INTERIOR BALLISTICS BY JAMES M. INGALLS Colonel United States Army, retired Formerly Instructor of Ballistics at the U. S. Artillery School ; Author of Treatises on Exterior and Interior Ballistics, Ballistic Machines, Ballistic Tables, Etc. THIRD EDITION NEW YORK JOHN WILEY & SONS LONDON: CHAPMAN & HALL, LIMITED 1912 COPYRIGHT 1912 BY JAMEvS M. INGALLS PREFACE TO THE EDITION OF 1894 (SECOND EDITION) WHEN, in the summer of 1889, it was decided by the Staff of the Artillery School to add to the curriculum a course of interior ballistics, the instructor of ballistics, knowing of no text-book on the subject in the English language entirely suited to the needs of the school, employed the time at his disposal before the arrival of the next class of student officers in studying up and arranging a course of instruction upon this subject, so important to the artillery officer. The text-book then planned was partially completed and printed on the Artillery School press, and has been tested by two classes of student officers. In the summer of 1893 the author again had leisure to work on the unfinished text-book, but in the meantime he had found so much of it which admitted of improvement that, with the encouragement of Lieutenant-Colonel Frank, Second Artillery, the Commandant of the School, it was decided to rewrite nearly the entire work as well as to complete it according to the original plan by the addition of the last two chapters. With the exception of portions of Chapters IV and V, the author claims no originality. He has simply culled from various sources what seemed to him desirable in an elementary text- book, arranged it all systematically from the same point of view and with a uniform notation. ARTILLERY SCHOOL, February 15, 1894. 258715 PREFACE TO THE THIRD EDITION THE second edition of this work was used as a text-book at the Artillery School until the School suspended operations at the outbreak of war with Spain, in April, 1898. This edition, having become exhausted, the author has been induced by the request of officers for whose wishes he has great respect, to prepare a new edition embodying the results-of some investiga- tions which were published in Volumes 24, 25 and 26 of the Journal of the United States Artillery, and which have been favorably received by Artillery Officers, both at home and abroad. The Journal articles have been rewritten, and many improve- ments attempted, suggested by friendly criticisms for which the author wishes to express his thanks. As most of the formulas of interior ballistics in the present state of our knowledge of the subject are more or less empirical in their nature, many applications of the formulas deduced in Chapter IV are given in the following chapter to show their agreement with the results of actual firing with guns of widely different calibers. In this connection it is gratifying to be able to quote from an article published in the Journal of the Royal Artillery, vol. 36, No. 9, by Captain J. H. Hardcastle, R.A., who states, with reference to the formulas of Chapter IV as applied to firing-practice with English guns loaded with cordite, that " After many dozens of calculations I can find no serious disagreement between the results of calculation and experiment." VI PREFACE In this paper Captain Hardcastle has very ingeniously adapted the formulas of Chapter IV to slide- rule operations, thereby lessening the labor of calculation somewhat, though at the expense of accuracy in some cases. For the benefit of those who are accustomed to use the slide rule in making logarithmic com- putations a supplemental table of the X functions has been added to Table I, omitting the function X & , which is not used in Captain Hardcastle's method. This work was prepared primarily for the officers of our Coast Artillery Corps; but it is hoped that gun-designers and powder-manufacturers may find in it something useful to them. The author desires to express his indebtedness to Lieut.- Colonel Ormand M. Lissak and Major Edward P. O'Hern of the Ordnance Department for valuable suggestions and for data employed in the " applications." Also to Captains Ennis and Bryant, for assistance in computing Table I. PROVIDENCE, R. I., September 20, 1911. TABLE OF CONTENTS CHAPTER I PAGE Definition and object. Early history of gunpowder. Robins' experi- ments and deductions. Mutton's experiments. D'Arcy's method. Ram ford's experiments with fired gunpowder. Rodman's inventions and experiments. Modern explosives. Density of powder. Inflam- mation and combustion of a grain of powder. Inflammation and combustion of a charge of powder I to 14 CHAPTER II Properties of perfect gases. Marriotte's law. Specific volume. Specific weight. Law of Gay-Lussac. Characteristic equation of the gaseous state. Thermal units. Mechanical equivalent of heat. Specific heat. Specific heat of a gas under constant pressure. Specific heat under constant volume. Numerical value of R for atmospheric air. Law of Dulong and Petit. Determination of specific heats. Ratio of specific heats. Relations between heat and work in the expansion of a per- fect gas. Isothermal expansion. Adiabatic expansion. Law of tem- peratures. Law of pressures and volumes. Examples. Theoretical work of an adiabatic expansion in the bore of a gun. Noble and Abel's researches on fired gunpowder in close vessels. Description of apparatus employed. Summary of results. Pressure in close ves- sels deduced from theoretical considerations. Value of the ratio of the non-gaseous products to the volume of the charge. Determina- tion of the force of the powder, and its interpretation. Theoretical determination of the temperature of explosion of gunpowder. Mean specific heat of the products of combustion. Pressure in the bore of a gun derived from theoretical considerations. Table of pressures. Theoretical work effected by gunpowder. Factor of effect. Actual work realized as expressed by muzzle energy 1 5 to 54 CHAPTER III Combustion of a grain of powder under constant atmospheric pressure. Notation. Definition of the vanishing surface. General expression for the burning surface of a grain of powder. Expression for the VIII TABLE OF CONTENTS PAGK volume consumed in terms of the thickness burned. Definition of the form characteristics. Fraction of grain burned. Applications. Spheres. Cubes. Strips. Solid cylinders. Pierced cylinders. Multiperf orated grains. General expression for surface of combus- tion of multiperforated grains. Maximum surface of combustion. Slivers. Expression for volume of slivers. Proposed ratio of dimen- sions of multiperforated grains to web thickness. Expression for weight of charge burned at any instant. Expressions for initial volume and surface of combustion of a charge of powder. Expression for specific gravity of grain. Initial surface of unit weight of powder. Volume of charge. Gravimetric density. Density of loading. Reduced length of initial air space. Working formulas for English and metric units. Examples 55 to 78 CHAPTER IV Combustion and work of a charge of powder in a gun. Introductory remarks. Sarrau's law of burning under a variable pressure and reason for adopting it. Expression connecting the velocity of burning of grain with velocity of projectile in bore. Expression for fraction of charge burned in terms of volumes of expansion of the gases generated. Expression for velocity of projectile while powder is burning. Velocity of projectile after powder is all burned. Pres- sure on base of projectile while powder is burning. Pressure after powder is burned. Expression for the initial pressure upon the sup- position that the powder was all burned before the projectile had moved from its seat, and the relation of this pressure to the force of the powder. Method of computing the X functions. Special formulas. Expressions for maximum pressure. Formula for velocity of combustion under atmospheric pressure. Working formulas. Eng- lish units. Metric units. Characteristics of a powder. Expressions for constants in terms of the characteristics for English and metric units. Expressions for force of powder when weights of charge and projectile vary 79 to 97 CHAPTER V Applications. Formulas which apply only while powder is burning. For- mulas which apply only after powder is all burned. Formulas which apply at instant of complete combustion. Discontinuity of pressure curve for certain forms of grain. Monomial formulas for velocity and pressure. Typical pressure and velocity curves. Example of TABLE OF CONTENTS ix PAGE monomial formulas, as applied to the 8-inch B. L. R. Comparison of computed velocities and maximum pressures with observed values. Determination of travel of projectile at point of maximum pressure, and also when powder is all burned. Expression for fraction of charge burned for any travel of projectile. Examples. Greatest efficiency when charge is all consumed at muzzle. Application to hypothetical 7-inch gun. Binomial formulas for velocity and pressure. Forms of grain for which binomial formulas must be employed. Methods for determining the constants from experimental firing. Applications to Sir Andrew Noble's experiments with a 6-inch gun. Description of the experiments. Discussion of the data for cordite, 0.4", 0.35", and 0.3" diameter. Remarks on the so-called "force of the powder" as deduced from the calculations. Examples. Applica- tion to the Hotchkiss 57-mm. rapid-firing gun. Data obtained by D'Arcy's method. New method for determining the form char- acteristics of the grains. Application to the magazine rifle, model of 1903. Powder characteristics. Formulas for designing guns for cordite, with application to a hypothetical 7-inch gun. Trinomial formulas. Grains for which these formulas are necessary. Spherical and cubical grains. Formulas for computing the constants. Ap- plication to Noble's experiments with ballistite in a 6-inch gun. Table of computed velocities and pressures. Remarks on the velocity and pressure curves. Examples. Multiperforated grains. Special formulas required for these grains. Discussion of the data obtained by the Ordnance Board with the 6-inch Brown wire gun. Remarks on the discontinuity of the pressure curves. Examples. Superiority of uniperforated to multiperforated grains. Application to the 14-inch rifle. Effect of increasing the volume of the chamber upon the maximum pressure. Better results can be obtained by lengthening the powder grains. Table of pressures. . . . 9810169 CHAPTER VI On the rifling of cannon. Advantages of rifling. The developed groove. Uniform twist. Increasing twist. General expression for pressure on the lands. Angular acceleration. Pressure for uniform twist. Increasing twist. Semi-cubical parabola. Common parabola. Rel- ative width of grooves and lands. Application to the lo-inch B. L. R., model of 1888. Application to the 14-inch gun. Retarding effect of a uniform twist of one turn in twenty-five calibers. . 170 to 186 Tables 18910215 INTERIOR BALLISTICS CHAPTER I INTRODUCTION Definition and Object. Interior ballistics treats of ihe\ formation, temperature and volume of the gases into which the powder charge, in the chamber of a gun, is converted by com- bustion, and the work performed by the expansion of these gases upon the gun, carriage and projectile. Its object is the deduction and discussion of rules and formulas for calculating the velocity, both of translation and of rotation, which the gases of a given weight of powder of known composition and quality are able to impart to a projectile and their reaction upon the gun and carriage. The discussion of the formulas deduced will bring out many important questions, such as the proper relation of weight of charge to weight of projectile and length of bore, the best size and shape of the powder grains for different guns and their effect upon the maximum and muzzle pressures, the velocity of recoil, etc. The most approved formulas for calculating the pressures upon the surf ace of the bore will be given ; but the methods which have been devised for building up the gun, so as best to resist these pressures, will not be entered upon here as their consideration belongs to another branch of the subject. Early History of Interior Ballistics. For more than five hundred years gunpowder an intimate mixture of nitre, sulphur and charcoal, was used almost exclusively as the pro- 2 INTERIOR BALLISTICS pelling agent in firearms; and though it has been entirely superseded within the last quarter of a century by gun-cotton, mtro-glycerine, and their various compounds, yet it possessed many admirable qualities which the modern powders do not as yet so fully enjoy. It ignited easily without deflagration; its effects were regular and sure; its manufacture was economical, rapid and comparatively safe; it produced but little erosion in the bore. Finally, it kept well in transportation, and in- definitely in properly ventilated magazines. It is on record that experiments made with gunpowder, manufactured more than two centuries before, showed that it had lost none of its ballistic qualities. The principal objection to gunpowder, as compared to nitrocellulose powders, are the dense volumes of smoke accompanying its explosion, the fouling of the bore, and the comparatively large charges required to give the desired muzzle velocity, necessitating an abnormal enlarge- ment of the powder chamber or an impracticable lengthening of the gun. Robins' Experiments and Deductions. The celebrated Ben- jamin Robins seems to have been the first investigator who had a tolerably correct idea of the circumstances relating to the action and force of fired gunpowder. In a paper which was read before the Royal Society in 1743 entitled, "New principles of gunnery," Robins described among other things some experiments he had made for determining the velocities of musket balls when fired with given charges of powder. These velocities were measured by means of the ballistic pendulum invented by Robins, "the idea of which is simply that the ball is discharged into a very large but movable block of wood, whose small velocity, in conse- quence of that blow, can be easily observed and accurately measured. Then, from this small velocity thus obtained, the large one of the ball is immediately derived from this simple proportion, viz., as the weight of the ball is to the sum of the weights of the ball and the block, so is the observed velocity of INTRODUCTION 3 the last to a fourth proportional, which is the velocity of the ball sought." * The deductions which Robins makes from these experiments, so far as they relate to interior ballistics, may be summarized as follows: (1) Gunpowder fired either in a vacuum or in air produces, by its combustion, a permanent elastic fluid or air. (2) The pressure exerted by this fluid is, cateris paribus, directly as its density. (3) The elasticity of the fluid is increased by the heat it has at the time of explosion. (4) The temperature of the fluid at the moment of combus- tion is at least equal to that of red-hot iron. (5) The maximum pressure exerted by the fluid is equal to about 1,000 atmospheres. (6) The weight of the permanent elastic fluid disengaged by the combustion is about three-tenths that of the powder, and its volume at ordinary atmospheric temperature and pressure is about 240 times that occupied by the charge. These deductions, considering the extremely erroneous ancj often absurd opinions that were entertained by those who thought upon the subject at all in Robins' time and even down to the close of the century show that Robins is well entitled to be called the "father of modern gunnery." Button's Experiments. Dr. Charles Hutton, professor of mathematics in the Royal Military Academy, Woolwich, con- tinued Robins' experiments at intervals from 1773 to 1791. He improved and greatly enlarged the ballistic pendulum so that it could receive the impact of i -pound balls, whereas that used by Robins was adapted for musket balls only. Button's experiments are given in detail in his thirty-fourth, thirty-fifth, thirty-sixth, and thirty-seventh tracts. They verify most of * Hutton's " Mathematical Tracts," vol. 3, p. 210 (Tract 37), London, 1812. 