Read an Excerpt
DETONATION
Theory and Experiment
By Wildon Fickett William C. Davis Dover Publications, Inc.
Copyright © 1979 Wildon Fickett and William C. Davis
All rights reserved.
ISBN: 978-0-486-14070-4
CHAPTER 1
INTRODUCTION
The rapid and violent form of combustion called detonation differs from other forms in that all the important energy transfer is by mass flow in strong compression waves, with negligible contributions from other processes like heat conduction which are so important in flames. The leading part of a detonation front is a strong shock wave propagating into the explosive. This shock heats the material by compressing it, thus triggering chemical reaction, and a balance is attained such that the chemical reaction supports the shock. In this process material is consumed 108 to 108 times faster than in a flame, making detonation easily distinguishable from other combustion processes. The very rapid energy conversion in explosives is the property that makes them useful. For example, a good solid explosive converts energy at a rate of 1010 watts per square centimeter of its detonation front. For perspective, this can be compared with the total electric generating capacity of the United States, about 4 × 1011 watts, or the solar energy intercepted by the earth, about 4 × 1016 watts. A 20-m square detonation wave operates at a power level equal to all the power the earth receives from the sun.
The most easily measured characteristic property of a detonation is the velocity at which the front propagates into the explosive. The front of a detonation wave initiated at one end of a large-diameter stick of explosive is found to approach a nearly plane shape and a constant velocity of propagation. Thus it seems reasonable to assume that a limiting velocity exists, and that in the limit the chemical reaction takes place in a steadily propagating zone in the explosive. A mathematically tractable problem related to physical reality in this limit is that of plane, steady detonation. It has been generally assumed that this problem's solution would describe detonation experiments as the results were extrapolated to infinite size. Therefore, experiments designed for comparison with theory have usually been measurements of a set of long cylinders of different diameters initiated at one end, and extrapolation of the results to infinite diameter. Unfortunately no other problem, not even the seemingly simple problem of a spherically expanding detonation, has been properly treated, so there can be no direct comparison (without extrapolation to infinite size) of experiment with theory.
In the absence of a theory for more complicated flow configurations, it has become customary to treat all detonation problems by assuming that the reaction zone differs inappreciably from its plane, steady limit, and to apply the corresponding theory to experiments directly. This assumption seems reasonable when the velocity has come close to the limiting value, which it does very quickly, and when the radius of curvature of the front is much larger than the reaction zone thickness. This approach has proved satisfactory for many engineering applications, particularly if some of the parameters, such as the equation of state, are chosen to fit measurements made on pieces similar to those being calculated. The situation is similar to most applications of theory in engineering. A mathematically tractable but not perfectly applicable treatment, with slight corrections based on previous experience, is often used. One difference is that in other fields the engineer's intuition is sometimes aided by the existence of exact solutions which, although too complicated for routine use, can be used to determine correction factors. Such exact solutions are not available to explosives engineers.
In addition to experimental results presented for comparison with the one-dimensional theory, we discuss some others in detail because they show a whole group of phenomena about which the theory has nothing to say. Although the detonation front is nearly plane and the velocity nearly constant as assumed in theory, close observation reveals a complex, three-dimensional, time-dependent structure. Discrepancies between the one-dimensional theory and experiment are probably largely attributable to the presence of this structure.
1A. HISTORY
In preparing this short history we have of course gone back and read the early papers, but we have not attempted the careful analysis of the professional historian, and have no doubt failed to give proper credit in some cases.
The phenomenon of detonation was first recognized by Berthelot and Vieille (1881, 1882), and by Mallard and Le Chatelier (1881), during studies of flame propagation. The elements of the simplest one-dimensional theory were formulated around the turn of the century by Chapman (1899) and by Jouguet (1905, 1917). Becker (1922) gives a good summary of the state of the subject in its early days.
Because we refer to the results of the Chapman-Jouguet theory throughout the following discussion, we digress now for a definition of terms and a short description of the results. (A more complete discussion is given in Chapter 2.) The entire flow is assumed to be one-dimensional and the front is treated as a discontinuity plane across which the conservation laws for shock waves apply, with the equation of state depending appropriately on the degree of chemical reaction. The chemical reaction is regarded as proceeding to completion within a short distance relative to the size of the charge so that it is, in effect, instantaneous. The state of the explosive products at the point of complete reaction is called the final state. As in the theory of shock waves, once the detonation velocity (the propagation velocity of the front) is specified, application of the laws of conservation of mass, momentum, and energy determines the final state behind the front (given the equation of state of the reaction products). The presence of the exothermic chemical reaction does, however, introduce a new and important property. In the nonreactive case, shocks of all strengths are possible, and in the limiting case of a zero-strength shock, the shock velocity is equal to the sound velocity. With an exothermic reaction, the conservation laws have no solution for shocks below a certain minimum velocity. For typical heats of reaction the minimum velocity is well above sound velocity.
