283-9216 

A Mathematical Model for the Prediction of the KSt for Metallic 
Dusts as a Function of the Particle Size Distribution 

Martina Silvia Scottona, Marco Barozzia, Marco Derudib, Sabrina Copellia,* 
a University of Insubria, Department of Science and High Technology, Via Valleggio 9, 22100, Como, Italy 
b Polytechnic of Milan, Department of Chemistry, Materials and Chemical Engineering,  
sabrina.copelli@uninsubria.it 

For several years, dust explosions have been one of the major causes of industrial accidents, spanning from 
metalworking to pharmaceuticals sectors. In accordance with the latest Chemical Safety Board (CSB) 
investigations, three out of four dust explosions in the United States involved metallic dusts (iron, titanium, 
zirconium and aluminum). Many chemical processes involve metal powders for their exceptional mechanical, 
optical and catalytic properties, such as the production of plastics, rubber, paints, coatings, inks, pesticides, 
detergents and even drugs. The severity of a dust explosion can be defined using experimental parameters 
such as the maximum explosion pressure (𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚), the maximum rate of pressure rise ((𝑑𝑑𝑃𝑃 𝑑𝑑𝑑𝑑⁄ )𝑚𝑚𝑚𝑚𝑚𝑚) and the 
deflagration index (𝐾𝐾𝑆𝑆𝑆𝑆), which are employed to predict the consequences of a dust explosion for a given 
scenario. Among these parameters, the deflagration index plays a fundamental role, as it is used for the design 
of deflagration nozzles aimed to protect industrial equipment and silos from internal dust explosions. The 
purpose of this work is to develop a mathematical model able to predict the 𝐾𝐾𝑆𝑆𝑆𝑆 value of metal powders as a 
function of chemical-physical data and the particle size distribution (𝐷𝐷50 was used as global information). The 
model structure is based on the writing and resolution of the material and energy balance equations on the 
single dust particle, also estimating the contribution of oxygen diffusion which, in the case of metal powders, 
greatly depends on both tortuosity and porosity. The results well agreed with experimental data, providing the 
basis for the development of more detailed models. 

1. Introduction 
The value of the deflagration index can be easily calculated through a series of tests carried out in the standard 
20-L apparatus. However, considering the intrinsic limits of this approach, the possibility of estimating the 𝐾𝐾𝑆𝑆𝑆𝑆 
value through a suitable mathematical model able to overcome at least some of the limits of the experimental 
approach has been studied in the literature. Indeed, several mathematical models have been proposed to 
predict the value of KSt and other explosion parameters, such as MIE, for organic combustible dusts. Fumagalli 
et al. (2017) have proposed a simple kinetic free model which used a few information about heat transfer 
properties of the analyzed dust and a single value of the deflagration index at a reference mean diameter for 
the evaluation of the deflagration index trend as a function of the mean diameter of the particle. Copelli et. al. 
(2021) have proposed a mathematical model able to theoretically estimate the MIE value of organic powders 
using very few easily accessible experimental information, such as granulometric and thermogravimetric 
analysis (TGA). Therefore, the aim of this work is to develop and validate a mathematical model (based on 
oxidation reaction kinetics) for the prediction of the 𝐾𝐾𝑆𝑆𝑆𝑆 value for metallic dusts as a function of their mean particle 
size. Several research have shown that metal dust explosions can be strongly conditioned by both mean 
diameter and size polydispersity (σD) of the dust (Castellanos et al., 2014). The present work considers only the 
first aspect (that is, the mean diameter), eliminating the polydispersity influence by considering highly peaked 
granulometric distributions. The model predicts the Kst value by solving material and energy balance equations 
for spherical dust particles wrapped by a single constant temperature fireball (generated by the ignitors 
explosion inside the 20 L sphere). The model was validated using literature experimental data for aluminum 
dust, a well-known explosible metal dust. 