4 INTERIOR BALLISTICS Robins' deductions, but with regard to Robins' estimate of the temperature of combustion and the maximum pressure Hutton says: "This was merely guessing at the degree of heat in the inflamed fluid, and, consequently, of its first strength, both which in fact are found to be much greater." * His own estimate of the temperature is double that of Robins, and he places the maximum pressure of fired gunpowder at 2,000 atmospheres. Hutton gives a formula for the velocity of a spherical projectile at any point of the bore, upon the assumption that the combus- tion of the charge is instantaneous and that the expansion of the gas follows Mariotte's law no account being taken of the loss of heat due to work performed a principle which at that time was unknown. D'Arcy's Method. In 1760 the chevalier D'Arcy sought to determine the law of pressure of the gas in the bore of a musket by measuring the velocity of the projectile at different points of the bore. This he accomplished by successively shortening the length of the barrel and measuring for each length the velocity of the bullet by means of a ballistic pendulum. Having obtained from these experiments the velocities of the bullets for several different lengths of travel, the corresponding accelerations could be calculated, and then the pressures, by multiplying the accelerations by the mass. This was the first attempt to determine the law of pressures dynamically. Rumford' s Experiments with Fired Gunpowder. The first attempt to measure directly the pressure of fired gunpowder was made, in 1792, by our countryman, the celebrated Count Rumford. A most interesting account of his experiments is given in his memoir entitled "Experiments to determine the force of fired gunpowder," f which must be regarded as the most important contribution to interior ballistics w r hich had been * Tracts, vol. 3, p. 211. t Philosophical Transactions, London, 1797, p*. 222; also "The Complete Works of Count Rumford," Boston, 1870, vol. I, p. 98. INTRODUCTION 5 made up to that time. The apparatus used by Rumford con- sisted of a small and very strong wrought-iron mortar (or eprouvette), which rested with its axis vertical upon a solid stone foundation. This mortar (or barrel, as Rumford calls it), was 2.78 inches long and 2.82 inches in diameter at its lower extremity and tapered slightly toward the muzzle. The bore (or chamber) was cylindrical, one-fourth of an inch in diameter and 2.13 inches deep. At the centre of the bottom of the barrel there was a projection 0.45 inch in diameter and 1.3 inches long, having an axial bore 0.07 inch in diameter connecting with the chamber above, but closed below, forming a sort of vent, but having no opening outside. By this arrangement the charge could be fired without any loss of gas through the vent by the application of a red-hot ball provided with a hole, into which the projecting vent-tube could be inserted, which latter would thus become in a short time sufficiently heated to ignite the powder. The upper part of the bore or muzzle was closed by a stopper made of compact, well-greased sole leather, which was forced into the bore, until its upper surface was flush with the face of the mortar, and upon this was placed the plane surface of a solid hemisphere of hard- ened steel, whose diameter was 1.16 inches. "Upon this hemisphere the weight made use of for confining the elastic fluid generated from the powder in its combustion reposed. This weight in all the experiments, except those which were made with very small charges of powder, was a piece of ordnance of greater or less dimensions or greater or less weight, according to the force of the charge, placed vertically upon its cascabel upon the steel hemisphere which closed the end of the barrel; and the same piece of ordnance, by 'having its bore filled by a greater or smaller number of bullets, as the occasion required, was made to serve for several experiments." * Rumford's Works, vol. i, p. 121. 6 INTERIOR BALLISTICS As one of the objects of Rumford's experiments was to determine the relation between the pressure of the powder gases and their density, he varied the charge, beginning with i grain, and for each charge placed a weight, which he judged was about equivalent to the resulting pressure, upon the hemisphere. If, on firing, the weight was lifted sufficiently to allow the gases to escape, it was increased for another equal charge; and this was repeated until a weight was found just sufficient to retain the gaseous products that is, so that the leathern stopper would not be thrown out of the bore, but only slightly lifted. The density of the powder gases could easily be determined by comparing the weight of the charge with the weight of powder required to completely fill the chamber and vent, which latter was about 25^2 grains troy. Rumford increased the charges a grain at a time from i grain to 18 grains, and from a mean of all the observed pressures he deduced the empirical formula, i +.0004* in which p is the pressure in atmospheres and x the density of loading to a scale of 1000 that is, for a full chamber x = 1000; for one-half full x = 500, and so on. This formula gives 29,178 atmospheres for the maximum pressure that is, when the powder entirely fills the space in which it is fired. In this case the value of x is 1000, and Rumford's pressure formula becomes p = 1.841 X 1000 I<4 = 29178 Nearly a century later Noble and Abel (see Chapter II) found by their experiments, which are entirely similar in charac- ter to those of Rumford, that the maximum pressure of fired gunpowder is but 6,554 atmospheres, or 43 tons per square inch; and this result has been accepted by all writers on interior INTRODUCTION 7 ballistics as being very near the truth. Their formula for the pressure in terms of Rumford's x is 2.818* ^ '" 1 0.00057* in which p and x are denned as before. If in this formula we make x = 1000, we have, as already stated, 2.818 X 1000 P = "" = 6554 i 0.57 For small densities of loading, Noble and Abel's formula gives greater pressures than Rumford's principally because the powder used by the later investigators was the stronger; but as the densities increase this is reversed. With a charge of 18 grains, for which x = 702, Noble and Abel's formula gives a pressure of 3,298 atmospheres, while Rumford's gives 8,140 atmospheres. To enable us to understand the cause of this great difference in the results obtained by these eminent savants (which is very instructive), we will go a little into detail. Two experiments were made by Rumford with a charge of 18 grains of powder. In the first of these a 24-pounder gun, weighing 8,08 1 pounds, was placed vertically on its cascabel upon the steel hemisphere closing the muzzle of the barrel. When ihe charge was fired "the weight was raised with a very sharp report, louder than that of a well-loaded musket." The barrel was again loaded with 18 grains as before, and enough shot were placed in the bore of the 24-pounder gun to increase its weight to 8,700 pounds. Upon firing the powder by applying the red-hot ball "the vent-tube of the barrel was burst, the explosion being attended with a very loud report." These experiments were the eighty-fourth and eighty-fifth, and closed the series. In the eighty-fourth experiment a weight of 8,081 pounds was actually raised by the explosion of 18 grains of powder (about one-fourth the service charge of the Springfield rifle), 8 INTERIOR BALLISTICS acting upon a circular area one-quarter of an inch in diameter. To raise this weight under the circumstances would require a pressure of more than 11,200 atmospheres, while, as we have seen, the actual pressure due to this density of loading, according to Noble and Abel's formula, is but 3,298 atmospheres. Evident- ly then the weight in this experiment was not raised by mere pressure; but we must attribute a great part of the observed effect (in consequence of the position of the charge at the bottom of the bore) to the energy with which the products of combustion impinged against the leathern stopper, which had only to be raised 0.13 inch (the thickness of the leather) to allow the gases to escape. In Noble and Abel's experiments there was no such blow from the products of combustion because the apparatus for determining the pressure (crusher gauge) was placed within the charge. Had the leathern stopper in Rumford's experiments been a little longer, it is probable that his conclusions would have been entirely different. Rodman's Inventions and Experiments. We have space only to mention the names of Gay-Lussac, Chevreul, Graham, Piobert, Cavalli, Mayevski, Otto, Neumann, and others, who did original work, of more or less value, for the science of interior ballistics prior to the year 1860. We will, however, dwell a few moments on the important work done by Captain (after- wards General) T. J. Rodman, of our own Ordnance Department, between the years 1857 and 1861.* The objects of Rodman's experiments were: First, to ascertain the pressure exerted upon different points of the bore of a 4 2 -pounder gun in firing under various circumstances. Second, to determine the press- ures in the y-inch, Q-inch, and n-inch guns when the charges of powder and the weight of projectiles were so proportioned that there should be the same weight of powder behind, and * " Experiments on Metal and Cannon and Qualities of Cannon Powder," by Captain T. J. Rodman, Boston, 1861. INTRODUCTION 9 the same weight of metal in front of each square inch of area of cross-section of the bore. Third, to determine the differences in pressure and muzzle velocity due to the variations in the size of the powder grains; and, fourth, to determine the pressures exerted by gunpowder burned in a close vessel for different densities of loading. For the purpose of carrying out these experiments Rodman, instead of using the system of varying weights employed by Rumford, invented what he called the "indenting apparatus/' which has since been extensively used, not only in this country but in all foreign countries, under the name of Rodman's pressure (or cutter) gauge; and which is too well known to require a description. The maximum pressure of gunpowder when exploded in its own space, as determined by Rodman by the bursting of shells filled with powder, ranged from 4,900 to 12,600 atmospheres; the mean of all the experiments giving 8,070 atmospheres, or 53 tons per square inch. These results, though much nearer the truth than those deduced by Rumford, are still about 25 per cent, greater than Noble and Abel's deductions; and this is undoubtedly due to the position of the pressure gauge, which was placed near the exterior surface of the shell, so that when the products of combustion had reached the gauge they had acquired a considerable energy which probably exaggerated the real pressure. The same causes, it will be remembered, vitiated Rumford's experiments. In both cases it was as if a charge of small shot had been fired with great velocity against the leathern stopper in the one case, or the end of the piston of the indenting tool in the other. General Rodman was the first person to suggest the proper shape for powder grains, in order to diminish the initial velocity of emission of gas and to more nearly equalize the pressure in the bore of the gun/ For this purpose he employed what he termed a "perforated cake cartridge" composed of disks of 10 INTERIOR BALLISTICS compressed powder from i to 2 inches thick and of a diameter to fit the bore. Rodman demonstrated that such a form of cartridge would relieve the initial strain by exposing a minimum surface at the beginning of combustion, while a greater volume of gas would be evolved from the increasing surfaces of the cylindrical perforations as the space behind the projectile be- came greater; and this would tend to distribute the pressure more uniformly along the bore. Rodman's experiments with this powder in the 15 -inch cast-iron gun which he had recently constructed for the government and which is without doubt the most effective and the best smooth-bore gun ever made fully confirmed his theory; but for many reasons he found it more convenient and equally satisfactory to build up the charge by layers of pierced hexagonal prisms about an inch in diameter fitting closely to one another, instead of having them of the diameter of the bore. The war of the rebellion which was inaugurated while General Rodman was in the midst of his discoveries and in- ventions, put an end forever to his investigations, but his ideas were speedily adopted in Europe, and his " prismatic powders," but slightly modified, are extensively used. Modern Explosives. Gun-cotton, made by immersing cleaned and dried cotton waste in a mixture of strong nitric and sulphuric acids, was discovered by Schonbein of Basel, in 1846, who immediately proposed to employ it as a substitute for gunpowder. General von Lenk made many experiments with gun-cotton by compressing it into cubes or cylinders, with the idea of employing it for artillery use. But all his efforts failed from the fact that, no matter how much it was compressed, it was still mechanically porous; and when ignited in a gun the flame and hot gases speedily penetrated the mass causing it to detonate, or, at least, to approach dangerously near to detona- tion. It was not until the discovery in the early eighties that gun-cotton could be dissolved or made into a paste, or colloid, INTRODUCTION 1 1 by acetone and other so-called solvents, that it was possible to employ it as a propellant. In this condition, when moulded into grains and thoroughly dried, it loses its mechanical porosity and burns from the surface in parallel layers, the grain retaining its