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Dust Explosions: Nature and Means to Prevent

2 Dec

The great book of Nature lies ever open before our eyes and the true philosophy is written in it…But we cannot read it unless we have first learned the language and the characters in which it is written…It is written in mathematical language and the characters are triangles, circles and other geometric figures.-Galileo Galilei, Florence 1623.

They that know the entire course of the development of science,  will, as a matter of course, judge more freely and more correctly of  the significance of any present scientific movement than they, who,  limited in their views to the age in which their own lives have been  spent, contemplate merely the momentary trend that the course of  intellectual events takes at the present moment.
-Ernst Mach, German Charles University, Prague 1883

Ignition is the process by which it occurs the propagation of a self-sustained combustion. An explosion (meaning a “sudden outburst”) is an exothermal process (i.e., liberate energy) that gives rise to a sudden increase of pressure when occuring at constant volume. It is accompanied by noise and a sudden release of a blast wave. An explosion may occurs in gases, dust or with solid explosives. In quite general terms, any kind of solid material that burn in air (or in oxygen rich environment) will burn more fastly with decreasing graininess. By order of importance we may say that firewood burn slowly, burns better when cut in smaller pieces, and burns fastly when cut in small particles.

Dust explosion in industry. Im credit:

We are concerned here with dust explosions, due to its importance not only industrially, but also in households and schools. The formation and ignition of dust clouds in industrial and agricultural environment is associated to extremely violent and damaging explosions, particularly in coal dust explosions in mines and dust explosions in grain elevators.

The dimensionless number that characterizes the possibility of a system to ignite is called the Darmkhöler number, D_a, which represents the ratio of the rate of heat production within the system due to exothermic chemical reactions to the rate of heat loss due conduction, convection or radiation.

A convenient measure of dust fineness is the specific surface area given by

if we assume the dust made of single cubes of edge length x [1]. The maximum rate of preassure increase in closed-bomb experiments gives a measure of the expected violence form an explosion of a dust cloud. There is a linear dependency between the time rate of preassure increase and the specific surface area, as shown in the Fig. below (extracted from Ref.[1]).

Not all materials can cause dust explosions. For example, silicates, sulphates, nitrates, carbonates and phosphates, and in general terms stable oxides. But the contrary, materials highly explosives are the following: natural organic materials (e.g., grain, linen, sugar); synthetic organic materials (e.g., plastics, organic pigments, pesticides); coal and peat; metals (e.g., aluminum, magnesium, zinc, iron). An example of a highly explosive bomb is the thermobaric weapon that works consuming the surrounding oxygen instead of the oxidizer-fuel mixture, and for that reason it is more destructive than usual bombs. One important parameter that determines the amount of heat liberated in a explosion is called the heat of combustion (see Table 1).

From the above Table 1, we may infer that Calcium (Ca) liberates more heat and coal contributes just one third of the former.

What is the concentrations of dust that may represent danger? If we denote by x the typical radius of a dust particle of density rho_p, and by L^3 the volume of the dust cloud, then we have

Experiments have determined the range of explosible dust concentrations in air at PTN conditions (that is, at normal temperature and atmospheric pressure) for natural organic dust, see Fig. below. The dust particles may fluctuate in a given container due to repulsive electrostatic forces exerted between them.

The last equation permits to estimate the dust concentration for which an explosion may occurs. Let us suppose that the particle concentration is ρ_p=1 g/cm^3 and we measure L/x=4 (by observation). Then it comes n_p=1.6 x 10^6, which is well above the dangerous threshold for explosion (see Fig.)

Although the above method is quite empiric, we may seek for a general mathematical theory to describe ignition and combustion. The most interesting framework is the one proposed by the Soviet school of physico chemical processes.

Zeldovich and Frank-Kamenetsky found a general rule in the frame of chemical kinetics, valid for atoms and small molecules, which states that the temperature required for a chemical process is of the order of 10% of the total energy required. This is an outcome of the Arrhenius activation energy [2]. The Arrhenius law is a law of the type (Eq.2)

where A is called the prefactor, R is the Universal gas constant (R=8.3144621(75) J/mol K), and Ea is the activation energy. The ratio Ea/RT is called the activation energy. Due to the exponential dependency, an increase of the temperature by a factor of 2 can increase the reaction rate by a factor of 10-12 orders of magnitude. This is a striking example of exponential phenomena in the natural world and from this mathematical function results the general concern about global warming.

Now, let us see understand what phenomena we intend to describe quantitatively. Let us begin by observe the propagation of a flame front.