487



2. Mathematical Model 
For a metal dust explosion, the oxidation mechanism has the greatest impact on the violence of the explosion 
itself and depends on both the particle temperature and the heterogeneous kinetics (Dufaud et al., 2010). Two 
different processes are in competition: oxygen diffusion through the oxide/metal layer (variable thickness) within 
the particle pores and heterogeneous oxidation kinetics. For big dust particles, oxygen diffusion through the 
pores can be considered as the rate determining step for metal dust explosion and, therefore, it is the main 
contributor to the deflagration index value. On the contrary, for very small dust particles, the diffusion is very 
fast, and the oxidation kinetics must be properly taken into account. The stoichiometry of the complete oxidation 
reaction, starting from pure metal species, M, is schematically described through this equation: 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑂𝑂2 ⇒ 
𝑎𝑎𝑚𝑚𝑂𝑂2𝑏𝑏. 
The explosion model is developed under the following assumptions: (1) one-dimensional, spherically symmetric 
dust particle whose volume element is equal to 𝜕𝜕𝜕𝜕 = 𝑟𝑟2 ⋅ 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 ⋅ 𝜕𝜕𝑟𝑟 ⋅ 𝜕𝜕𝜃𝜃 ⋅ 𝜕𝜕𝜕𝜕, and surface element is equal to 
𝜕𝜕𝜕𝜕 = 𝑟𝑟2 ⋅ 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 ⋅ 𝜕𝜕𝜃𝜃 ⋅ 𝜕𝜕𝜕𝜕;  (2) negligible resistance to mass transfer; (3) no secondary reactions of the oxides 
produced; (4) local thermal equilibrium; (5) unsteady-state assumption for the gaseous phase and negligible 
convective fluxes (only diffusive fluxes are taken into account in the case of metal powders); (6) particles with 
constant total volume, 𝜕𝜕𝑆𝑆 (presence of a metallic skeleton with a constant density 𝜌𝜌𝑀𝑀, non-constant porosity 𝜀𝜀 
because of the reduction of the pores diameter due to the formation of oxide layers onto the metallic skeleton - 
conversely to what happens for organic particles for which the porosity increases). Basing on the last assumption 
(6), it is necessary to write material balances for both the metallic phase (M) and the gaseous oxygen; obviously 
a certain amount of metallic particle could not react with oxygen. 

2.1 Material balance equations on the metallic phase 

The material balance equation on the metallic phase can be written as: 

𝜕𝜕𝑠𝑠𝑀𝑀
𝜕𝜕𝑑𝑑

= −𝑎𝑎 ∙ 𝑘𝑘 ⋅ 𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎 ∙ 𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎 ⋅ 𝜕𝜕𝑆𝑆     𝑤𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒:
𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎 =

𝑠𝑠𝑀𝑀
𝜕𝜕𝑆𝑆

𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎 =
𝑠𝑠𝑂𝑂2
𝜕𝜕𝑆𝑆

 
 

(1) 

where the apparent concentrations of metal and oxygen (𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎 = 𝑠𝑠𝑀𝑀 𝜕𝜕𝑆𝑆⁄  and 𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎 = 𝑠𝑠𝑂𝑂2 𝜕𝜕𝑆𝑆⁄ , respectively) 
were introduced referring to the total volume of the particle (i.e., including empty and full spaces). 
Considering assumption (6), Eq.(1) can be re-written as: 

�

𝜕𝜕𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎
𝜕𝜕𝑑𝑑

= −𝑎𝑎 ∙ 𝑘𝑘 ⋅ 𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎 ∙ 𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎

𝐼𝐼. 𝐶𝐶.    𝑑𝑑 = 0      𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎,0 =
𝜌𝜌𝑀𝑀
𝑃𝑃𝑎𝑎𝑀𝑀

∙ (1 − 𝜀𝜀0) 
 (2) 

where 𝜀𝜀 = (𝜕𝜕𝑆𝑆 − 𝜕𝜕𝑆𝑆) 𝜕𝜕𝑆𝑆⁄  is the porosity of the dust particle (variable during the process), 𝜀𝜀0 is the initial porosity 
of the dust particle (which is constant), and 𝜕𝜕𝑆𝑆 is the volume occupied by the metal/oxides phase. It is possible 
to make dimensionless Eq. (1) by introducing the following dimensionless variables 𝜇𝜇 = 𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎 𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎,0⁄  and 𝜂𝜂 =
𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎 𝑐𝑐𝑂𝑂2,𝑆𝑆⁄ ; where the first is the apparent dimensionless metal concentration and the second is the apparent 
dimensionless oxygen concentration. Finally, the dimensionless material balance on the solid phase results: 

�
𝜕𝜕𝜇𝜇

𝜕𝜕𝑑𝑑
=  −𝛼𝛼 ∙ 𝐶𝐶𝑂𝑂2,𝜕𝜕 ∙ 𝑘𝑘 ∙ 𝜇𝜇 ∙ 𝜂𝜂

𝐼𝐼. 𝐶𝐶.      𝑑𝑑 = 0     𝜇𝜇 = 1
 (3) 