original form until completely consumed. Gun-cotton is mixed in certain proportions with nitro- glycerine to form nearly all the powders employed for war purposes. For example, the powder used in the British army and navy (called cordite), consists of 65 per cent, of gun-cot- ton, 30 per cent, of nitro-glycerine and 5 per cent, of mineral jelly or vaseline, this latter being used as a preservative. This is also very nearly the composition of the powder used in the United States army and navy. For a full account of the properties, manufacture and uses of gun-cotton and nitro- glycerine, the reader is referred to General Weaver's "Notes on Explosives." Density of Powder. By density of a powder is meant its specific gravity, or the ratio of the weight of a given volume of the powder to the weight of an equal volume of water at the standard temperature. It is sometimes referred to as mercurial density, since it may be determined by art apparatus which utilizes the property of mercury of filling the interstices between the grains without penetrating into the pores or uniting chemi- cally with the powder. The density varies somewhat according to the pressure to which the grains were subjected during the manufacture and ranges from about 1.56 to 1.65. Inflammation and Combustion of a Grain of Powder. Inflammation is the spreading of the flame over the free surface of the grain from the point of ignition. Combustion is the propagation of the burning into the interior of the grain. Igni- tion is produced by the sudden elevation of the temperature of a small portion of the grain to about 180 C. (in the case of cordite) either by contact with an ignited body, by mechanical shock or friction, or by detonation of a fulminate. The velocity 12 INTERIOR BALLISTICS of inflammation depends upon the nature of the source of heat which ignites it, upon the state of the surface of the grain and upon its density and dryness. The combustion of a grain takes place in successive concentric layers, and in free air equal thicknesses are burned in equal times. As the mass of gas disengaged in any given time is proportional to the quantity of powder burned during the same time, and, therefore, propor- tional to the surface of inflammation, it follows that the emission of gas is largely influenced by the form of the grain. For example, if the grain is spherical the surface of inflammation decreases rapidly up to the end of its burning where it is zero. On the other hand, the surface of inflammation (or of combus- tion) of a multi-perforated grain increases until it is nearly consumed. Inflammation and Combustion of a Charge of Powder. The inflammation of a charge of powder involves the trans- mission of the flame from one grain to another. Its velocity depends not only upon the inflammability of the grain but also upon the facility with which the gases first generated are able to penetrate the charge. This is assisted by a proper arrangement of the grains composing the charge and also by placing an igniter of fine rifle powder at each end of the cartridge. The com- bustion of a charge composed of grains of the same form and dimensions should, from what has been said, practically termi- nate at the same time with each or any grain; and, therefore, the time of combustion of a charge increases with the size of the grains, and is in all cases with service powders much longer than the time of inflammation. If a charge of powder be confined in a close vessel and ignited, its combustion takes place silently, and permanent gases and a certain amount of solid matter are produced which can be collected for analysis by opening the vessel, as in the experiments of Noble and Abel described in Chapter II. In this case no work is performed by the gases, and the accompanying phe- INTRODUCTION 13 nomena are comparatively simple. But if the combustion takes place in a chamber of which one of the walls is capable of moving under the tension of the gases, which condition is realized in cannon, the resulting phenomena are much more complicated, as a little consideration will show. When the charge of powder in the chamber of a loaded gun is ignited at both ends of the cartridge, all the grains will be inflamed practically simultaneously. The first gaseous products formed will expand into the air-spaces of the chamber and almost immediately acquire a tension sufficient to start and overcome the forcing of the projectile. This latter once in motion will encounter no resistances in the bore comparable with those which opposed its start, and its velocity will rapidly in- crease under the continued action of the pressure of the gases. This pressure will also increase at first; for, though the displace- ment of the projectile gives a greater space for the expanding gases, this is more than compensated for by a more abundant disengagement of gas. But the pressure soon reaches its maximum; for if, on the one hand, the disengagement of gas is accelerated by the increase of pressure, on the other hand the increasing velocity of the projectile offers more and more space for the gases to expand in. The velocity itself would soon reach a maximum if the bore were sufficiently prolonged; for in addition to the friction and the resistance of the air, both of which retard the motion of the projectile, the propulsive force decreases by the expansion and cooling of the gases. Therefore the retarding forces will in time predominate and the projectile be brought to rest. Its velocity starting from zero passes to its maximum and if the gun terminated at this point the projectile would leave the bore with the greatest velocity the charge was capable of communicating to it. So far only charges in general have been considered. Take, now, a charge composed of small grains of slight density. The initial surface of inflammation will be very great and the emission 14 INTERIOR BALLISTICS of gas correspondingly abundant. The pressure will increase rapidly, and, in consequence, the velocity of combustion. It results from this that the grains will be consumed nearly as soon as inflamed, and this before the projectile has had time to be displaced by a sensible amount. Hence all the gases of the charge, disengaged almost instantaneously, will be confined an instant within the chamber; their tension will be very great, and they will exert upon the walls of the gun a sudden and violent force which may rupture the metal, and which in all cases will produce upon the gun and carriage shocks which are destructive to the system and prejudicial to accuracy of fire. On the other hand the projectile will be thrown quickly forward, as by a blow from a hammer. If, on the contrary, the charge is made up of large grains of great density, the total weight of gas emitted will be the same as before; but the mode of emission will be different. The initial surface of inflammation will be less, and the initial tension of the gas not so great. The combustion will take place more slowly, and will be only partially completed when the projectile shall have begun to move. The pressure of the gases will attain a maximum less than in the preceding case, but the pressure will decrease more slowly. Under the continued action of this pressure, the velocity of the projectile will be rapidly accelerated and at the muzzle will differ but little from that obtained by the fine powder, without producing upon the gun and carriage the destructive effects mentioned above. CHAPTER II PROPERTIES OF PERFECT GASES Mariotte's Law. When a mass of gas is subjected to pressure the volume diminishes until the increased tension just balances the pressure; and it was found by experiment that if the tem- perature of the gas remains constant, the tension, or pressure, is inversely proportional to the volume. Thus, if Vi and v 2 represent different volumes of the same mass of gas and pi and p 2 the corresponding tensions, or pressures, then if the tem- perature is the same for both volumes we have the proportion: Hence Vi pi = v- 2 p- 2 = constant. That is, for every mass of gas at invariable temperature the product of the volume and tension is constant. This law is generally called Mariotte's law, though it was first discovered by the English chemist Robert Boyle, in 1662, and verified by Mario tte in 1679. Specific Volume. The specific volume of a gas is the volume of unit weight at zero temperature and under the normal atmos- pheric pressure. Designate the specific volume by v and the normal atmospheric pressure by p . Then we have by Mariotte's law Specific Weight. The specific weight of a gas is the weight of unit volume at zero temperature and under the pressure p . It is therefore the reciprocal of the specific volume v . 15 1 6 INTERIOR BALLISTICS Law of Gay-Lussac. The coefficient of expansion of a gas is the same for all gases, and is sensibly constant for all tem- peratures and pressures. Let, as before, v be the specific volume, v t the volume at temperature / and a the coefficient of expansion. Then the variation of volume by Gay-Lussac's law will be expressed by the equation Vt - v o = atv ; whence v t = V (i + at) The value of the coefficient a is approximately - - for 2 73 each degree centigrade. The last equation may, therefore, be written ' i H ^ Characteristic Equation of the Gaseous State. The last equation, which expresses Gay-Lussac's law, may be combined with Mariotte's law, introducing the pressure p. The problem may be enunciated as follows : Having given the specific volume of a gas v to determine its volume v t at a temperature t under the corresponding pressure p t . Let x be the volume v t would become at o C., under the pressure p t . Then by Gay-Lussac's law v t = x ( i + a t) and by Mariotte's law Pt* = P O V O ', whence eliminating x, Pt /= PoVo (i + = (273 + c . oo. Since - - is constant, put 273. 7? **. K = ~ 273 PROPERTIES OF PERFECT GASES 17 whence p t v t = (273 + /); or, dropping the subscripts as no longer necessary, pv = R(2J3 + t) The temperature (273 + /) is called the absolute tempera- ture, and is reckoned from a zero placed 273 degrees below the zero of the centigrade scale. Calling the absolute temperature T there results finally pv = RT ...... ( i) which is called the characteristic equation of the gaseous state. It is simply another expression of Mariotte's law in which the temperature of the gas is introduced. Equation (i) expresses the relation existing between the pressure, volume and absolute temperature of a unit weight of gas. For any number y units of weight occupying the same volume v the relation evidently becomes pv=yRT ...... (2) A gas supposed to obey exactly the law expressed in equation (i) is called a perfect gas, or is said to be theoretically in the perfectly gaseous state. This condition represents an ideal toward which gases approach more nearly as their state of rarefaction increases. Of all gases, hydrogen approximates most closely to such an hypothetical substance, though at ordinary temperatures the simple gases, nitrogen, oxygen and atmospheric air, may for most practical purposes be considered perfect gases. Thermal Unit. The heat required to raise the temperature of unit weight of water at the freezing point one degree of the thermometer is called a thermal unit. There are two thermal units in general use, namely: the British thermal unit (B. T. U.), which is the heat required to raise the temperature of one pound 1 8 INTERIOR BALLISTICS of water from 32 F. to 33 F.; and the French thermal unit (called a calorie), which is the heat required to raise the tem- perature of one kilogram of water from o C. to i C. There is still another thermal unit of frequent use, namely: the heat required to raise the temperature of one pound of water from o C. to i C., and which may be designated as the pound- centigrade (P. C.) unit. Mechanical Equivalent of Heat. The mechanical equivalent of heat is the work equivalent of a thermal unit, and will be designated by E. According to Rowland the value of E is 778 foot-pounds for a B. T. U. Since a degree of the centi- grade scale is of a degree of the Fahrenheit scale, we have for a P. C. thermal unit, E = X 778 = 1400.4 foot-pounds. 3 Also since there are 3.280869 feet in a metre, the value of E for a calorie is 1400 4 3. 280869 = 42<5 ' 84 kil g ram - metres - Specific Heat. The quantity of heat, expressed in thermal units, which must be imparted to a unit weight of any sub- stance to increase its temperature one degree of the thermometer, or the quantity of heat given up by the substance while its temperature is lowered one degree, is called its specific heat. The specific heat of different substances varies greatly. Thus, if a pound of mercury and a pound of water receive the same quantity of heat the temperature of the former will be much greater that the latter. Indeed, it requires about 32 times as much heat to raise the temperature of water i as it does to raise the temperature of mercury by the same amount. The heat imparted to a substance is expended in three different ways: i. Increasing the temperature, which may be called vibration work; 2. In doing internal or disgregation work; 3. In doing external work by expansion. If it were PROPERTIES OF PERFECT GASES IQ possible to eliminate the two latter, we should get the true specific heat, or the heat necessary to increase the temperature simply. For a perfect gas, however, the disgregation work is zero, and for all substances the disgregation work is small in comparison with the vibration work. The specific heat of a gas may be determined in two different ways, giving results which are of fundamental importance in thermodynamics, namely: Specific heat under constant pressure, and specific heat under constant volume. Specific Heat of a Gas Under Constant Pressure. To fix the ideas suppose a unit weight of gas to be confined in a spherical envelope capable of expanding without the expenditure of work and which allows no heat the gas may have to escape, and to be in" equilibrium with the constant pressure of the at- mosphere. Under these conditions let a certain quantity of heat be applied to the gas just sufficient to raise its temperature one degree of the thermometer after it has expanded until equilibrium is again restored. This quantity of heat, in thermal units (designated by C p ), is called specific heat under constant pressure. Specific Heat Under Constant Volume. Next repeat the experiment just described, but replacing the elastic envelope, which by hypothesis permitted the gas to expand freely, by a rigid envelope, thus keeping the volume of the gas constant while heat is applied. It will now be found that there will less heat be required to raise the temperature of the gas one degree. The quantity of heat required in this case is called the specific heat under constant volume, and in terms of the thermal unit employed, is designated by C v . The number of molecules of gas being the same in both experiments and the