Shock waves and detonation

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KVANT 2008/№1

TO

100TH ANNIVERSARY

I.K.KIKOINA

Shock waves and detonation

L. BELOPUKHOV

THE ARTICLE “EXPLOSION” SAID THAT any explosion is necessarily accompanied by the formation of shock waves in space and that a shock wave is an indispensable element of the detonation explosive process in an exploding gas (or dust-air) environment and in a condensed explosive.

A little about gas dynamics

Shock waves are a very interesting natural phenomenon. Their existence was predicted by many scientists, in particular the famous mathematician Riemann. His name is associated primarily with the development of the ideas of his teacher Gauss in the field of geometry. In 1854, the 28-year-old Riemann gave the general idea of ​​mathematical space, and then developed in detail one of the non-Euclidean geometries that has since bore his name. But Riemann made contributions to many areas of mathematics. Thus, in his work on the theory of differential equations, he discovered (on paper) the possibility of shock waves. It was on this occasion that the popular saying arose: “a phenomenon arose at the tip of a theorist’s pen.” The differential equations that Riemann studied represent the mathematical apparatus of a science called gas dynamics. It studies, as the name suggests, the movement of air masses, in particular storms and hurricanes, the flow of gas through pipes, the movement of gas in turbines and rocket engine nozzles, and much more. The basic gas-dynamic equation is Newton’s law applied to an element of a gaseous medium (in this case it is called Euler’s equation). But since a gas element changes its density and often its temperature when moving, it is necessary to use energy (thermodynamic) equations in addition to power ones. One of the simple consequences of solving the system of gas dynamics equations is the wave equation, in which the wave speed for an ideal gas depends only on temperature. These are sound waves, i.e. longitudinal vibrations of the gas element. The speed of sound is v =

$\gamma RT, Μ$

Continuation. The beginning – the article “Explosion” – see «Kvant» No. 6 for 2007.

where T is the absolute temperature, M is the molar mass, R is the universal gas constant, $\gamma$ is the adiabatic index, equal to the ratio of heat capacities at constant pressure and constant temperature. For air at temperatures less than 1000 K, the adiabatic index is 1.4. The speed of sound is v = 20 T (m s), which gives 342 m/s for standard conditions. If we take into account the non-ideal nature of the gas, the possibility of high temperatures at which the adiabatic exponent is not constant and is expressed by a complex function of state parameters, then the system of gas-thermodynamic equations becomes quite complex. Since gas dynamics is not interested in individual atoms or molecules, but considers the gas element, which in theory can be an infinitesimal quantity, then in the differential equations of gas dynamics and in their solutions all state parameters are continuous functions of coordinates and time. But in the 1860s, Riemann showed that there was a special solution to the system of gas-thermodynamic equations. Not a general solution, the methods for obtaining which he developed, and not particular solutions that follow from the general solution using initial and boundary conditions, but a special solution that cannot be obtained by standard mathematical methods. This special solution represents a discontinuity in the functions of pressure, density and velocity of the medium, i.e. an instantaneous, widthless jump in parameters. This jump is called a shock wave. The word “wave” does not accurately convey the nature of the phenomenon, since in the usual understanding a wave is characterized by periodicity, frequency, and a jump is a one-time sharp change. It would be more correct to call the phenomenon a shock pulse.

Emergence of a shock wave

The formation of a shock wave is easiest to imagine using the example of the plane movement of a gas in a pipe into which an accelerating piston is moving. When the piston begins to move and begins to move and compress the adjacent layer of gas, the news of this event (an increase in pressure and density) spreads through the gas at the speed of sound. A “piece” of compression wave (quarter wave) appears in the gas with