Learn how to do your own experiment with this video:

What we better have to describe energetic processes in nature is the time-dependent governing equation for a given i-species Eq.3(e.g., Fuel):

Here, w_i is the specific mass of the i-species, w_i=m_i/m, with m the total mass of our system (e.g., fuel + oxydizer), D_i^M is the  diffusion coeffient of species i into the mixture of other species (they can be read in Tables or calculated through analytical formulas).

The energy equation is also necessary, which may be written under the general form, Eq.4:

In this Eq. is the mass density () of the mixture (e.g., fuel+oxydizer), v is the mean mass velocity of the center of mass of the mixture,   is the specific heat capacity of the specie i, λ is the heat conductivity of the mixture. The above equation is based on the “Analytical theory of Heat”, proposed by the French scientist Joseph Fourier [4] (see also footnote 1).

These two equations can be re-written under the general form of a partial differential equation:

The time derivative dY/dt represents the temporal changes of the variable Y at a given position z, the term in A represents the molecular transport associated to diffusion and heat conduction, the term with B represents the flow and the last term C contains the effects associated to chemical reactions occurring locally (at a given z).

If we consider the simplest case, where no chemical reaction and no molecular transport is present, that is, putting A=C=0, then from the above it follows the equation

This equation represents a travelling wave propagating with velocity v. A simple analytical solution exists given by the expression:

The shape of a travelling wave does not alter with time and we represent graphically in the fig. this process.

Bill Gates is committed to support investigation in a new kind of travelling wave nuclear reactor, read here how it works and see the movie below.

And now, what about ignition processes? Once again, we may stress the real complexity of the problem, but thankfully another mogul of chemical physics come to rescue us, proposing a simple (and effective) model, the Frank-Kamenetski’s model of thermal explosions. For example, in spherical geometry the energy conservation equation can be written under the form:

Exercice: Re-write the above equation under the form (see p. 144 Ref.[3]):

Here, T_w is the wall temperature of the container where is supposed to be the explosive mixture. And again, it can be shown mathematically that the above equations has stationary solutions for any δ < δ_crit, with δ_crit=3.32 in spherical geometry. This means that for a given system (fuel+oxidizer) and a given wall temperature and container with size r_0, explosion will not occur provided we maintain δ < δ_crit.

Yakov Zel’dovich, Andrei Sakharov and David Frank-Kamenetski in the town of Sarov, mid 50ths. Im credit:”%5D

This short compilation may help the reader to get a glimpse of the complexity of the processes of combustion, ignition and means to control these phenomena. More deep information can be obtained in our proposed references.

[1] Dust Explosions in the Process Industries, Rolph K. Eckhoff ()Gulf Professional Publishing, Amsterdam, 2003)

[2] Alexander Fridman, Plasma Chemistry, p. 5 (Cambridge University Press, Cambridge, 2008)

[3] Combustion: Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation (Springer, )Berlin, 2006)

[4] Analytical Theory of Heat, by Joseph Fourier (Cambridge University Press, London, 1878)


(1)-Joseph Fourier was also a precursor on Global Warming Studies, see this site here.

Detectors in Optical Astronomy

15 Oct

“Astronomy compels the soul to look upwards, and leads us from this world to another.” – Plato (4277-3477 BC), The Republic.

“Correctly understood, the stars were proof of a higher design in the Cosmos.”Plato, in Exploring Ancient: A Survey of Ancient and Cultural Astronomy, by David H Kelley, Eugene F. Milone, Anthony F. Aveny (Springer, New Yoork, 2011) 

Humankind always has sustained, since the dawn of times, a major interest in observing the sky and gaining new knowledge and insights on the working mechanism of Nature (and for some, a glimpse of God). For this reason astronomy is the oldest science developed by humankind and we have been using the discoveries made by the use of this science to improve our civilization, embedding this knowlegde in our culture, art, religion, and our present notions of space and time.

But, how can we access to this knowledge, understand how the universe looks like? The first mean we have is just the bare eye able to separate image details with about 3 minutes of arc. This is equivalent to to 1/10 of the moon diameter, or the airplane wing span (10 m) flying at 10 km high.


What is called angular resolution is the smallest detail detected by a telescope (or an eye), which is limited by aberrations and diffraction pattern (a series of concentric rings of light and darkness due to interference).

Fig. 1 -- Eye structure.