temperatures being equal, it follows that the quantity of heat absorbed by the gas, or the vibration work, is the same in both experiments. But in the experiment made under constant volume the heat absorbed is necessarily equal 20 INTERIOR BALLISTICS to the total heat supplied, namely, C v thermal units, since the envelope is considered impermeable to heat. Therefore in the first experiment there is a loss of heat equal to C p C v thermal units. This last heat must then have been expended in over- coming the atmospheric pressure in expanding; and the work done will be found by multiplying C p C v by the mechanical equivalent of heat. That is, for an increase of one degree of temperature, Work of expansion = (C p - C v ) E. The work of overcoming a constant resistance is measured by the product of the resistance into the path described. In the case of the expanding gas just considered the constant re- sistance is the atmospheric resistance p \ and the path described is measured by the increase of volume of the gas. To determine this latter Gay-Lussac's law gives for the centigrade scale tV Vt VQ = ' - 2 73 and therefore for an increase of temperature of one degree there is an increase of volume equal to ^0/273. The work of expansion for one degree is, therefore, * 2 73 The quantity R is, then, the external work of expansion performed under atmospheric pressure by unit weight of gas when its temperature is raised one degree centigrade. But this work of expansion has already been found equal to (C p C v ) E. There results, therefore, the important equation (C V -C V }E = ^ = R . . . . (3) for the centigrade scale of temperature. For the Fahrenheit scale the equation becomes (C p -C V )E = ^- fa 1 ) 491.4 PROPERTIES OF PERFECT GASES 21 Numerical Value of R. The numerical value of R for any particular gas depends upon the units of length and weight adopted, the atmospheric pressure, the specific weight of the gas and the scale of temperature. Throughout this chapter the foot and pound will be employed for the units of length and weight, respectively; and generally the centigrade scale of temperature will be used. The adopted value of the atmos- pheric pressure is Po = 10333 kgs. per m. 2 log = 4-01423- p = 2116.3 Ibs. per ft. 2 lo g = 3-3 2 55 8 - p = 14.6967 Ibs. per in. 3 log = 1.16722. As an example, find the numerical value of R for atmospheric air. The specific weight of this gas, according to the best authorities, is 0.080704 Ibs. The specific volume is the recip- rocal of this; or V = 12.3909 c. ft. Therefore, 2116.3 X 12.3000 R = JV = 96.056 foot-pounds. 273 Therefore, for one pound of this gas, p v = 96.056 T] and for y pounds p v = 96.056 y T. Law of Dulong and Petit. The product of the specific heat of a perfect gas under constant volume, by its density, is a constant number. By the density of a gas is meant its specific weight expressed in terms of the specific weight of atmospheric air taken as unity. If C va is the specific heat of air at constant volume and C v and d the specific heat at constant volume, and density, respectively, of any other gas, then in accordance with this law, C,d = C m . Determination of Specific Heats. The specific volume and the specific heat at constant pressure of a gas can both be 22 INTERIOR BALLISTICS determined with great accuracy by experiment; but the specific heat under constant volume is almost impossible to measure directly on account of the dissipation of heat through the sides of the vessel containing the gas. It can, however, be computed by equation (3) which gives C v = C p - - (4) By means of this equation and the direct determination of specific heats under constant pressure, Regnault has deduced the following law for perfect gases : The specific heats under constant pressure and constant volume are independent of the pressure and volume. The following table gives the specific weights, volumes and heats of those gases which approximate most nearly to the theoretically perfect gas. The values of R were computed by (i) and those of C v by (4). The temperature is supposed to be o C., and the barometer to stand at 760 mm. = 29.922 in. : Gas Specific Weight Specific Volume R C P C v Atmospheric air Pounds 0.080704 Cubic Feet 12.3909 96 . 056 0.23751 o. 16892 Nitrogen 0.078394 12.7569 98.887 0.24380 O.I73I9 Oxygen Hydrogen 0.089230 o OOSSQO II .2070 178 8910 86.878 1386.8 0.21751 3 . 40900 0.15547 2.41873 Ratio of Specific Heats. In the study of interior ballistics the values of C p and C v for the gases given off by the explosion of the charge are of little importance. It suffices generally to know their ratio which is constant for perfect gases and approxi- mately so for all gases at the high temperature of explosion. That this ratio is constant for perfect gases may be shown as follows : Since PROPERTIES OF PERFECT GASES 23 R = P V = P = P 273 273 w 273 d w a in which w a is the specific weight of atmospheric air, we shall have for two gases distinguished by accents, the relation that is, the values of R for two perfect gases are inversely as their densities. But by the law of Dulong and Petit we have C, d" R' T^r = ~r f ~n, (as shown above). Therefore R' R" -~r = r^r = constant. ^ V ^ V Therefore from equation (4), c* /? -:- = i + -T^-E = constant = n (say). C v C^zi If we compute n by means of atmospheric air, we shall have 96.056 n = i + -77) - = 1.406. 0.16892 X 1400.4 Relations Between Heat and Work in the Expansion of Perfect Gases. The relations which exist between the varia- tions of the volume and pressure of a given weight of gas and the heat necessary to produce them, may now be determined from equation (i) as follows: This equation is pv = RT and contains three arbitrary variables p, v and T. If we suppose an element of heat, d q, to be applied to the gas, the temperature will generally be augmented by an elementary amount d T, and this may be accomplished in three different ways : i. The volume may increase by the element dv without altering the pressure. 2. The pressure may increase by d p 24 INTERIOR BALLISTICS while the volume remains constant. 3. The volume and pressure may both vary at the same time. We will consider each of these cases separately. i. Differentiating (i), supposing p constant, we have and therefore the quantity of heat communicated to the gas will be, in thermal units, from the definition of specific heat, ,/ r IT C pP dv dq = CpdT 2. If, the volume v remaining constant, the pressure is varied by d p, we shall have, proceeding as before, A r ir C v vd P a q = L v a 1 5 3. If the volume and pressure vary together, the corre- sponding element of heat will be the sum of the partial variations given above. That is dq = -j(C p pdv + C v vdp) ... (5) The differential of (i) is RdT = pdv + vdp; . . . . (6) whence, eliminating v d p between (5) and (6), there results dq = C v dT + Cp ~ R Cv pdv ... (7) Whence, since C p , C v and R are constants for the same gas, /c c r dT+-^ J pdv. The first integral represents the change of temperature and the second the external work of expansion. Denoting by 7\ and T the initial and final temperatures of the expanding gas and by W the external work, we have q = (7\ - T) C v + -^ W . . . (8) PROPERTIES OF PERFECT GASES 25 Isothermal Expansion. If we suppose the initial temperature TI to remain constant, that is, that just sufficient heat is im- parted to the gas while it expands to maintain its initial tem- perature, equation (8) becomes We see in this case that the quantity of heat absorbed by the gas is proportional to the external work done. The quantity r> -~r - ^r is, therefore, the ratio of the effective work of a unit weight of gas to the quantity of heat absorbed, or the mechanical equivalent of heat, E. Therefore E = c^c, a result already established by another method. The work performed, therefore, by the isothermal expansion of unit weight of gas is given by the equation W = E q = 1400.4 q foot-pounds, ... (9) where q is expressed in P. C. thermal units. The work of an isothermal expansion may also be expressed in terms of the initial and final volumes or pressures. Thus, substituting in the general equation of the work of expansion, W =-- fpdv, the value of p from (i) and integrating between the limits Vi and v, we have W = R 7\ log, ^ = #1 i log, ~ v (10) where v is the greater volume and Vi the less. Since from (i) ^__!i Vi ' p 26 INTERIOR BALLISTICS we also have W = p l v l log e ^ ..... (n) in which pi is the greater tension and p the less. The reciprocal of E may be called the heat equivalent of work, that is, the quantity of heat equivalent to a unit of work. Therefore from (9), (10) and (n), we have W Pl Vl W v ' q ~- = ~E : ~E~ log ^| p^ Pl ' ' ' (> loge p j Equations (10) and (n), by inverting the ratios of volumes or pressures, evidently hold good when the initial volume Vi and initial tension pi are changed by compression under constant temperature into the less volume v and greater tension p. Adiabatic Expansion. If a gas expands and performs work in an envelope impermeable to heat, so that it neither receives nor gives up heat during the expansion, the transformation is said to be adiabatic. In such an expansion the temperature and tension of the gas both diminish and the work performed must be less than for an isothermal expansion, other things being equal. For an adiabatic expansion, q is zero in (8) and, therefore, since the temperature diminishes, P F = 7 r(r I -r)i l^p I-'? = _*_ . .... = C V E (iV- T)\ Therefore in an adiabatic expansion the work done is pro- portional to the fall of temperature. Next consider equation (7), where, if we make dq zero, it becomes PROPERTIES OF PERFECT GASES 27 /? T which, by dividing by C v and replacing p by its value , re- duces to _ dT dv ~T ~ ~ ~v Integrating between limits, we have T T, " Again, making d q zero in (5), we have o = C p p d v + C v v d p, which may be written (dividing by C v p v) d v dp Integrating between limits, we have Combining (14) and (15) gives the important relations - <> By means of (16) the work of an adiabatic expansion given by (13) may be expressed either in terms of the initial and terminal volumes, or of the initial and terminal pressures. Thus, since the last of equations (13) may be written, 28 INTERIOR BALLISTICS EXAMPLES. 1 . Determine the volume of 5 pounds of oxygen at a pressure of 50 pounds per square inch by the gauge, and at a temperature of 60 C. The real pressure is gauge pressure plus the atmospheric pressure = 50 + 14.6967 = 64.6967 Ibs. per in. 2 Therefore, p = 144 X 64.6967 Ibs. per ft. 2 T = 273 + 60 = 333. R = 86.878. Therefore from (2), 5 X 86.878 X 333 ,, 3 144 X 64.6967 2. One pound of atmospheric air occupying a volume of one cubic foot has a tension of 50,000 Ibs. per ft. 2 What is its temperature by the Fahrenheit scale ? For the centigrade scale we have R = 96.056, v = i, p = 50,000 and y = i. Therefore T = ~ = 5 2O. 54 C. - 9 68. 97 F. .*. / = 968.97 - 491.4 = 477-57 F. 3. A gas-receiver having a volume of 3 cubic feet contains half a pound of oxygen at 70 F. What is the pressure by the gauge ? Here y = ^, v = 3, R = 86.876, t = 21 - C., and T = 294!. Therefore, 86.876 X 2941- ^ = % v 7 v TA,~ ~ J 4-697 = 14-876 Ibs. per in. A 3 A 144 4. A spherical balloon 20 feet in diameter is to be inflated with hydrogen at 60 F., when the barometer stands at 30.2 in., so that gas may not be lost on account of expansion when the PROPERTIES OF PERFECT GASES 29 balloon has risen till the barometer stands at 19.6 in., and the temperature falls to 40 F. How many pounds and how many cubic feet of gas are to be run in? Here v = -- * X io 3 = 4188.8 ft. 3 = *9- 6 X *" 6 -3 = I3 86.8 lbs. per ft.' 29.9215 T = 277! C. R = 1386.8. p V ' y = ~D~T = I 5-9 2 lbs. To determine the number of cubic feet of gas run. in, we have yR T v = S-j = 2827.4 ft. 3 , where 30. 2 X 2116.3 29.9215 = 2136.0. 2P= -1 (60 _ 32) + 273 = 2 88f. 5. "The balloon in which Wellman intends to seek the North Pole has a capacity of 2 24,244 cubic feet, and weighs, with its car and machinery, 6,600 lbs. What will be its lifting capacity when filled with hydrogen at 10 C. and 760 mm. of the ba- rometer ?" (Lissak's " Ordnance and Gunnery," p. 61.) The balloon, when inflated, will hold at 10 C., 17,458 lbs. of air and 1,209 Mbs. f hydrogen. Its lifting capacity will. therefore, be 17,458 (1,209 + 6,600) = 9,649 lbs. 6. Two pounds of air expand adiabatically from an initial temperature of 60 F., and a pressure of 65.3 lbs. per in. 2 to a pressure of 50 lbs. per in. 2 Determine the initial and terminal volumes, the terminal temperature and the external work done. 30 INTERIOR BALLISTICS Here p, = 144 X 65.3 = 9403-2; p = 50 X 144 == 7200; T! = 288! C.; R = 96.056; y = 2. Take n = 1.4 .'.0i= T - = 5-8954 ft. 3 ri T = T\~) 7 = 267.37 C. = 481.266 F. :.t = 21.87 F. W = ^ - (Ti- T) = 10177 ft.-lbs. 7. Compute the work of expansion of 2 pounds of air at temperature 100 C., which expands adiabatically until it doubles its volume. Also determine the temperature after expansion and the ratio of the initial and terminal pressures. Answers: W = 43378 ft.-lbs. / = i9.68 C. P = 0.3789 />L 8. A mass of air occupying a volume of 3 ft. 3 expands adiabati- cally from an initial temperature of 70 F., and pressure of 85 Ibs. per in. 2 , until external work of 8,000 ft.