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 ofthe significance of any present scientific movement than they, who,limited in their views to the age in which their own lives have beenspent, contemplate merely the momentary trend that the course ofintellectual 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.

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: http://www.sakharov-center.ru”%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.

REF:

[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)

FOOTNOTES:

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

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