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KVANT 2008/№1

Thus, to destroy a large structure, sometimes a small charge placed directly inside (or next to) the supporting structure is sufficient. And people are killed either by the shell of an explosive device (limon grenade) or by small metal objects specially placed in the device. In nuclear explosions, damaging factors such as thermal (radiation from a fireball) and radiation are added to the effect of the shock wave. Shock waves do not only occur in air. At very high pressures, the ability of a solid to maintain its shape is lost, and the substance “flows” like a gas. But, of course, a different relationship between pressure and density leads to a different shock compression equation than the Hugoniot adiabat for a gas. The highest pressure that a chemical explosion in contact with a solid can create is about 3 H $10^{8}$ Pa (or 300,000 atm). If you surround a solid ball with an explosive, when it explodes, a spherical converging shock wave will appear in the solid body, which, reflecting “from itself,” will give a record pressure in the substance - up to $10^{12}$ Pa (or 10 million atm). At the same time, however, the temperature also increases, which complicates the scientific study of matter at such ultra-high parameters. Compression of a solid by a converging wave is used in the design of nuclear bombs, which can (and do) find use for peaceful purposes, and not for the purpose of destroying culture.

Detonation and its occurrence

Detonation is a special wave (more precisely, pulse) phenomenon when a chemical reaction propagates through a substance at supersonic speed. For example, the speed of the detonation wave in TNT is 7 km/s, and in the no less infamous RDX in our time - about 9 km/s. There are explosives with a detonation speed of 13 km/s. And the speed of sound in substances of this type (more precisely, the speed of propagation of longitudinal elastic vibrations) is about 3 km/s. How to explain such high rates of propagation of chemical reactions? Rapid combustion occurs in gas mixtures. In this case, the temperature transfer from the combustible mixture to the one that has not yet ignited occurs due to thermal conductivity and due to the diffusion of active reaction centers - free radicals. The transfer speeds of particles and temperatures are much lower than the speed of sound, so the burning speed in gases is about 20 m/s, and the burning speed of solid mixtures (powders) is about 100 m/s. The high detonation speed is explained by other reasons. Detonation is not only a chemical phenomenon, but to a large extent also a gas-dynamic one. It is explained by the propagation of shock waves. In a shock wave, where pressure and temperature increase, the rate of chemical reaction increases sharply. The energy released in this case (speed of movement hour-

Tits) plays the role of a piston, compressing the yet unreacted substance by impact (and not the usual adiabatic) method. Thus, the detonation wave supports itself. The processes of detonation in condensed media are interesting. For many explosives it begins only under certain and sufficiently strong impacts. TNT blocks (bricks that look like soap) can be set on fire, and they will burn, smoking like bad firewood (partisans often used them for kindling in wet weather). In order for the TNT to detonate, it is necessary to apply a shock effect - first produce an explosion in the detonator capsule. The shock wave of this “provocateur” will, in turn, cause the TNT block to detonate. And the detonator capsule contains another, much more sensitive explosive that can detonate with a small impact (mercury fulminate). Nowadays, electric detonators are more often used, which contain a special substance - lead azide, which easily detonates from heat (from a wire through which current is passed). Much in detonation processes still remains mysterious and awaits its researchers.

Explosion-savior

Shock waves and detonation processes also play an important role in nuclear explosions. These explosive phenomena will be the subject of the next article in the “explosive” series. I am sure that someday the time will come for the globalization of most of the earth’s states and the danger of nuclear war will disappear forever, and nuclear explosions will again be successfully used for a wide variety of peaceful purposes. Perhaps it is nuclear explosions that will save humanity from a global catastrophe - a collision of our planet with an asteroid or comet. Specialists from the Marshall Space Flight Center, part of NASA, have already developed a model of a device designed for this purpose. The interceptor ship will have up to six nuclear warheads, which, alternately exploding on the surface of the asteroid, will give impulses that transfer it to another trajectory that is safe for the Earth. One of the first targets of interceptor ships may be the asteroid Apophis, which, according to preliminary calculations, will fly past the Earth in 2029 at a distance of about 37,000 km, which approximately corresponds to the altitude of geostationary orbit. During the next “visit”, he will be able to fall to Earth under the influence of gravity. The diameter of this asteroid is about 400 km, the mass is about 2 H $10^{20}$ kg. This is larger than the entire Australian mainland. The scale of the disaster is impossible to even imagine - it is unlikely that life on Earth will be able to survive after this. So an explosion-warrior, who can also be an explosion-worker, can also become an explosion-savior.

(To be continued)