-lbs. has been done. Compute the terminal volume, pressure, temperature, and weight of air. Answers: v = 3.768 ft. 3 p = 61.78 Ibs. per in. 2 t = 2 3 .86 F. y = i. 3 Ibs. Theoretical Work of an Adiabatic Expansion in the Bore of a Gun. If, in the first of equations (17), we replace C v E 7\, by its equal - it becomes for y pounds of gas ni n i ( v PROPERTIES OF PERFECT GASES 3! This equation gives the work of y pounds of gas at the initial temperature 7\, expanding from the initial volume v to volume v. Suppose the mass of gas to occupy the chamber of a gun with the projectile at its firing seat; and to expand by forcing the projectile along the bore. In this case v\ will be the volume of the chamber, which is an enlargement of the bore, and is measured by what is called the reduced length of the chamber; that is, by the length of a cylinder whose cross-section is the same as the bore and whose volume is that of the chamber. If u is the reduced length required, V c the volume of the chamber and d the diameter corresponding to the area of cross-section of bore, and which on account of the rifling is slightly greater than the caliber, we evidently have ~d 2 The variable volume v is the volume of the chamber plus the volume of the bore in rear of the projectile after it has moved any distance u; and is, therefore, measured by u -f u. Therefore the above expression for the work of expansion be- comes W = y ^\i - n-i ( There is some uncertainty as to the proper value of n for the gases of fired powder. As we have seen, the value of this ratio for perfect gases is approximately 1.4; and it has been generally assumed that at the high temperature of combustion of powder the gases formed may be regarded as possessing all the properties of perfect gases ; and therefore most of the earlier writers on interior ballistics employed this value of n in their deductions. But more recent experiments have shown that this value is too great, but have not fixed its true value. The experiments of Noble and Abel with the gases of fired gunpowder, at or near the temperature of combustion, made n = i^ nearly; 32 INTERIOR BALLISTICS and this is the value which, for want of a better, we will adopt in what follows. Introducing this value of n into the above expression for the work of expansion; and making R T l = f and the ratio u/u = x, we have s/*|'--TrnFl The work of expansion in the bore of a gun is expended in many ways, but chiefly in the energy of translation imparted to the projectile. If we assume that the entire work is thus expended, we shall have W W ( (l +JC 3 ) It is evident from (2) that / is the pressure per unit of surface of unit weight of gas at temperature 7\. The ratio x is the number of volumes of expansion of y pounds of gas due to the travel u. The assumption that the work of expansion is measured by the energy of translation of the projectile does not change the form of the second member of (19); and it is evident that by giving to / a suitable value determined by experiment, the equality expressed in (19) may be strictly true. But in this case / ceases to have the value R 7\ and becomes simply an experimental coefficient. In English units (pound and foot) , / would be theoretically the pressure in pounds per square foot of one pound of gas at temperature T\ confined in a volume of one cubic foot. In metric units (kilogramme and decimetre), / would be defined as above, making the proper change of units. We may deduce a second approximation to the velocity impressed upon the projectile by the expansion of the gas by taking into account the work performed upon the gun and carriage, as well as upon the projectile. We will suppose the gun mounted upon a free-recoil carriage. Let M be the mass PROPERTIES OF PERFECT GASES 33 of the gun and carnage, V their velocity at any period of mo- tion and m the mass of the projectile. The expression for the work of expansion will now be 2 W = m IT + M V\ . . . . . (20) A second equation between the velocities v and V can be deduced by equating the momenta of the system proiected upon the axis of the gun. We thus obtain m v = M V ...... (21) Eliminating V from (20) and (21) there results 2W This expression for v 2 is the same as that given by (19) with the exception of the small fraction m/M which can be safely neglected in comparison with unity. Similarly it may be shown that the work expended upon the projectile in giving it rota- tion about its axis is small in comparison with the work of translation. Noble and Abel's Researches on Fired Gunpowder. Noble and Abel's experiments on the explosion of gunpowder in close vessels were given to the world in two memoirs which were read before the Royal Society in 1874 and 1879, respectively. These experiments have an important bearing upon the subject of interior ballistics, since they furnish the most reliable values we possess of the temperature of combustion of fired gun- powder, the mean specific heat of the products of combustion (solid as well as gaseous), the ratio of solid to gaseous products, and, lastly, what is known as the force of the powder, all of which are important factors in computing the work done by the gases of a charge of gunpowder exploded in the chamber of a gun. The vessels in which the explosions were produced were of two sizes, the smaller one for moderate charges and for experi- 34 INTERIOR BALLISTICS ments connected with the measurement or analysis of the gases, while in the larger one Captain Noble states that he has succeeded in absolutely retaining the products of combustion of a charge of 23 pounds of gunpowder.* These vessels consisted of a steel barrel open at both ends, the two open ends being closed by carefully fitted screw plugs (firing plug and crusher plug), furnished with gas checks to prevent any escape of gas past the screw. In the firing plug was a conical hole closed from within by a steel cone which was ground into its place with great exactness, and which, when the cylinder was prepared for firing, was covered with very fine tissue paper to give it electrical insula- tion from the rest of the apparatus. The two wires from a Leclanche battery were attached, the one to the insulated cone and the other to the firing plug, and were connected within the powder chamber by a fine platinum wire passing through a glass tube filled with mealed powder. This platinum wire became heated when the electric current passed through it, and the charge was thus fired. At the opposite end of the cylinder from the firing plug was another plug fitted with a crusher gauge for determining the pressure of the gases. The vessel was also provided with an arrangement for collecting the gases after an explosion for analysis, measurement of quantity, or for other purposes. Results of the Experiments. It was found that about I 57 per cent, by weight of the products of combustion were non- gaseous, consisting principally of potassium carbonate, potassium j sulphate, and potassium sulphide, the first named greatly preponderating. The remaining 43 per cent, were permanent gases, principally C0 2 , CO and N. These gases, when brought to a temperature of o C., and under the normal atmospheric pressure of 760 millimetres, occupied about 280 times the volume of the unexploded powder. * Lecture on Internal Ballistics, by Captain Noble, London, 1892, p. 12. PROPERTIES OF PERFECT GASES 35 Pressure in Close Vessels, Deduced from Theoretical Con- siderations. The expression for the pressure of the gases developed by the combustion of gunpowder in a close vessel is deduced upon the following suppositions : i st. That a portion of the products of combustion is in a liquid state. 2d. That the pressure due to the permanent gases can only be calculated by deducting the volume of the liquid products from the volume of the vessel. Upon these hypotheses the expression for the pressure may be deduced as follows : Let A B C D be a section of a close vessel of volume v in which a given charge of powder is exploded. Let A E F D represent the space (vi) occupied by the charge, and A G H D the space (v 2 ) occupied by the non-gaseous products. Let A t be the so-called density of the products of combustion, that is AI = -- ; and B OL the ratio of the non-gaseous prod- ucts to the volume of the charge, or a = V, 2-. The gases after ex- v D plosion will occupy the space v v 2 = v AI z; = v (i a AI). Let pi be the pressure that would be developed if the volume of the vessel were A E F D (or z;,). In this case the density of the products of combustion (A t ) (the charge remaining the same) would be unity; and the space occupied by the gases would be Vi v 2 = Vi (i a) = A t v (i ). Now if p is the pressure when the volume of the vessel is t>, we have by Mariotte's law (assuming that the temperature is the same for all densities of the products of combustion), 36 INTERIOR BALLISTICS A I T^(I a r ) or, making we have I - a The factor /is called the force of the powder. Value of the Ratio . Let p 2 and p s be the pressures in the same vessel produced by two different charges, and A 2 and A 3 the corresponding densities of the products of combustion. Then from equation (23) (assuming/ to be the same for all values of AO, A-, and whence by division, I-A 2 ^ 3 A 2 * Therefore 3 A 2 -/> 2 A 3 ) = A A by means of which the mean value of a can be determined when a sufficient number of pressures, corresponding to different values of AI, have been found by experiment. The value of finally adopted by Noble and Abel is 0.57. Determination of the Force of the Powder. To determine/ we have from equation (23), i .=57 A, f-t- A ; **1 from which / may be found by means of a single measured pressure corresponding to a given density of the products of PROPERTIES OF PERFECT GASES 37 combustion. When A t = i, that is, when the vessel is completely filled by the charge, p was found to be 43 tons per square inch, and therefore / = 43 (i .57) = 18.49 tons or 41417.6 pounds per square inch. Therefore Noble and Abel's formula for the pressure in a close vessel is, for different densities of the products of combustion, A! p = 18.49 ~~ r tons per sq. in. ~t = 41417.6 - - Ibs. per sq. in. 1 ~ -57 **i To transform this equation so that it shall express the press- ure in kilos per dm. 2 we may employ a simple rule which, as it is of frequent use, is here inserted for convenience : RULE: To reduce a pressure expressed in tons per square inch to the same pressure expressed in kilos per dm. 2 , add to the logarithm of the former the constant logarithm 4.1972544 and the sum is the logarithm of the pressure required. If the pressure to be reduced is in pounds per in. 2 then the constant logarithm to be added is 0.8470064. Applying this rule the expression for the pressure of the products of combustion of a charge of gunpowder fired in a close vessel is found to be p = 291200 -.57 ! . ' . / 291200 kilos per dm. 2 It will be seen from the definition given to A t that it is the density of loading as defined in Chapter III when the gravi- metric density of the powder is unity, that is, when a kilo of the powder fills a volume of a dm. 3 ; or, what is the same thing, when a pound occupies a volume of 27.68 cubic inches; and in this case, when A t is unity the charge just fills the receptacle. Noble and Abel were careful to keep the gravimetric density of the powder they experimented with as near unity as possible. 38 INTERIOR BALLISTICS Interpretation of f . It will be seen from Equation (23) that the quantity designated by / is the pressure of the gases when (i- a that is, when the space occupied by the gases is equal to the volume of the charge, which requires that the vessel should have i -j- OL units of volume. Thus if the kilogramme and litre are the units of weight and volume, respectively, the volume of the vessel must be 1.57 litres in order that the gases may occupy a volume of one litre, and have a tension equal to /. From this / may be denned to be the pressure of the gases of unit weight of powder occupying unit volume at the temperature of combustion r,. If e is the weight of gas furnished by the combustion of unit weight of powder we have from Equation (2), p! 1)i = R Tij and if Vi is the unit of volume, there results p l =f= RT 1 ..... (25) If the pound is the unit of weight the unit of volume is 27.68 cubic inches. In this case the definition of / requires that the volume of the vessel should be 1.57 X 27.68 = 43459 cubic inches. The value of e, according to Noble and Abel, is 0.43; and therefore the pressure of unit weight of the gases of fired gun- powder at temperature TI is 0-43* From this it follows that the pressure of one pound of the gases of fired gunpowder at temperature of combustion, confined in a volume of 27.68 cubic inches, is 41417-6 - = 96320 Ibs. per square inch. PROPERTIES OF PERFECT GASES 39 Also, the pressure of one pound of the gases of the paragraph immediately preceding, confined in a volume of one cubic foot, is, in pounds per square foot, 06320 X 27.68 = 222180 &g. 12 If the gravimetric density of the powder be unity, and y and v be taken in pounds and cubic inches, respectively, then Equa- tion (23) becomes Solving with reference to y and to v gives pv 2 7 .68 (a ; and v = 27 .6 & y(a P+ f) (2g) These equations are useful in questions involving the bursting of shells, etc. Theoretical Determination of the Temperature of Explosion of Gunpowder. Having determined the value of / from the experiments, we can deduce the temperature of explosion by means of the formula T! 2 ?3 f According to Noble and Abel's experiments, if the gravi- metric density of the powder is such that a kilogramme occupies one litre, the gases furnished by its combustion will fill a volume of 280 litres at o C. under the normal atmospheric pressure of 103.33 kgs. per square decimetre. We therefore have 280 v ~ y * and Po = I0 3-33 40 INTERIOR BALLISTICS whence 273 X 291200 C ' > 103.33 X 280 This is the absolute temperature of combustion of gunpowder according to Noble and Abel's latest deductions from their ex- periments. Subtracting 273 from this temperature we have temperature of explosion = 2475 C. (4487 F.). Mean Specific Heat of the Products of Combustion. From equation (8), we have when W = o, that is, when no external work is performed, Q = C, (T, - 273) in which Q is the heat of combustion; that is, the quantity of heat that unit of weight of the explosive substance evolves, under constant volume, when the final temperature of the products of combustion is o C. From this equation we find c - 273 The heat of combustion was determined by Noble and Abel in the following manner: "A charge of powder was weighed and placed in one of the smaller cylinders, which was kept for some hours in a room of very uniform temperature. When the apparatus was through- out of the same temperature, the thermometer was read, the cylinder closed, and the charge exploded. 1 ' Immediately after explosion the cylinder was placed in a calorimeter containing a given weight of water at a measured temperature, the vessel being carefully protected from radiation, and its calorific value in water having been previously deter- mined. ' ' The uniform transmission of heat through the entire volume of water was maintained by agitation of the liquid, and the thermometer was read every five minutes until the maximum PROPERTIES OF PERFECT GASES 41 was reached. The observations were then continued for an equal time to determine the loss of heat in the calorimeter due to radiation, etc.; the amount so determined was added to the maximum temperature." In this way the heat of combustion of R. L. G. and F. G. powders was found to be 705 heat-units; that is, the combustion of a unit weight of the powder liberated sufficient heat to raise the temperature of 705 unit- weights of water i C. We there- fore have This result is accepted by Noble and Abel, and also by Sarrau, as a very close approximation to the mean specific heat of the entire products of combustion. If we assume that the mean specific heat of gunpowder of different compositions is constant, we can compute the temperatures of combustion when the heat of combustion has been determined by the calorimeter, by the formula T Q 0.285 in which T is given by the centigrade scale. Pressure in the Bores of Guns Derived from Theoretical Considerations. "At an early stage in our researches, when we found, contrary to our expectation, that the elastic pressure de- duced from experiments in close vessels did not differ greatly (where the powder might be considered entirely consumed, or nearly so) from those deduced from experiments in the bores of guns themselves, we came to the conclusion that this departure from our expectation was probably due to the heat stored up in the liquid residue. In fact, instead of the expansion of the per- manent gases taking place without addition of heat, the residue, in the finely divided state in which it must be on the ignition of the charge, may be considered a source of heat of the most per- 42 INTERIOR BALLISTICS feet character, and available for compensating the cooling effect due to the expansion of the gases on production of work. "The question, then, that we now have to consider is What will be the conditions of expansion of the permanent gases when dilating in the bore of a gun and drawing heat, during their ex- pansion, from the non-gaseous portions in a very finely divided state?"