2.2 Material balance equation on oxygen 

In order to write the indefinite material balance equation on the oxygen present in the gas phase of the dust 
particle, it is necessary to define the apparent oxygen concentration as 𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎 = 𝑐𝑐𝑂𝑂2 ⋅ 𝜀𝜀 and, then, it is possible 
to insert such a definition in the complete oxygen balance equation.  
Moreover, considering both under unsteady state conditions and negligible convective fluxes for the gaseous 
phase (hp: (5)), it is possible to obtain: 

488



⎩
⎪
⎪
⎨

⎪
⎪
⎧
𝜕𝜕𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎
𝜕𝜕𝑑𝑑

= ℘ ⋅
𝜕𝜕
𝜕𝜕𝜕𝜕

�
𝜕𝜕𝑐𝑐𝑂𝑂2,𝑎𝑎𝑝𝑝𝑝𝑝
𝜕𝜕𝜕𝜕

� − 𝛽𝛽 ∙ 𝑘𝑘 ⋅ 𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎 ∙ 𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎

𝐼𝐼. 𝐶𝐶.       𝑑𝑑 = 0,          𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎 = 0 

𝐵𝐵. 𝐶𝐶.         𝜕𝜕 = 0,        
𝜕𝜕𝑐𝑐𝑂𝑂2,𝑎𝑎𝑝𝑝𝑝𝑝
𝜕𝜕𝜕𝜕

�
𝑚𝑚=0,𝑆𝑆

= 0 

𝜕𝜕 = 𝑅𝑅,        𝑐𝑐𝑂𝑂2,𝑎𝑎𝑝𝑝𝑝𝑝(𝑑𝑑, 𝜕𝜕 = 𝑅𝑅) = 𝑐𝑐𝑂𝑂2,𝑆𝑆 

 (4) 

Such an equation must be made dimensionless and transferred in a spherical coordinates system before being 
solved with Eq. (3). Finally, it results: 

⎩
⎪⎪
⎨

⎪⎪
⎧𝜕𝜕𝜂𝜂
𝜕𝜕𝑑𝑑

= ℘ ⋅ �
𝜕𝜕2𝜂𝜂
𝜕𝜕𝑟𝑟2

+
2
𝑟𝑟
∙ �
𝜕𝜕𝜂𝜂
𝜕𝜕𝑟𝑟
�� − 𝛽𝛽 ∙ 𝑘𝑘 ⋅ 𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎,0 ∙ 𝜇𝜇 ∙ 𝜂𝜂

𝐼𝐼. 𝐶𝐶.       𝑑𝑑 = 0,          𝜂𝜂 = 0 

𝐵𝐵. 𝐶𝐶.     𝑟𝑟 = 0,   
𝜕𝜕𝜂𝜂
𝜕𝜕𝑟𝑟
�
𝑟𝑟=0,𝑆𝑆

= 0 

𝑟𝑟 = 𝑅𝑅,        𝜂𝜂(𝑑𝑑, 𝑟𝑟 = 𝑅𝑅) = 1 

 (5) 

As it is possible to notice, Eq. (5) considers only the oxygen diffusion within the dust particle and the reactive 
term due to the aluminum oxidation: the convective fluxes were not considered because the high tortuosity of 
the intraparticle environment.  
One of the most important parameters to be determined with a quite high degree of accuracy is the oxygen 
diffusion coefficients within the dust particle, ℘. Such a parameter can be calculated considering the combined 
contribution of molecular and Knudsen diffusion inside the particle.  
Particularly, knowing the specific surface area of the particles, 𝑎𝑎𝑠𝑠, and assuming the presence of a single 
equivalent cylindrical pore of diameter 𝑑𝑑𝑎𝑎, it is possible to write the following system of algebraic equations: 

⎩
⎨

⎧𝑎𝑎𝑠𝑠 =
𝜋𝜋𝑑𝑑𝑎𝑎𝐿𝐿

𝜌𝜌(1 − 𝜀𝜀)𝜕𝜕𝑆𝑆

𝜀𝜀 𝜕𝜕𝑆𝑆 =
𝜋𝜋𝑑𝑑𝑎𝑎2 

4
∙ 𝐿𝐿

 (5b) 

The resulting equivalent diameter of the single pore is: 

𝑑𝑑𝑎𝑎 =
4𝜀𝜀

𝑎𝑎𝑠𝑠 𝜌𝜌(1 − 𝜀𝜀)
 (5c) 

Such a value can be considered representative for the calculation of the Knudsen diffusion of oxygen inside the 
particle. 

