BURNING O F DOUBLE-BASE ROCKET POTDERS
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T H E THEORY OF T H E BCRXING OF DOUBLE-BASE ROCKET POWDERS' 0. K. R I C E
ASD
ROBERT G I S E L L l
Department of Chemistry, University of ivorth Carolina, Chapel Hill, S o r t h Carolina Received A p r i l 20, 1949 I. INTRODUCTION
The burning of double-base rocket powder offers a number of features of scientific interest, These powders are essentially homogeneous mixtures of nitrocellulose and nitroglycerin with small additions of other material. They are self-combustible, that is, one portion of the powder is oxidized by another portion, which may even be a part of the same molecule, no external source of oxygen being required. In the experiments that we shall consider, the burning takes place in an inert atmosphere. It consists of a rather complex series of reactions, starting at the solid surface or within the solid, and continuing in the gas phase. Gas is ejected normally to the surface, and in the ordinary type of burning the surface recedes in a direction perpendicular to itself. The rate at which the burning proceeds normal to the surface, and the way in which it is affected by t,he room temperature, the pressure exerted by the surrounding inert gas, and the composition are the subjects of the present paper. The qualitative features of the burning process are summarized in figure 1, which also serves to fix the terminology. In figure 1 it is assumed that the solid stick of powder is burning from the end only, as in the experimental arrangement of Crawford and his collaborators ( 5 ) , upon whose data we largely rely. We designate distances along the axis of the stick of powder by x, the powder surface being at x = 0. In the steady state, the powder stick may be assumed to be fed into the flame so that the surface at x = 0 remains stationary. Then M grams of material cross any cross section (< z < m ) per square centimeter per second, M being the burning rate. The burning of the powder appears to occur in three stages, all with evolution of heat. The first stage occurs at the surface of, or just within, the solid powder, and results in the formation of unsaturated fragments which are ejected into the gas phase normal to the surface. In the second stage, these fragments react in the gas phase (at average distance xl) in an assumed second-order reaction. This paper is based in large part on material presented in OSRD Reports 5224 (June, 1945) and 5574 (November, 1945) and some earlier monthly reports by 0 . K. Rice and Robert Ginell which were prepared under Contract OEZfsr-9if3 between the Office of Scientific Research and Development and the University of Xorth Carolina. A preliminary report on this work was presented a t the Symposium on Kinetics of Propellants which was held under the auspices of the Division of Physical and Inorganic Chemistry a t the 112th Meeting of the American Chemical Society, S e w York City, September 15, 1947. Since then the material on the effect of diffusion has been added and the calculations have been reworked. * Present address: Brooklyn College, Brooklyn, Xew York.
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0 . K . RICE AND ROBERT GINELL
These two stages together constitute what has been called "fizz burning"; the gases produced are still capable of further reaction. At low pressures (from one to a few atmospheres) the only heat reaching the surface is that produced by the reaction at the surface, which is thus self-sustaining at a definite (limiting low-pressure) rate.3 At higher pressures heat generated from the gas phase reaction reaches the surface and the reaction rate then increases with pressure. The reactions thus far considered produce no visible light. The third stage of the reaction consists of the flame, which appears only at pressures exceeding about 20 atm., and apparently does not affect the rate of reaction below about 100 atm. The flame exhibits a very striking appearance. The edge of the luminous zone is fairly sharp; it is located in figure 1 at r2. The
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DiLt
TimeJ(t) Temperatures(TJ Lowest possible T'S Holeculm wtr.(WJ
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TIT' ri: 1
Thmnai DiffurivitieS(x) i, 7, +
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FIG.1 FIG.2 FIG.1. Schematic sketch of a stick of burning powder, illustrating notation and use of subscripts. The quantities in the last three lines with subscripts s, 1, and 3 designate properties of the gas just t o t h e right of t h e position indicated. Primes are to be added t o t h e H symbols for specific temperatures Ti,Ti,and T;. For the relation between T I , r;, and t i see subsection "Effect of Diffusion" of Section IV. The following list will be a guide t o other frequently used symbols (the number in the parentheses gives the equation where first used): A (E),Bo (48), Bb (47a), B, (47b), Bd (47~1,D, (15), E, (12), kl (13), K (39), nz u (3), 21 ( ~ ) , Z (221, I 22 (411, Z 3 (44), (36), M (11, M i (28)) M i (28), nz (35), P (61, PI ( l o ) , (46a), 8%(42), (45), X ( l ) , p (a), r3 (44). (In general T i = T ; ;see followingequa tion 45.) FIG.2. Schematic diagram t o illustrate construction of t h e fizz curve (line DE is used in calculating the effect of the flame; see Section V I I I ) .
o3
base of the flame at the burning end of a long cylindrical stick of powder appears no larger in diameter than the powder. Beyond this, however, the flame flares out, frequently exhibiting signs of turbulence. This appearance suggests that the second gas-phase reaction resembles a branching-chain explosion. We can imagine that as the gas leaves the fizz-burning zone there develops a concentration of some active particles. When this concentration reaches some defia Below 1 a t m . the rate falls off again. This phenomenon is not treated here, b u t has been considered by Parr and Crawford (17). In some cases t h e burning-rate pressure curve above 1 atm. seems to merge with t h a t below 1 atm., without showing the approach to constancy in the region just above 1 atm. This, in fact, is the case with the powder considered in detail here (see figure 4). However, there are enough cases in which the burning rate-pressure curve appears to flatten out at low pressures (above 1 a t m . ) , so t h a t we believe t h a t this is a general phenomenon which is accidentally masked in some cases.