* Let c t be the specific heat of the non-gaseous portion of the charge, which we can assume, without material error, to be con- stant. We shall then have c t d T for the elementary quantity of heat yielded to the gases per unit of weight of liquid residue. If there are w t units of weight of liquid residue it will yield to the gases w v c t d T units of heat; and if there are w 2 units of weight of gas we shall have in heat-units, in which that is, is the ratio between the weights of the non-gaseous and gaseous portions of the charge. The negative sign is given to the second member because T decreases while q increases. Substituting the above value of d q in Equation (7), it be- comes -(C v + p Cl )dT = C -^pdv . . . . (29) and this combined with Equation (6), gives, by a slight reduction, - (ft c, + )? -(fa+CJ^ . . . (30) Since C p , C VJ Ci and p are, by hypothesis, constant during the expansion, the integration of Equation (30) between the limits v 2 * Noble and Abel, Researches, etc., page 98. PROPERTIES OF PERFECT GASES 43 and v s the former being the initial volume occupied by the per- manent gases and the latter their volume after the projectile has been displaced by a distance u, gives in which r = Equation (31), it will be seen, becomes identical with Equa- tion (15), when /? = o; that is, when there is no liquid residue. To introduce Vi and v, that is the volumes occupied by the charge and the entire volume in the rear of the projectile, into Equation (30) in place of v 2 and z> 3 , proceed as follows: Let ABC D a?! E F G H A C EG represent the chamber of the gun, which we will suppose filled with powder without compression, and further that one pound of the powder fills a space of 27.68 cubic inches. The gravimetric density and density of loading are each unity; and if Vi is the volume of the chamber, it follows that Vi = 27.68 w. co being the weight of charge. Suppose the powder to be entirely consumed before the pro- jectile moves any perceptible distance; and that the non-gaseous products fill the space A B E F, whose volume is a z>,. The gases, 44 INTERIOR BALLISTICS therefore, which by their expansion give motion to the projectile will occupy the space B C F G before perceptible motion begins. The volume of the space B C F G is evidently v 2 = v t a ^ = Vi (i a). Let D H be the base of the projectile after it has moved a distance ; and, designating the volume A D E H by v, we evidently have v 3 = v a v lt Substituting these values of v 2 and v s in Equation (31) gives In this equation ^ is the pressure produced by the combustion of a charge of powder in a close vessel when the density of load- ing is unity. The values of the constants are given by Noble and Abel as follows:* pi = 43 tons per square inch a = 0.57 ^ = 1-2957 C p = 0.2324 C v = 0.1762 c t = 0.45 Vi = 27.68 ! (v aVj)" whence, integrating, we have W = ^- i -. r i Multiplying and dividing the second member by hi (i )]' ', we have (r - PROPERTIES OF PERFECT GASES 49 If, in this last equation, pi be expressed in kilogrammes per square decimetre, and Vi be made unity (one litre), the work will be expressed in decimetre-kilogrammes per kilogramme of powder burned. To express the work in foot-tons per pound of powder burned, we must make Vi = 27.68 cubic inches; and then, since pi is given in tons per square inch, divide the result by 12, the number of inches in a foot. Making these substitu- tions and replacing <* and r by their values already given, we have, in foot-tons, ( / 43^ N 1 - 1 W = 576.369 i - L ' D / or, in terms of A 1? i = 576.369! i - 0.93946 (-^^) j" (37) Substituting in Equation (37) from Equation (35) we have or, since, according to Noble and Abel, ro i = 2 55 we have W = 0.23008 (Tt-T) ..... (38) which gives the work in terms of the loss of temperature of the products of combustion. Table III gives the work of expansion of the gases of one pound of gunpowder of the normal type and free from moisture, computed by Equation (37) . By means of the work given in this table, and by the use of a proper factor of effect determined by experiment, Noble and Abel consider that the actual work of a given charge of powder upon a projectile may be computed with considerable accuracy. Their method of using this table will be clearly seen by the following extract: "If we wish to know the maximum work of a given charge, 4 50 INTERIOR BALLISTICS fired in a gun with such capacity of bore that the charge suffered five expansions (A! = 0.2) during the motion of the projectile in the gun, the density of loading being unity, the table shows us that for every pound in the charge, an energy of 91.4 foot- tons will as a maximum be generated. "If the factor of effect for the powder and gun be known, the above values, multiplied by that factor, will give the energy per pound that may be expected to be realized in the projectile. "But it rarely happens, especially with the very large charges used in the most recent guns that densities of loading so high as unity are employed; and in such cases, from the total energy realizable must be deducted the energy which the powder would have generated, had it expanded from a density of unity to that actually occupied by the charge. Thus in the example above given, if we suppose the charge instead of a density of loading of unity to have a density of 0.8, we see from Table 3, that from the 91.4 foot-tons above given, there must be subtracted 19.23 foot- tons; leaving 72.17 foot- tons as the maximum energy realiz- able under the given conditions, per pound of the charge." To apply these principles practically for muzzle "velocities, let, as before, Vi be the volume occupied by the charge, in cubic inches. v the total volume of bore and chamber, in cubic inches. V b the volume of the bore. V c the volume of the chamber, in cubic inches. Then v=V b +V c ; and, if the gravimetric density of the powder be unity, Vi = 27.68 co, where co is the weight of the charge in pounds. Therefore the * Noble and Abel, Researches, page 176. PROPERTIES OF PERFECT GASES 51 number of volumes of expansion of the products of combustion will be, at the muzzle, JL s .i _A_ ,i. l\ ~ A! 27.68 co "V which may be written, if the gravimetric density of the powder be unity, = 0.0361263-^+ A! u A in which A is the density of loading as denned in Chapter III. If the gravimetric density of the powder be not unity, let v 2 be the volume in cubic inches of one pound of powder not pressed together except by its own weight; and let 27.68 - = m\ ^ then we have in all cases, ^ = m \ 0.0361263^+^ in which is the number of volumes of expansion of the prod- ucts of combustion. Let W 2 be the work taken from Noble and Abel's table (Table III) of the gases of one pound of powder for a given value of , and Wi the work due to the expansion -. Also, let F be the factor of effect. Then if we assume that the work of expansion is all expressed in the energy of translation of the projectile, we shall have approximately, - FW * ...... (39) in which w is the weight of the projectile and W = W 2 - W l From (39) the muzzle velocity v may be computed when the 52 INTERIOR BALLISTICS factor of effect is known; or, we may determine the factor of effect when the muzzle velocity has been measured by a chrono- graph. These two equations reduced to practical forms are the following : = 379-57 \IFW- .... (40) \ w and F = 0.000006041 -fjT . . . . (41) w & As an illustrative application of these formulas to interior ballistics take the following data from Noble and Abel's second memoir, relative to the English 8-inch gun: It was found by firing a charge of 70 pounds of a certain brand of pebble powder, with a projectile weighing 180 pounds, that a muzzle velocity of 1694 foot-seconds was obtained. What was the factor of effect (F) pertaining to this gun and brand of powder ? For this particular gun and charge we have w = 70 pounds, w = 180 pounds, A! = 0.1634, A = 0.605 andw = i. In Noble and Abel's table of work (Table III) the first column gives values of , increasing by a common difference, while the second column con- tains the corresponding values of A t . By a simple interpolation we find for the values of A t and A given above, W 2 = 99.4 and Wi = 37.6; whence W = 61.8 foot-tons. Substituting these values in Equation (41) we have 1 80 X (i694) 2 F = 0.000006041 2 5~~ = 0.8287. 70 X 61.08 That is, the actual work realized, as expressed and measured by the projectile's energy of translation, as it emerges from the bore, is nearly 83 per cent, of the theoretical maximum work which the powder gases are capable of performing, leaving but 17 per cent, for the other work done by the gases, namely, the work expended upon the charge, the gun and carriage, and in PROPERTIES OF PERFECT GASES 53 giving rotation to the projectile; the work expended in overcom- ing passive resistances, such as forcing the rotating band into the groove, the subsequent friction as the projectile moves along the bore, and the resistance of the air in front of the projectile; and lastly, the heat communicated to the walls of the gun. It is very difficult to evaluate these non-useful energies, but it is prob- able that they do not consume more than 17 per cent, of the maximum work of the gases. Longridge finds by an elaborate calculation that this lost work in a lo-inch B. L. Woolwich gun amounts to 30 per cent, of the maximum work; * but it is believed that he has greatly overestimated the work required to give motion to the products of combustion. Colonel Pashkievitsch makes the lost work rather less than 17 per cent, of that expressed in the energy of translation of the projectile.! To test the correctness of Equation (40) for determining muzzle velocities we will apply it to the same gun by means of which the factor of effect was determined, increasing the charge from 70 to 90 pounds, and again to 100 pounds, and compare the computed velocities with those measured with a chronograph. For a charge of 90 pounds of powder we have A! = 0.210 and A = 0.780; whence W 2 = 89.3, Wi= 20.86, and W = 68.44 1.8287X90X68.44 = 2Q2i f '\ 180 The measured velocity with this charge was 2027 foot-seconds. In a similar way we find by the formula that for a charge of 100 pounds v = 2174 foot-seconds, while the measured velocity was 2182 foot-seconds. The differences between the computed and observed velocities in these examples are about one-third of one * " Internal Ballistics." By Atkinson Longridge. London, 1889. Chapter V. f " Interior Ballistics. " By Colonel Pashkievitsch. Translated from the Russian by Captain Tasker H. Bliss, U. S. Army. Washington, 1892. 54 INTERIOR BALLISTICS per cent., and are well within the limits of probable error in measuring them. The factor of effect increases with the caliber of the gun, as is shown by experiment. Thus with the English lo-inch gun fired with charges of 130 and 140 pounds of the pebble powder we have been considering, the factor of effect is 0.855; while with the n-inch gun, and charge of 235 pounds, the factor of effect is 0.89. CHAPTER III COMBUSTION UNDER CONSTANT PRESSURE Combustion of a Grain of Powder Under Constant Atmos- pheric Pressure. In what follows it is assumed that the powder grain is of some regular geometrical form to which the elementary rules of mensuration can be applied. It will also be assumed as the result of observation, that the combustion of the grain takes place simultaneously on all sides and that, under the constant pressure of the atmosphere, parallel layers of equal thickness are burned away in equal successive intervals of time that is, that the velocity of combustion under constant pressure is uniform. The form and dimension of each grain of powder constituting the charge are of the utmost importance, as upon them depends the proper distribution of the mean effective pressure within the bore. If the initial surface of combustion of the charge be large and the web thickness of the grains small, then the maxi- mum pressure will be excessive and the muzzle velocity inade- quate. On the other hand, if the web thickness be too great the chase pressure may prove destructive to the gun. More than one of our heavy guns it is believed have been wrecked during the past ten years simply from excessive web thickness. Many forms of grain have been adopted by different manu- facturers in this and foreign countries, but they may all be divided into two general groups, viz.: those burning with a continuously decreasing surface, and those in which the surface of combustion may increase (or decrease) to a certain stage, the grain then breaking up into other forms entirely dissimilar to the original and which are then consumed with a rapidly de- creasing surface. To the first group belong spherical, cubical, ribbon-shaped, and indeed all solid grains of whatever form, 55 56 INTERIOR BALLISTICS and cylindrical grains with an axial perforation. To the latter group belong pierced prismatic and the so-called multiperforated grains employed by both our army and navy. Notation. Let / = thickness of layer burned in time /. 1 = one-half the least dimension of the grain. Since com- bustion takes place on all sides of a grain at once, it may be assumed that when / = 1 all grains of the first group are totally consumed. This, of course, is not the case with m. p. grains. S = the total initial surface of combustion of the grain. S = surface of combustion at time t, corresponding to /. S' = the total burning surface when I = I ' y that is, when the grain, as a grain, is about to disappear. This surface may be called the vanishing surface of combustion. V = the initial volume of a grain. V = volume of grain burned at time t. That is, the volume comprised between the surfaces S and 5. V k = fraction of grain burned in time /. That is, k = TT- * o The general expression for the burning surface of a grain of powder moulded into any one of the simple geometrical forms adopted by powder manufacturers may take the form, S = S + a I + bF (i) where / is the thickness of layer burned from instant of ignition. At that instant / is zero and 5 the initial surface of combustion S . In the course of burning when / is about to become 1 , S is about to become S f . Therefore S-' = S + al + bi: (2) In these two equations a and b are constants for the same form of grain, whose values will be deduced later. The general expression for the volume consumed while a thickness / is burned away, is COMBUSTION UNDER CONSTANT PRESSURE 57 whence substituting for 5 its general value from (i) and inte- grating, The initial volume V is evidently what V becomes when the grain is completely consumed, that is, when / = 1 . Therefore This, of course, gives the entire original volume only for those grains which are completely consumed when I = 1 , or, in other words, when the web thickness is burned. It need hardly be said that it does not apply to m. p. grains. In this latter case, it gives the original volume minus the " slivers," so called. If, in (4), we substitute for S its value from (2), namely, S = S' - a 1 - b 1 2 it becomes F.