2.3 Energy balance equation on the solid particle 

The energy balance equation for a single dust particle is referred to its total volume, 𝜕𝜕𝑆𝑆. Hypothesizing that the 
heat capacity of the solid phase (S) is constant and much greater than that one of the gaseous phase (oxygen), 
it is possible to obtain the general form: 

𝜌𝜌𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎 ⋅ 𝑐𝑐𝑎𝑎,𝑀𝑀
𝜕𝜕𝜕𝜕
𝜕𝜕𝑑𝑑

= −𝛻𝛻 × (ℎ ⋅ �⃗�𝑣 + �⃗�𝑞) + 𝛥𝛥𝐻𝐻𝑜𝑜𝑚𝑚 ⋅ 𝑟𝑟𝑜𝑜𝑚𝑚 (6) 

where: 𝜌𝜌𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎 = 𝜌𝜌𝑀𝑀 ⋅ (1 − 𝜀𝜀̄) is an effective particle density taking into account that the porosity is varying from 
an initial value to a smaller one; ℎ = 𝜌𝜌𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎 ⋅ 𝑐𝑐𝑎𝑎,𝑂𝑂2 ⋅ 𝜕𝜕 is the enthalpy of the volatiles per unit of volume (we refer 
only to the gaseous phase because the solid phase can not be subjected to convective fluxes), J/m3; �⃗�𝑞 = −�̄�𝜆 ⋅
𝛻𝛻𝜕𝜕 is the conductive heat flux, W/m2 (not only associated with temperature gradients); �̄�𝜆 is the effective thermal 
conductivity of the system, W/(m K); 𝛥𝛥𝐻𝐻𝑜𝑜𝑚𝑚 is the exothermic reaction enthalpy for the oxidation reaction, [J/kmol] 
or [J/kg]. Inserting the previous definitions, developing all the terms considering hp (5) and considering constant 
the thermal conductivity, it results (in spherical coordinates): 

489



⎩
⎪
⎪
⎨

⎪
⎪
⎧ 𝜌𝜌𝑀𝑀 ∙ (1 − 𝜀𝜀)̅ ∙ 𝑐𝑐𝑎𝑎,𝑀𝑀 ∙

𝜕𝜕𝜕𝜕
𝜕𝜕𝑑𝑑

= �̅�𝜆
𝜕𝜕2𝜕𝜕
𝜕𝜕𝑑𝑑2

+
2 ∙ �̅�𝜆
𝑟𝑟

∙
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟

+ ∆𝐻𝐻𝑜𝑜𝑚𝑚 ∙ 𝑟𝑟𝑜𝑜𝑚𝑚
𝐼𝐼. 𝐶𝐶.   𝑑𝑑 = 0,    𝜕𝜕 = 𝜕𝜕0

𝐵𝐵. 𝐶𝐶   𝑟𝑟 = 0,     �̅�𝜆 ∙
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
�
𝑟𝑟=0

= 0

𝑟𝑟 = 𝑅𝑅,    �̅�𝜆 ∙
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
�
𝑟𝑟=𝑅𝑅

= −ℎ𝑐𝑐 ∙ (𝜕𝜕|𝑟𝑟=𝑅𝑅 − 𝜕𝜕𝑒𝑒𝑚𝑚𝑆𝑆) − 𝜀𝜀 ∙ 𝜎𝜎(𝜕𝜕4|𝑟𝑟=𝑅𝑅 − 𝜕𝜕𝑒𝑒𝑚𝑚𝑆𝑆4 )

 (6b) 

where �̅�𝜆 = 𝜆𝜆 ∙ (1 − 𝜀𝜀)̅ (since the porosity of the particle varies during time, it was necessary to introduce 𝜀𝜀̄ which 
is a mean porosity). As the porosity tends to decrease during the oxidation process, 𝜀𝜀̄ can be considered equal 
to 0 as a first approximation. Introducing all the variables, the following system is obtained: 