Given the detonation velocity D as a parameter, the conservation laws have no final-state solution for D less than this minimum value, one solution for D equal to the minimum value, and two solutions for all larger values of D. The solution for the minimum value of D is called the Chapman-Jouguet or CJ state. For larger values of D, one solution has pressure greater than CJ pressure and is called the strong solution; the other has pressure lower than the CJ pressure and is called the weak solution. In the coordinate frame moving with the front, the flow is subsonic at the strong point and supersonic at the weak point. Therefore a detonation whose final state is at the strong point may be overtaken by a following rarefaction and reduced in strength, whereas one whose final state is at the weak point runs away from a following rarefaction, leaving an ever-growing, constant-state region between the end of the reaction zone and the head of the rarefaction wave.
The rear boundary condition determines the detonation velocity actually realized. With the usual rear boundary conditions, such as a rigid wall or no confinement at all at the initiating plane, a rarefaction wave must follow the detonation front to reduce the forward material velocity at the final-state point to that required at the rear boundary. The strong solution must be rejected, then, because of its vulnerability to degradation by this following rarefaction wave. The weak solution is also rejected on rather intuitive and arbitrary grounds. The only solution left is that defined by the so-called Chapman-Jouguet hypothesis: that the steady detonation velocity is the minimum velocity consistent with the conservation conditions. At this velocity the flow immediately behind the front is sonic in a coordinate system attached to the front, so that the head of the rarefaction wave moves at precisely the speed of the front.
This simple theory was apparently an immediate success. The earliest workers were able to predict detonation velocities in gases to within one or two percent, even with the crude values of thermodynamic functions available then. Had they been able to measure density or pressure, they would have found appreciable deviations. Even so, the simple theory works remarkably well.
The first indication that real detonations might be more complex than is postulated by the simple theory came with discovery of the spin phenomenon by Campbell and Woodhead (1927). Their smear-camera photographs of certain detonating mixtures showed an undulating front with striations behind it. The most likely explanation of this front is that it contains a region of higher than average temperature and luminosity which rotates around the axis of the tube as the detonation advances. This spin phenomenon was observed in systems near the detonation limit, where the available energy and the rate of reaction are barely sufficient to maintain propagation in the chosen tube diameter. We use the term "marginal" to describe such a system, with no sharp dividing line between marginal and non-marginal systems. With the techniques then available, the appearance of spin appeared to be confined to such marginal systems. Jost (1946) gives an account of the early work in this field.
A fundamental advance that removed the special postulates and plausibility arguments of the simplest theory was made independently by Zeldovich (1940) in Russia, von Neumann (1942) in the United States, and Doering (1943) in Germany, and their treatment has come to be called the ZND model of detonation. It is based on the Euler equations of hydrodynamics, that is, the inviscid flow equations in which transport effects and dissipative processes other than the chemical reaction are neglected. The flow is assumed to be one-dimensional, and the shock is treated as a discontinuity, but now as one in which no chemical reaction occurs. The reaction is assumed to be triggered by the passage of the shock in the material and to proceed at a finite rate thereafter. It is represented by a single forward rate process that proceeds to completion. In the coordinate system attached to the shock, the flow equations have a solution that is steady throughout the zone of chemical reaction. The state at any point inside this zone is related to that of the unreacted material ahead by the same conservation laws that apply to a jump discontinuity, with the equation of state evaluated for the degree of reaction at the point. The state at the end of the reaction zone is, of course, included and thus obeys exactly the same conservation laws as apply to instantaneous reaction, so the reasons for applying the CJ hypothesis are unchanged. The CJ detonation velocity is then independent of the form of the reaction rate law, as in the simple theory, and can be calculated from the algebraic conservation laws and the complete-reaction equation of state. Fortunately, conditions in gas detonations (pressure under 100 atm, temperature under 7000 K) are such that the products are accurately described by the ideal gas equation of state with very small corrections. The standard tables of thermodynamic functions are applicable. Thus the equation of state is known, and a priori calculations can be made exactly.