-S').-4V-^.' .... (5) o From (4) and (5) we readily find ... (6) and b =j-,(S. + S') --T^ .... (7) ^o l o These equations give a and b when S , S' and V can be computed by the rules of mensuration. It will be observed that a is a linear quantity while b is of zero order of magnitude. These properties afford tests, as far as they go, as to whether the work of deducing a and b in any particular case has been correctly performed. 58 INTERIOR BALLISTICS Fraction of Grain Burned for any Value of 1. We have by definition k = _L = 2 _ _3 This may be transformed into V ju, M a_t _L , *v L V 'I - " o v o Put for convenience, Then 7 ( / 72 ) .... (9) For all grains of the first group k becomes unity when / = 1 , that is, when the grain is all burned ; in this case (9) reduces to i = a (i + \+ /*) (10) This relation always subsists between these numerical con- stants and serves to test the correctness of their derivation in any case. The following relations which are easily established will be useful : aV S' = (i + 2\ + 3 tiS -$.-f-jT (aX+3J); (n) or, more generally, We also have a (X + 2 /.) = ~- - i. ' n COMBUSTION UNDER CONSTANT PRESSURE 59 Therefore for all grains whose vanishing surface (S') is zero, we have I + 2\+ 3 fJ. = O. and a (\+ 2f*) =- i (12) Applications. We will now apply these formulas to a dis- cussion of various forms of grain now in use or which may come into use. i. Sphere. For a spherical grain 1 is evidently the radius. Then by mensuration S = 4^lo Substituting these in (6) and (7) we readily find a = 8 TC 1 and b = 4 n. Therefore from (i) and, therefore, 5 is a decreasing function of /. From (8) we find a = 3, X = - i and fJ. = -; o and these substituted in (9) give /\ 3 / i r- j / i\ T + ~ 71 = I - ( I - T) In S "ft I x I'n' which is the fraction of grain burned in terms of the thickness of the layer /. If we divide the thickness of web (radius of grain) into five equal parts the following table may be computed, which will be useful for comparing this form of grain with others to be given: 6o INTERIOR BALLISTICS / I. k. First Differences. 0.0 o.ooo O.2 0.488 0.488 0.4 0.784 0.296 0.6 0.936 0.152 0.8 0.992 0.056 I.O I.OOO O.OO8 The second column gives the entire fraction of grain burned and the third column the fraction of grain burned for each layer. It will be observed that nearly one-half the initial volume of the grain is in the first layer. 2. Parallelopipedon. Let 2 1 be the least dimension of the parallelopipedon and m and n the other two dimensions. Then, by the rules of mensuration, S = 4l m + 4l n + 2mn S' = 2 (m - 2 1 ) (n - 2 Q = 2 m n - 4 1 m - 4 1 n + 8 / 2 V = 2l mn Substituting these values of S , S f and V in (6) and (7), gives a = 8 (2 1 + m -\- n) and b = 24. Making the following substitutions, viz. : 2 L 2 I - u i u - = x and = y m n in which x and y are generally less than unity, we have, finally, x + y + xy ocy y It may be noted that these values of a, X, // satisfy equation (10). COMBUSTION UNDER CONSTANT PRESSURE 6 1 There are three special parallelopipedons worthy of separate notice : (a) Cube. The cubical form has been used for ballistite and for some other powders. For this form we evidently have x = y = i. Therefore a = 3; \ = i; p = -. o These are the same as were found for spherical grains, as might have been inferred. They also apply approximately to sphero-hexagonal, mammoth and rifle powders (old style). (b) Square Flat Grains. For these grains (still used with certain rapid-firing guns), m and n are equal and greater than 2 1 . Therefore x and y are equal and less than unity. There- fore, If these grains are very thin, x becomes a very small fraction and may be omitted in comparison with unity. In this case X and IJL are approximately zero and a unity. This gives or, a constant emission of gas during the burning; but the grain would be consumed in a very short interval of time. (c] Grains Made Into Long Slender Strips (or "Ribbons"), with Rectangular Cross-Section. These grains are approximately those of the new English powder called "axite." Also of the French "B N" powders, and others. If we suppose the width of the strip to be five times, and the length one-hundred and fifty times, its thickness (which corresponds nearly with the " B N " powders) , we shall have x = and y = -- . Therefore 5 a = 1.207; ^ = 0.172; /z = o.ooi; 62 INTERIOR BALLISTICS and the expression for k becomes I k = 1. 2O7T--J I 0.172 In ( The following table illustrates the progressiveness of this particular grain : I lo k. First Differences. 0.0 o.ooo 0.2 0.233 0-233 0.4 0.450 0.217 0.6 0.650 0.200 0.8 0.833 0.183 I.O I. 000 o. 167 I. 000 These strips, made up into compact bundles or fagots to form the charge, seem well adapted for rapid-firing guns of moderate caliber. In the application of the expression for k for computing velocities and pressures in the gun, fj, may be regarded as zero, and thus greatly shorten the calculations with- out impairing their accuracy. If the cross-section of the strip is square, we shall have 2 I, m = 2 1 , x = i and y = , n being the length of the strip. Therefore, in this case, 1 + 2 y y a = 2 + y; X = ; ft = - . 2 + y ' 2 + y If the strip be very long in comparison with the linear COMBUSTION UNDER CONSTANT PRESSURE 63 dimension of cross-section, y may be considered zero, and we have a = 2 ; X = --;/ = o. Therefore k = 2-\I ~--} = I -(l -^ 3. Solid Cylinder. For this form of grain there are two cases to be considered: (a) When the diameter of cross-section of the cylinder is the least dimension, (b) When the length of the cylinder is the least dimension. That is, a cylinder proper and a circular disk. (a) Cylinder Proper. In accordance with the notation adopted, 1 will be the radius and m the length of the cylinder. We have by mensuration, S = 2 TT (1 m + / 2 ); S f = o; V = * 1 2 m\ whence a = 2 TT (4 1 -f m) and b = 6 n. 2 1 Putting, as before, - = x, there results These are the same expressions for , X, /* as was found for a strip with square cross-section, as might readily be inferred. If x be small in comparison with unity, that is, if the grains are long slender cylinders (thread like), like cordite, we have very approximately, a = 2 ; X= - j;/* = o; and, as before shown, IV k =-(-) 64 INTERIOR BALLISTICS The following table was computed by this formula: J_ k. First Differences. 0.0 O.2 0.00 0.36 0.36 0.4 0.64 0.28 0.6 0.84 O.2O 0.8 0.96 0.12 0.04 I.O I. 00 I .00 Comparing this table with that given for "strips," it will be seen that the burning of cordite is not so progressive as that of axite. If the length of the solid cylindrical grain be the same as its diameter, then x = i ; and we have as for spherical and cubical grains. (b) Circular Disk. With this form of grain the thickness becomes the least dimension instead of the diameter. Let 2 1 be the thickness of the disk and R its radius. Then S = 2 TT R (2 1 + #>; S' = 2 K (R - / ) 2 ; V = 2 n 1 R\ Whence a = 4 TT (2 R + 1 ) and b = 6 n. 2 I Therefore making -^ = ~ = x, we have, as has already COMBUSTION UNDER CONSTANT PRESSURE 65 been found for square flat grains, x (2 + x) x 2 a = i + 2 #: X = I + 2JC ' I + 20C 4. Cylinder with Axial Perforation. Let R = radius of grain, r = radius of perforation, and m = its length. We then have i 2l = R-r, and R + r = 2(R~1 ). . ' . R 2 - r 2 = 4 1 (R - Q. By the rules of mensuration, we find, after reduction, S = 2 TT m (R + r ] + 2 TT (R- r) =4x( m + 2 1 ) (R - 1 \ s f - 4 *(- Wit* -4) V =4*l (R-lo) Therefore a = 16 T: (7? / ) and b = o. 2 / Making, as before, x = - - we have Therefore As an example of this form of grain, suppose the length to be three hundred times the thickness of web. Then x = - ; a = - ; X = - ; fi = o. 300' 300' 301" The expression for k is _ _ 300 / / 301 / 5 ^ ( 300 300 / 66 INTERIOR BALLISTICS The following table was computed by this formula : I c k. First Differences. o o O.QQOO O.2 0.2005 0.2005 0.4 0.4O08 0.2003 0.6 0.6008 O.2OOO 0.8 0.80O5 0.1997 O IQQS I .0 I .OOOO I. OOOO This form of grain is very progressive, much more so than any other form that has been proposed, and seems well adapted for guns of all calibers. The first differences show that for all practical purposes the emission of gas may be considered constant during the entire burning of the grain. From (n) we have, when /* = o, Therefore in this example, when x = -- .we have 3 and the burning surface during the entire combustion lies be- tween its initial value S and its final value -- S . \J 5. Multiperf orated Grains. These grains, which are used exclusively with the heavy artillery of the army and navy of the United States, are cylindrical in form and have seven equal longitudinal perforations, one of which coincides with the axis COMBUSTION UNDER CONSTANT PRESSURE 6 7 of the grain, while the others are disposed symmetrically about the axis, their centres joined forming a regular hexagon. The web thickness (2 Q is the distance between any two adjacent circumferences ; and therefore, if R is the radius of the grain and r the radius of each of the perforations, we have the relation 2l - R ~ 3r 2 From the geometry of the grain as denned above we have the following relations: S = 2 r: [F - 7 r> + m(R + 7 r)} . . . (13) S' = S + 4*lo(3- 2(R + 7 r) - 9 1 ) . (14) V = r. m (R 2 - 7 r) = {S - 2 r, m (R + 7 r) } (15) V' = / S + 27r/ 2 ( 3 m- 2(R + 7 r) -61 ) . (16) In these expressions S and V are the initial surface of combustion and volume, respectively, while S f and V' are the vanishing surface and volume burned, when / is about to become 1 and the grain to break up into slivers. If we substitute the values of S , S' and V' from the above equations in (6) and ( 7 ), they reduce to a = 4 TT (3 m - 2 (R -f 7 r)) and b = - 36 TT These values of a and b, in equations (8), give __ _ ( \ ' R 2 - 7 r 3 + m (R + 7 r) These values of a, X, and ^ satisfy the equation of condition a (i + X + ,) = i, 68 INTERIOR BALLISTICS since when I = 1 the volume V' has been consumed. When this occurs, the original form of the grain disappears and there remain twelve slender, three-cornered pieces with curved sides technically called "slivers." These of course must be treated differently. In the applications of these formulas given in Chapter V, the form characteristics of the slivers are assumed to be OL = 2, X = and /* = o, with good results. The form characteristics deduced in (17), (i 8), and (19), if substituted in (9), will give the fraction of volume V ' burned. But what is required in practice is the fraction of the entire grain (or charge). This is found by employing V instead of V . By this means we find and this value of OL will be used in all the applications. The expressions for A and /*, being independent of the volume (see equations (8)), are those deduced above. Substituting the form characteristics in (9) and making l = lo we shall have the fraction of the entire grain burned when the web thickness is burned. Calling this fraction k r it will be found that ,_!.( (n-^(R + ,r + 3 U m ( R 2 - 7 r 2 ) This expression for k' would also be obtained by dividing (16) by (15). It will be seen that for the same web thickness a and k' decrease as m increases, but within moderate limits, their limit- ing values, when m is infinite, being R 2 - 7 r 3 COMBUSTION UNDER CONSTANT PRESSURE 69 and , _ 2l (R + jr+3l ) R 2 - 7 r* For the grains employed in the United States service, the D ratio varies but little from n. If we adopt this ratio, the expressions for the form characteristics a, X, /* and k' become 12 2/ 19 ' m 2(m-6 1 X = 19/0 + 6 w 6m k > = 19 19 m We also have R = 5.5 1 and r = 0.5 1 . If, in addition, we make m = n 1 we have 12 2 <* = 1 19 n a ( - 6) 6w + 19 4 M = - + 19 6 19 19 n It may be noted that the limiting values of these form characteristics, as the length of the grain is indefinitely increased, are, a = ; X = ; /z = o and k f = . J 9 3 19 70 INTERIOR BALLISTICS Also that X is zero when n = 6 and becomes negative when the grain is still further shortened. It will be seen that the percentage of slivers can never be greater than about 16. For the grains in use n is approximately 26, which gives a = = 0.70850 247 / * 8 X = = 0.22857 35 / = - -~^ = - 0.022857 *' = ^7 = - 85425 There seems to be no valid reason why these, or other simple ratios, for R/r and m/l should not be adopted by powder manufacturers for all sizes of m.p. grains, making the diameter of the grain and perforations, and also its length, depend upon the web thickness adopted for a particular gun. For example, the web thickness adopted for the i4-inch gun is 0.1454 inch. Therefore the dimensions of the grains would be Diameter = 5.5 X 0.1454 = 0.7997 in. Diameter of perforation = 0.1454/2 = 0.0727 in. Length = 13 X 0.1454 = 1.89 in. These dimensions are practically the same as those of the actual grains. From eqjation (26') of this chapter it will be seen that the initial surface of one pound of these grains would vary inversely as the web thickness. For these grains, equations (13) to (16) reduce to S = $2$*l* S'= 729 *l? V = 741 * lo COMBUSTION UNDER CONSTANT PRESSURE 71 The vanishing surface is therefore about 39 per cent, greater than the initial surface. Captain Hamilton has shown conclusively that the m.p. grains now in use are much too short to secure a proper alignment in the powder chamber, and that this lack of alignment conduces to excessive pressure.* If we make n = 200, that is, make the length of the grains 100 times the web thickness, we should have a = 0.64158 X = 0.31829 /z = 0.00328 k' = 0.84368 This value of n would make the length of the grains for the i4-inch gun 14.52 inches; which would not only secure a good alignment of the grains in the containing bag, but would also give a much less initial surface of combustion to the charge and would thus reduce the maximum pressure. The general expression for the surface of combustion of an m.p. grain with 7 perforations, in terms of the thickness of web burned, is by (i), 5 = S + 4 * (3 ~ 2 (R + 7 ')) I ~ 36 n P Differentiating twice, we have = 4 TT (3 m - 2 (R + 7 r)) - 72 nl There is, therefore, a maximum value of S which occurs when 3 m - 2 (R + 7 r) I = 18 * Journal U. S. Artillery, July-August, 1908, page 9. INTERIOR BALLISTICS and the maximum surface of combustion is 7r( 3 w- 2(# From these formulas are easily deduced the following: 1. When 3 m 2 (R + 7 r) = o, S is a decreasing function of / during the entire burning of the web thickness. 