⎩
⎪
⎪
⎨

⎪
⎪
⎧
𝜕𝜕𝜕𝜕
𝜕𝜕𝑑𝑑

=
𝜆𝜆

𝜌𝜌𝑀𝑀 ∙ 𝑐𝑐𝑎𝑎,𝑀𝑀
∙ �
𝜕𝜕2𝜕𝜕
𝜕𝜕𝑟𝑟2

+
2
𝑟𝑟
∙
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
� +

∆𝐻𝐻𝑜𝑜𝑚𝑚
𝜌𝜌𝑀𝑀 ∙ 𝑐𝑐𝑎𝑎,𝑀𝑀 ∙ (1 − 𝜀𝜀)̅

∙ 𝑘𝑘 ∙ 𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎 ∙ 𝑐𝑐𝑂𝑂2,𝑚𝑚𝑎𝑎𝑎𝑎

𝐼𝐼. 𝐶𝐶            𝑑𝑑 = 0,         𝜕𝜕 = 𝜕𝜕0

𝐵𝐵. 𝐶𝐶.   𝑟𝑟 = 0,   𝜆𝜆 ∙
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
�
𝑟𝑟=0

= 0

  𝑟𝑟 = 𝑅𝑅,     𝜆𝜆 ∙
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
�
𝑟𝑟=𝑅𝑅

= −ℎ𝑐𝑐 ∙ (𝜕𝜕|𝑟𝑟=𝑅𝑅 − 𝜕𝜕𝑒𝑒𝑚𝑚𝑆𝑆) − 𝜀𝜀 ∙ 𝜎𝜎 ∙ (𝜕𝜕4|𝑟𝑟=𝑅𝑅 − 𝜕𝜕𝑒𝑒𝑚𝑚𝑆𝑆4 )

 (7) 

And, using the previously introduced dimensionless variables: 

Equations (3), (5) and (8) constitute a system of Partial Differential Equations (PDEs) that must be numerically 
solved applying suitable mathematical methods. In this work, the Method Of Lines (Schiesser, 1991), which 
solve systems of partial differential equations where the derivatives with respect to both time and space are 
present by discretizing the spatial derivatives and solving the resulting differential algebraic equation (DAEs) 
system, was chosen. 

2.4 Deflagration index determination 

As a first approximation, it is possible to assume that all the changes in the explosion violence (that is, in the KSt 
value) observed by varying the particles size distribution can be well represented by the maximum rate of 
temperature rise at the particle boundary.  
Such a rate of temperature rise is influenced by different phenomena: 1) the convective heat flux coming from 
the "fireball" generated by the explosion of the ignitors, which heats the air present into the 20 L sphere; 2) the 
diffusion of oxygen inside the particle; and, 3) the heterogeneous oxidation reaction taking place into the particle 
itself.  
According to a previously proposed correlation to determine the deflagration index of an organic dust as a 
function of the mean particle diameter (Copelli et al., 2019), the correlation developed in the present work uses 
one experimental datum, the KSt of a reference particle size distribution of the analyzed dust, together with the 
maximum rate of temperature rise at the particle boundary for both the reference and the analyzed dust. 
The following correlation can be therefore used: 

𝐾𝐾𝑆𝑆𝑆𝑆(𝐷𝐷𝑚𝑚) =
�𝜕𝜕𝜕𝜕𝜕𝜕𝑑𝑑�𝑚𝑚𝑚𝑚𝑚𝑚
�𝜕𝜕𝜕𝜕𝜕𝜕𝑑𝑑�𝐷𝐷𝑚𝑚

∙ 𝐾𝐾𝑆𝑆𝑆𝑆�𝐷𝐷𝑟𝑟𝑒𝑒𝑟𝑟� (9) 

where 𝐷𝐷𝑟𝑟𝑒𝑒𝑟𝑟 is a reference average diameter, 𝐾𝐾𝑆𝑆𝑆𝑆(𝐷𝐷𝑟𝑟𝑒𝑒𝑟𝑟) is the experimental value of the deflagration index 
measured at that average diameter, (𝜕𝜕𝜕𝜕 𝜕𝜕𝑑𝑑⁄ )𝑚𝑚𝑚𝑚𝑚𝑚 is the derivative of the temperature with respect to time at the 
external boundary of a dust with the granulometric distribution equal to Dref. Likewise, KSt(Dm) is the value of the 
deflagration index predicted for a dust with granulometric distribution equal to Dm and (𝜕𝜕𝜕𝜕 𝜕𝜕𝑑𝑑⁄ )𝐷𝐷𝑚𝑚is the derivative 
of the temperature, with respect to time, at the external boundary of a dust with granulometric distribution equal 
to Dm. It should be noted that this method, conversely to the previously proposed for organic dusts, do not make 
possible to completely ignore the information on the chemical kinetics of the process (the heterogeneous 
oxidation) as they are only partially contained in the experimental value of KSt(Dref).  