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nite value, which occurs at a definite distance x2 from the surface in the steady state, the gas proceeds to react, this reaction occuring at an average distance 2 3 . This resembles the nature of the phenomena observed in branching-chain explosions in gases contained in ordinary reaction vessels; for example, gas mixtures can be made which \Till show a definite induction period, during which no apparent reaction takes place and at the end of which the mixture suddenly explodes. As will appear later (see equation 39 and accompanying discussion) another reason for believing that the initiation of the flame may involve a branching-chain explosion is the great dependence of x2 on the pressure. At the lorest pressures, where only t,he surface reaction occurs, the temperature of the surface is designated as T , . At big$ pressures, where heat gets back to the surface from the gas, T , rises above T , . If only fizz burning occurred the gases would emerge from the fizz zone with temperature T I . At high pressures, around 190 atm., n-here heat from the flame gets back to the fizz zone, TI rises above T I .The temperature at Rhich the flame starts is taken as T 2 (this is in general higher than T1 because the flame does not start where the fizz zone ends) and the gases finally emerge at 2'3, if the full heat of reaction is developed. Over the pressure range in which the flame develops the heat of reaction rises from that characteristic of fizz burning alone, around 500 cal./g., to its full value, about 800 to 1300 cal./g., depending on the powder. 11. TYPES OF THEORY
There are two distinct possibilities as regards the gas-phase reactions, which will make a considerable difference in the nature of the theory to be applied: ( I ) If the energy of activation of a gas-phase reaction is small, the average distance from the surface and the average time a t which it will occur will be determined by a reaction rate which will not depend strongly on the temperature. (2) If the energy of activation is high the average distance and time will not depend directly on the rate constant, but rather on the time required for any particular portion of the gas to be carried into a region where the temperature is high enough for reaction to occur a t an appreciable speed. Case 1 \Till be the theory considered in this paper. Among recent developments of this type may be mentioned the work of Daniels (12). Case 2 has been considered in detail by Boys and Corner (2). In this case, the gas-phase reaction is completely rate-determining; the zone of reaction merely moves toward or away from the surface, so that the temperature of the latter is adjusted to eject reactive fragments into the gas, a t the required rate for maintenance of a steady state. This theory makes the slope of the log M us. log p curve ( p = pressure) equal to one-half the order of the gas-phase reaction ( 2 ; 19, pp. 70 ff.), while the theory of Case 1 indicates that the slope of the curve may be a much more complicated matter. We shall present some reasons for supposing that Case 1 corresponds more closely to the experimental facts, considering separately the fizz reaction and the flame reaction. The $fizz. reaction: According to the theory of Boys and Corner, at low pres-
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sures the reaction would recede from the surfa:e and the temperature of the surface would drop. But it cannot drop below T,, and at the pressure at which this temperature was reached there would be a sudden change of the ratepressure law, since below this pressure the rate would simply remain constant. Below this pressure, the rate of reaction at the surface would be too fast for the reaction in the gas phase, the zone of the latter would, therefore, be pushed out to infinity, and the gas-phase part of fizz burning would cease. There appears, however, to be no particular reason to suppose that it does stop or change ite character. Crawford, Huggett, and McBrady (5) have conducted a rather extensive investigation of the burning of a cordite powder at low pressures. This powder starts its increase in burning rate at around 4 or 5 atm., but does not show a marked increase up to 6 atm. The heat of burning does not show any sudden or very great change in the region of low pressures, up to the pressure a t which the effects of flame burning begin to be felt. The percentages of carbon monoxide and carbon dioxide in the product gases change only gradually through this pressure region and up to high pressures. The percentage of hydrogen does show an apparent increase of 50 to 100 per cent between about 3 and 5 atm., and then changes gradually up into the higher pressure range. It is difficult to know whether to assign any particular significance to this, but on the whole there does not at any point appear to be any very abrupt and marked change in the character of the burning in the region in which such would be expected if Case 2 applied. The $ame reaction: In this case the situation seems to be even more clear-cut. The flame exists and approaches the surface according to a regular law in the range of pressures from 20 to 100 atm., where it has little or no effect on the rate of burning, presumably because no heat reaches the surface. (This conclusion results from study of a catalyzed powder, in which the flame is much closer to the surface, but which burns at about the same rate as an uncatalyzed powder until the flame approaches t o a distance corresponding to that in the uncatalyzed powder at 100 atm.; see Section VIII). If no heat from the flame reaches the surface, and the rate of reaction is controlled by the fizz burning, the rate will in general be either too fast for the flame, or too slow, assuming the position of the flame to be controlled by the mechanism of Case 2. In the former case it would recede to infinity; in the latter it would approach the surface and proceed to control the rate. As it does neither of these things, we conclude that the position of the flame is controlled by a reaction starting at the surface in the fizz zone, as envisioned in Case 1. 111. THE LAW OF STEADY HEAT FLOW
The problem which we have to solve is essentially a problem in heat conduction, chemical kinetics, and diffusion. When a steady state has been attained (in which the powder stick is fed into the flame so that the position of the burning surface is fixed at x = 0), we may make use of the “law of steady heat flow,” established by Boys and Corner (2) and independently discovered, though not published, by J. E. Mayer. This law states that the amount of heat
BURXIKG OF DOUBLE-BASE ROCKET POWDERS
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crossing any given plane x per unit time is the same as that crossing any other; otherwise there would be accumulations of heat and changes in temperature. Heat is transferred by mass transport and by conduction. The heat crossing any x per unit time per unit cross section from left t o right by mass transport is M H , where H is the heat content per gram; by conduction it is -XdT/dx, where X is the heat conductivity. The total is thus M H - XdT/dx. Now dT/dx is zero at z