The work of Berets, Greene, and Kistiakowsky (1950) ushered in an era of extensive comparison of the one-dimensional theory with experimental measurements on gaseous systems. They repeated both the detonation velocity measurements and the calculations made earlier by Lewis and Friauf (1930) on a number of hydrogen-oxygen mixtures, using more recent thermodynamic data. Away from the detonation limits the agreement was good, the calculated velocities being about one percent higher than the experimental ones. In the next few years there were significant improvements in both computational and experimental techniques. Electronic calculating machines permitted extensive and precise calculations including a determination of the chemical equilibrium composition, a very laborious procedure when done by hand. Duff, Knight, and Rink (1958) measured the density by x-ray absorption; Edwards, Jones and Price (1963) measured the pressure using piezoelectric gauges; and Fay and Opel (1958) and Edwards, Jones, and Price (1963) measured the Mach number of the flow by schlieren photography of the Mach lines emanating from disturbances at the tube walls. Careful measurements of the detonation velocity were made by Peek and Thrap (1957) and by Brochet, Manson, Rouze, and Struck (1963). White (1961) made a very comprehensive study of the hydrogen-oxygen system with carbon monoxide and other diluents. In addition to measuring pressure and detonation velocity, he made extensive use of spark interferograms to measure density changes, and visible light photometry to measure a composition product and to detect temperature rises in compression and shock waves generated behind the detonation.
The results of all this work suggested that the effective state point at the end of the reaction zone is in the vicinity of a weak solution of the conservation equations. The measured pressures and densities are ten to fifteen percent below the calculated CJ values, the flow is supersonic with Mach number about ten percent above the calculated CJ values, and the detonation velocities are one half to one percent above, in approximate agreement with the conservation laws. The departures from calculated values became less puzzling when detailed concurrent studies showed that the front was not smooth, but contained a complicated time-dependent fine structure. The measured state values thus represent some sort of average, inherent in the design of the experiment and the apparatus.
Experiments in condensed explosives are much more difficult and costly, and fewer state variables are amenable to measurement. More serious is the lack of knowledge of the product equation of state, precluding a priori calculation with any useful degree of accuracy. (The many published determinations of the constants in approximate theoretical forms amount to little more than complicated exercises in empirical curve fitting.)
Fortunately an indirect but exact method of testing the CJ hypothesis comes to the rescue. It requires no knowledge of the equation of state other than the assumption that it exists. Following earlier work by Jones (1949), Stanyukovich (1955) and Manson (1958) pointed out that, using only very general assumptions, essentially those of the simple ZND theory, with no knowledge of the equation of state, the derivatives dD/dρoand dD/dEo determine the detonation pressure. Wood and Fickett (1963) turned the argument around by suggesting that the applicability of the theory could be tested by comparing the pressure calculated from the measured derivatives with that found by the more conventional method of measuring the velocity imparted to a metal plate driven by the explosive. Davis, Craig, and Ramsay (1965) tested nitromethane and liquid and solid TNT and showed that, well outside of the experimental error, the theory does not describe these explosives properly. Unfortunately this indirect method of testing the theory does not indicate the actual state at the end of the reaction zone.
While experiment and theory were being extensively compared, the theory of the one-dimensional, steady reaction zone was extended to include other effects, beginning with the work of Kirkwood and Wood (1954). They considered a fluid that obeyed the Euler equations, that is, one in which transport effects are neglected, but in which an arbitrary number of chemical reactions may occur, each proceeding both forward and backward so that a state of chemical equilibrium can be attained. Others considered a fluid described by the Navier-Stokes equations, that is, one in which heat conduction, diffusion, and viscosity are allowed but which has only a single forward chemical reaction. The work of Kirkwood and Wood was extended to include slightly two-dimensional flow caused by boundary layer or edge effects treated in the quasi-one-dimensional (nozzle) approximation.
All of these problems exhibit mathematical complexities not found in the earlier theoretical treatments. In each case, the flow equations are reduced, under the steady state assumption, to a set of autonomous, first-order, ordinary, differential equations with the detonation velocity as a parameter. Without drastic and unwarranted assumptions to simplify the equations, analytical solutions cannot be obtained, and the properties of a solution must be obtained indirectly by studying the topology of the phase plane, with particular attention to the critical points. The qualitative behavior of the integral curve (a solution of the set of differential equations) starting at the initial point (the state immediately behind the shock) must be ascertained for each value of the detonation velocity. The equations have some mathematical similarity to those encountered in the nonlinear mechanics of discrete systems with more than one degree of freedom. An additional complication introduced by the possibility of chemical equilibrium is the appearance of thermodynamic derivatives at both fixed and equilibrium compositions, the most important being the so-called frozen and equilibrium sound speeds. Although this complicates the analysis considerably, the effect on the results is relatively small, particularly for condensed explosives.
(Continues...)
Excerpted from DETONATION by Wildon Fickett William C. Davis. Copyright © 1979 Wildon Fickett and William C. Davis. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