⎩
⎪⎪
⎨

⎪⎪
⎧
𝜕𝜕𝜕𝜕
𝜕𝜕𝑑𝑑

=
𝜆𝜆

𝜌𝜌𝑀𝑀𝑐𝑐𝑎𝑎,𝑀𝑀
 ∙ �

𝜕𝜕2𝜕𝜕
𝜕𝜕𝑟𝑟2

+
2
𝑟𝑟
∙
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
� +

∆𝐻𝐻𝑜𝑜𝑚𝑚
𝜌𝜌𝑀𝑀 ∙ 𝐶𝐶𝑃𝑃,𝑀𝑀 ∙ (1 − 𝜀𝜀)̅

𝑘𝑘 ⋅ 𝑐𝑐𝑀𝑀,𝑚𝑚𝑎𝑎𝑎𝑎,0 ∙ 𝑐𝑐𝑂𝑂2,𝑆𝑆 ∙ 𝜇𝜇 ∙ 𝜂𝜂

𝐵𝐵. 𝐶𝐶.          𝑟𝑟 = 0,     𝜆𝜆 ∙
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
�
𝑟𝑟=0

= 0 

 𝑟𝑟 = 𝑅𝑅,      𝜆𝜆 ∙
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
�
𝑟𝑟=𝑅𝑅

= −ℎ𝑐𝑐 ∙ (𝜕𝜕|𝑟𝑟=𝑅𝑅 − 𝜕𝜕𝑒𝑒𝑚𝑚𝑆𝑆)  − 𝜀𝜀 ∙ 𝜎𝜎 ∙ (𝜕𝜕|𝑟𝑟=𝑅𝑅4 − 𝜕𝜕𝑒𝑒𝑚𝑚𝑆𝑆4 )

 (8) 

490



In practice, having the experimental value of KSt of a metallic powder of known particles size distribution is 
possible, using Eq. (9), to predict the value of KSt of the same powder with different particle size distributions, 
using only the estimated rate of the maximum heating rate of the particle at its external boundary for both 
diameters.  

3. Validation and discussion 
The mathematical model here presented was validated considering different aluminum granulometric 
distributions and using 𝐷𝐷50 as global information.  
This metal, generally, possesses particle size distributions highly peaked around the mean value; therefore, 𝐷𝐷50 
can be used to represent the entire particle size distribution with respect to the determination of the deflagration 
index value.  
Six diameters (5, 9, 15, 20, 25 and 30 μm), characterized by σD values of 0.95, 1.17, 1.48, 1.87, and 2.51, were 
tested. For what concerns the last two distributions, it is possible to notice a quite high degree of polydispersity; 
anyway, it was decided to test them in order to evaluate the limitations of the present model. 
It is important to consider that such distributions refer to quite big aluminum particles, therefore the 
corresponding values of the deflagration index will be quite low and it will be quite difficult to be within the 
acceptable ranges. 
In order to determine the maximum rates of temperature increase at the boundary of the particles for both the 
reference and the analyzed dust, Eq. (3), (5) and (8) were solved considering a constant temperature of the gas 
surrounding the dust particles equal to 500 [K], that is the stoichiometric temperature reached by the air in the 
20 L sphere when the ignitors only are burst. 
Such an assumption ca be justified because the heterogenous oxidation of aluminum is very fast and highly 
exothermic and it requires quite low temperatures to be triggered. Moreover, the heat generated by the oxidation 
quickly rises the temperature of the dust particle, also igniting a strong radiating effect towards the other metallic 
dusts: this effect was not considered within this preliminary work. Such a piece of information is considered to 
be summarized in the experimental value of the deflagration index present into Eq. (9). 
The results obtained are shown in the parity plot of Figure 1.  