Why it is not possible to obtain an angular resolution with infinite value? Because unfortunately any optical detector (eye, telescope, camera,…) gives of a point light source a diffuse image of what it is called a diffraction pattern (Fig.2) caused by the light waves diffracted at the fringes of what is called the “apertures” (diaphragms that confine the circular beam area). Therefore, the view of the smallest point of the object emmiting light is limited by this pattern, which may be described by parameters such as the diameter of the inner structure (defined as the first zero of light intensity), and other parameters. But to simplify, usually it is prefered to use a new definition, actually a convention proposed firstly by Ernst Abbe in 1874:


Here, d is the size of the finest detail that can be resolved with a camera, lambda is the light wavelenght, and NA denotes the numerical aperture of the objective lens, which can be defined by the ratio:


This means that a telescope with 12 cm of diameter can separate at maximum stars distant one second of arc. The formula is this one:

AR(angular resolution)=250,000 x lambda/diameter

AR=250,000 x 6500Angstrom/0.12= 1.35 seconds of arc

We used for lambda the value 6500 angstrom=10^(-10) meters, a common value in the optical range. And what size eye (AR=1′, one minute) would you need to detect radio waves (lambda=0.1 m) ?

d=250,000 x 0.1/ 1’=450 m !

Sunset at the fingertips of my friend Americo Jones. The bridge links Lisbon to the South of Portugal and it is similar to the Golden Gate Bridge that cross San Francisco Bay.


That is, your eyes should be apart 450 meters to detect radio waves…, for which the atmosphere is transparent. Unfortunately, our eyes cannot strecht so much apart. Otherwise, human eyes, working together with the human brain, are extraordinary matural detectors in its range of color sensitivity, sensitivity to dim light and adaptability. Painters are highly sensitive to colors, see this article about impressionists French painters, and the eye was the principal help classical astronomers had before and served to observe planets and the most brilliant stars in the sky.


Hence, this represents a theoretical resolution, since atmospheric turbulence interposed between the star and the observer forbid this possibility. This is why telescopes are built at the top of mountains in order to gain a factor of 3 or 4 in this resolution. For example, at Mauna Kéa, Hawaii, at 4200 meters of altitude, it was built the world’s largest observatory for optical, infrared, and sub-millimeter astronomy. Among the equipment, France built a telescope 360 meters of diameter, so huge in order to increase the luminosity coming from the stars.

Mauna Kéa, Hawaii, tha world's largest observatory for optical astronomy. Image credit:

In the actuality, astrophysicists need more sensitive detectors to observe quantitatively very faint stars. But notice that at the end it is again the eyes that remain the ultimate detectors, since astronomers need to look at the pictures…


The next detector used in astronomoy is the analog camera. Although less sensitive than the eye, its advantage stays in the possibility to let the shutter open for a long time of exposure, it is simple to handle and possess a great capacity to stock information. But its main disadvantage is the difficulty to digitalize the image for computer analysis. For example a black-and-white roll film has one side more brilliant than the other.  The least brilliant side has an emulsion of gelatin with suspended array of silver halide crystals which determine the sensitive of the film to light exposure. The reaction taking place is the following:

Ag Br + h f -> Ag^+ + Br + e^- [when exposed to light with energy hf, Bromure is liberated and retained in the gelatine]

Ag^+ + e^- -> Ag  [the liberated silver atom is sequentially converted in metallic silver by electronc capture.]

In particular, the size of the grains determine the time of exposure necessary: larger grains needs faster exposure but gives a grainier appearance; smaller needs more extended time of exposure but the images are finer looking. The ISO factor traduces the graininess of the roll film, appearing as a multiple of 10 or 100. For example, lower ISO numbers produce finer grain but slower film, and vice versa.


However, nowadays the amateur and professional have acess to electronics registreurs, which has as precurseurs the electronographic camera invented by Prof. Lallemand and his team at the Observatoire de Paris in 1967. Although this detector was capable to detect just one photon and has a magnetic focalisation of the photo-electrons, it has a low performance. with the advent of integrated circuits (IC) and semiconductors, at 1970’s Williard S. Boyle (received in 2009 the Nobel Prize for this invention) and Smith invented a camera called CCD (charged coupled device) where successive metallic electrodes were capable to locally confine electric charge into silicium. See how it works here.

This operational mode allows that pratically each photon incident on the telescope can be enregistered and its properties made CCD cameras of great use in telescopes at the surface of Earth and in satellites.

After this short digression on the means to watch the sky by yourself, I invite you to watch these wonderful programs broadcast by BBC and presented by Sir Patrick Moore (here) along the great British traditions of scientific divulgation.

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