 

Figure 1: Experimental KSt value vs Simulated KSt value for aluminum dust 

Observing Figure 1, it is possible to notice that all the predicted 𝐾𝐾𝑠𝑠𝑆𝑆 values are quite close to the experimental 
ones and, in all cases but one, fall within the maximum and minimum acceptable limits, independently on the 
occurrence of either overestimation or underestimation of the experimental 𝐾𝐾𝑠𝑠𝑆𝑆.  

491



These inaccuracies in the deflagration index prediction are not a major problem as all the underestimates are 
acceptable because they perfectly meet the experimental errors; while, the overestimations do not represent a 
problem from the safety point of view, in fact they lead to conservative predictions.  
Furthermore, in accordance with all the studies conducted so far, the predicted values also indicate that the 
increase in the particles size leads to a reduction in the explosion severity. 

4. Conclusions 
The aim of this work was to develop a mathematical model able to predict the 𝐾𝐾𝑆𝑆𝑆𝑆 value of metal powders as a 
function of a few chemical-physical data and the particle size distribution (using only 𝐷𝐷50 as global information). 
The validation of the model was done by comparison among the model estimated values of the deflagration 
index and the experimental data available in the literature for aluminum dust, which is one of the most used 
metal powder in the process industries. 
As shown by the results, a good degree of accuracy in the prediction of the deflagration index was reached. The 
comparison among predicted and experimental data demonstrated that the proposed model was able to predict 
with a good accuracy the value of the deflagration index for aluminum powders having different particle sizes 
and distributions.  
This mathematical model can be considered innovative because it is able to estimate the contribution of diffusion 
of oxygen inside the particle pores, which, in case of metal powders, greatly depends on tortuosity and porosity. 
It also considers the kinetics of metal oxidation, and it is able to predict the 𝐾𝐾𝑆𝑆𝑆𝑆 value of a metallic dust with a 
given particle size distribution, regardless of the powder concentration used, which means a substantial 
reduction in time and costs related to experimental testing. 
One of the criticalities of the model is the underprediction of the deflagration index for very small mean particles 
diameters, those ones implying the highest values of the 𝐾𝐾𝑆𝑆𝑆𝑆. 
Anyway, preliminary results showed a good agreement between literature 𝐾𝐾𝑆𝑆𝑆𝑆 values and model predictions, as 
they are always within the acceptability range related to the unavoidable experimental uncertainties.  
Such an achievement, once further validated on a wider range of explosive metal dusts, could lead to an 
improvement in the risk assessment panorama for powder handling operations by making the risk evaluation 
process quicker and cheaper. 

References 

Castellanos, D., Carreto-Vazquez, V.H., Mashuga,C.V., Trottier, R., Mejia, A.F., Mannan, M.S., 2014, The effect 
of particle size polydispersity on the explosibility characteristics of aluminum dust, Powder Technology, 254, 
331–337. 

Copelli, S., Barozzi, M., Scotton, M.S., Fumagalli, A., Derudi, M., Rota, R., 2019, A Predictive Model for the 
Estimation of the Deflagration Index of Organic Dusts, Process safety and environmental protection, 126, 
329–338. 

Copelli, S., Scotton, M.S., Barozzi, M., Derudi, M., Rota, R., 2021, A Practical Tool for Predicting the Minimum 
Ignition Energy of Organic Dusts, Industrial & engineering chemistry research 60 (29), 10807–10813. 

CSB U.S. Chemical Safety and Hazard Investigation Board, 2005, Hayes Lemmerz International Dust Explosion 
and Fire <csb.gov/hayes-lemmerz-dust-explosions-and-fire> accessed 02.02.2022 

CSB U.S. Chemical Safety and Hazard Investigation Board, 2014, AL Solutions Fatal Dust Explosion 
<csb.gov/assets/1/20/final_case__study_7.161.pdf?15288> accessed 02.02.2022 

Dufaud, O., Traoré, M., Perrin, L., Chazelet, S., Thomas, D., 2010, Experimental investigation and modelling of 
aluminum dusts explosions in the 20 L sphere, Journal of Loss Prevention in the Process Industries, 23, 
226–236. 

Fumagalli, A., Derudi, M., Rota, R., Snoeys, J., Copelli, S., 2017, Prediction of KSt reduction with the particle 
diameter increase for organic dust, Journal of Loss Prevention in the Process Industries 50, 67-74. 

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	lp-2022-abstract-218.pdf
	A Mathematical Model for the Prediction of the KSt for Metallic Dusts as a Function of the Particle Size Distribution