2. When 3 m 2 (R + 7 r) is equal to, or greater than, iBl the grain burns with an increasing surface. 3. When 3 m 2 (R + 7 r) lies between o and 18 1 the sur- face of combustion is at first increasing and then decreasing. Expression for Weight of Charge Burned. If we assume that the entire charge is ignited at the same instant, which is practically the case with an igniter at both ends of the cartridge, the combustion of the charge will be expressed by the same function that applies to a single grain. Therefore if y is the weight of the charge burned at any period of the combustion and o> the weight of the entire charge, we may assume the equality (since the weights are proportional to the volumes) In this equation a is always positive from its definition, S I viz.: a = ~TF^- It varies in value from 3 (spheres and cubes) * o to less than unity (service multiperf orated grains). The smaller OL is, cczteris paribus, the less will be the maximum pressure for a given charge. Of the other characteristics, X and fij either may be positive, negative, or zero, but not both at the same time. Expressions for Initial Volume and Surface of Combustion of a Charge of Powder. Let N be the number of grains in unit weight of powder, V the volume of unit weight of water, and the specific gravity of the powder. Then, from the definition of specific gravity, COMBUSTION UNDER CONSTANT PRESSURE 73 V NV ' (23) since we may assume that the weights are proportional to the volumes. The number of grains in unit weight of powder can be counted, and, with the carefully moulded grains now in use, V can be calculated with great accuracy. Thus (23) can be employed to determine the specific gravity of a powder when it is not given by the manufacturer, as is usually the case. For the large grains designed for seacoast guns the number of grains in 100 units should be counted, estimating the fraction of a grain in excess. For small-arms powder, if the specific gravity of the mass of which the grains are made is known, the number of grains in unit weight may be computed by the formula V N = Jy-^^ (24) The units to be used in these and other formulas that will be deduced will be considered later. Initial Surface of Unit Weight of Powder and of the Entire Charge. Let S be the initial surface of the grains of unit weight of powder. Then if S is the surface of one grain, we have, by (24) Si = NS.= j^. ..... (25) But by (8) ^o / l o Therefore d 1 1 for one unit weight of powder; and for w units weight, 74 INTERIOR BALLISTICS This simple formula was first published in the Journal U. S. Artillery for November-December, 1905. It shows that for two charges of equal weight and made up of grains of the same density and thickness of web, but of dissimilar forms, the entire surfaces of all the grains in the two charges are proportional to the corresponding values of . It also shows that if the initial surfaces of two charges of equal weight but made up of grains of dissimilar forms, are to be the same, the web thicknesses must be inversely as the values of <*. For example, if the two charges are made up, the one of cubes and the other of long slender cylinders (axite and cordite), the web thickness of the former must be one-half greater than the latter to obtain the same initial surface for each charge. These principles are important since the maximum pressure in a gun varies very nearly with the initial surface of the charge. Volume of Entire Charge. Let Vs. be the volume of a charge of o> units weight supposed to be reduced to a single homogeneous grain. For a single grain of unit weight (23) gives V - d and for d> units *VF ^ Gravimetric Density. Gravimetric density is the density of a charge of powder when the spaces between the grains are considered. It is measured by the ratio of the weight of any given volume of the powder grains to the weight of the same volume of water. Since one pound of water fills 27.68 cubic inches we may say that the gravimetric density of a powder is the weight in pounds of 27.68 cubic inches of the powder not pressed together except by its own weight. Or, if we take a cubic foot as the unit and designate the gravimetric density by ?% the weight of a cubic foot of the powder grains by ', and by w the weight of COMBUSTION UNDER CONSTANT PRESSURE 75 a cubic foot of water, we shall have by definition, w 1728/27.68 62.427* It is evident that f will vary not only with the density of the individual grains but also with the volume of the interstices between them; and this latter varies with the general form of the grains, or, in other words, with their ability to pack closely or the reverse. It is evident that the maximum value of f is the weight of a cubic foot of solid powder, in which case the above ratio would be the specific gravity of the powder, designated by 8. The gravimetric density is therefore always less than the specific gravity. For modern powders gravimetric density is of very little importance. Density of Loading. Density of loading is defined to be the "ratio of the weight of charge to the weight of a volume of water just sufficient to fill the powder chamber." Let A be the density of loading and V c the volume of the powder chamber. Since V is the volume of unit weight of water it is evident that V c / V is the weight of a volume of water equal to the volume of the chamber. Hence by definition, From (27) we have, V and this substituted in (28) , gives ^> / A = ~y~ ( 2 9) c From this last equation the density of loading may be defined as the ratio of the volume of the powder grains supposed to be reduced to a single grain, to the volume of the chamber, multi- plied by the density of the powder. If V^ = V CJ that is, if the 76 INTERIOR BALLISTICS chamber is filled by a single grain, then A = d ; and this is the superior limit of density of loading. The inferior limit is, of course, zero, namely, when V& = o. If the density of loading is unity it follows from (28), that V c = F 7 ' that is, the weight of charge equals the weight of water that would fill the chamber. Reduced Length of Initial Air Space. By initial air space is meant that portion of the volume of the chamber not occupied by the powder grains constituting the charge. The reduced length of the. initial air space is the length of a cylinder whose cross-section is the same as that of the bore, and whose volume is equal to the initial air space. Denote this length by z and the area of cross-section of the bore by &>. Then as V c V^ is the volume of the air space we have Substituting for V c and V& their values from (28) and (27), we have Zo= ~VvA~ Put i i d -A A Ad Then (30) Working Formulas for English and French Units. The English units used with formulas (23) to (30), inclusive, are the pound and inch. Therefore V = 26.78 cubic inches, nearly. COMBUSTION UNDER CONSTANT PRESSURE 77 The French units employed with the same formulas are the kilogramme and decimetre. For these units we have V f = i cubic decimetre. The two sets of formulas in working form are therefore: d N V, A 7. English 27.68 Units (230 (240 a * (->fr'\ French * J Units (23") (24") (26") (27") (28") IW " NV 27.68 ~NV N ' " dV 27.68 ^ w. a co 27.68* (20 ) > " 37 4. The volume of the chamber of the 1 2-inch rifle is 17487 cubic inches. If the charge is 400 Ibs. what is the density of loading? Ans.: A = 0.633. CHAPTER IV COMBUSTION AND WORK OF A CHARGE OF POWDER IN A GUN IT has been established by experiment that a grain of modern powder burns in concentric, parallel layers, and that the velocity of combustion under constant pressure is uniform. Let 1 be one-half the web thickness of a grain and r the time of burning this thickness under the constant pressure of the atmosphere. We then have, since the web burns on both sides, 7 = velocity of combustion = constant = v c (say) . (i) In the bore of a gun, however, the pressure surrounding the grain is very far from being constant and greatly exceeds the atmospheric pressure. All writers on interior ballistics agree that the velocity of combustion may be regarded at each instant as proportional to some power of the pressure; but they differ widely among themselves as to what this power is. Sainte- Robert, Vieille, Gossot, and Liouville give reasons (based, how- ever, upon experiments made with a small quantity of powder exploded in an eprouvette of a few cubic inches capacity) for 2 Q adopting the exponent -. Centervall makes the exponent for "Nobel N K" powder. Sebert and Hugoniot, from ob- servations of the recoil of a lo-cm. gun mounted on a free-recoil carriage, deduced a law of burning directly proportional to the pressure. This law is the most simple of all and allows an easy and complete integration of the equations entering into the problem.* But simple as is this law of Sebert and Hugoniot, * See Journal U. S. Artillery, vol. 7, pp. 62-82. 79 80 INTERIOR BALLISTICS we prefer to make use of Sarrau's law of the square root of the pressure, because the resulting formulas are easily worked and give results which "agree very well with facts" as stated by Sarrau, and as has been repeatedly shown by the writer and others. Sarrau's law of burning under a variable pressure p leads directly to the equation, dl 1 P\" ... in which p is the atmospheric pressure and / the thickness of layer burned in time t. It will be assumed that the variable pressure p in the bore is measured by the energy of translation imparted to the projectile (which is many times the sum of all the other energies entering into the problem) ; and it will be taken for granted that all the other work done by the expansion of the powder gas may be accounted for by giving suitable values to the constants so as to satisfy the firing data by means of which they are deter- mined. This procedure will be fully illustrated further on. If p is the variable pressure per unit of surface upon the base of the projectile at any instant, & the area of the base, and u the corresponding distance travelled by the projectile from its firing seat, we have from the principle of energy and work, in which w is the weight of the projectile. But from mechanics and calculus, d 2 u dv d v du dv id (v 2 ) dt 2 ~ dt ~ du dt V du "~ 2 du ' in which v from now on represents velocity. Therefore w d(*) COMBUSTION OF A CHARGE OF POWDER IN A GUN 8l Combining (2) and (3), we have *! = !*.( w --\* ( po' \ d x v Integrating between the limits o and x, we have In order to perform the integration indicated, we must know the relation existing between v and x, that is between the velocity of the projectile in the bore at any instant and the corresponding 6 82 INTERIOR BALLISTICS number of volumes of expansion of the gas. We get this relation from (19) Chapter II, which is w From this equation we deduce by simple differentiation dx v Substituting this in (7) and making r* d x x > = J voT^FTcrr we have It will be observed that X is a function of a ratio and is in- dependent of any unit, and may therefore be tabulated with x as the argument. If we put K I- --- ) ..... (n) r V6# a / we have j = KX. ...... (12) 1 Substituting the value of Ill from (12) in (22), Chapter III, we have k =-1 = aKX (i + \KX + v(KX o y-) . (13) CO an equation which gives the fraction of the charge burned at any instant in terms of the volumes of expansion of the gases gener- ated. When the powder is all burned in the gun (if it be all burned before the projectile leaves the bore), we have y = co and I = 1 . COMBUSTION OF A CHARGE OF POWDER IN A GUN 83 If, therefore, we distinguish X by a dash when / = 1 0) (12) becomes KX = i, ...... (14) and (13) reduces to i = a(i + X + /0 a fundamental relation established in Chapter III. Substituting the value of K from (14) in (13), we have, while the powder is burning, the relation X X Expression for Velocity of Projectile while the Powder is Burning. Substituting the value of y from (15) in (8) and making *'=*('- (FT^ji) ^ we have co X j ( X fXo\ 2 \ tf =6gaf- '=-Ji+X =- + M = ) \. - w Xo ( X \A / ) This equation holds only while the powder is burning and ceases to be true when X > X . Velocity of Projectile when y = co. When X = X and, therefore, X l = Xi, equation (17) reduces to V = 6g/-=r<*(i + X + //); w X 2 or, since " (i + X + /) = i, it becomes This equation is, of course, the same as (8) from which it is 84 INTERIOR BALLISTICS derived as is evident from (16). Putting A"t _- ^ ~ X2> or, generally, the expression for v 2 becomes It should be remembered that all symbols employed in this work affected with a dash refer to the position of the projectile, either in the bore on in the bore prolonged, when the powder has all been burned, and therefore where y = o>. From (19), we have , , v 2 w 6 gf = = r ; (20) and this substituted in (17), gives .^vMi+xfs+>Y|fyt () For convenience, put A-l Then, finally, while the powder is burning, A-l J*- ** ^ tf=MXi{i+NX +N'X *} . . . (22) Velocity of Projectile after Powder is all Burned. The velocity of the projectile after the powder is all burned is given by (8), substituting & for y. Reducing by means of (16), (18), and (20), and denoting velocity after the powder is all burned by capital F, equation (8) becomes COMBUSTION OF A CHARGE OF POWDER IN A GUN 85 The velocity then after the powder is all burned varies directly as the square root of X 2 . From (16) and (18), we have and therefore the superior limit as x (or ) increases indefi- nitely is unity. On this supposition (23) becomes r- = = = V? (say) .... (25) A 2 We may regard Vi then as the theoretical limiting velocity after an infinite travel. In terms of V\ (23) becomes F 2 =7 1 2 JT 3 ...... (26) Since from (25) v 2 = X 2 V\ ! , therefore and, therefore, ,, F t 2 ^ : -= (27) Pressure on Base of Projectile while Powder is Burning. Differentiating (17) with respect to the independent variable x and putting for simplicity we have Therefore, from (5) W Combining the constants outside the brackets into one multiplier by making - - M', 86 INTERIOR BALLISTICS we have the following expression for the pressure per unit of surface, on the base of the projectile : Pressure after the Powder is all Burned. Differentiating (26) with reference to x and substituting the differential co- efficient in (5) we have, employing capital P to express pressure in this case, wV* dX 2 2 g co z dx But from (24) dx 3 (i +x)*' wV, 2 p' do) 6 g oj 2 P P' \^ u / (*T\ whence, putting we have finally If we make x = o in (31), we have p = f. (32) Therefore P' is the pressure per unit of surface at the origin supposing the powder to be all burned before the projectile moves from its seat. Relation Between f and P'. From (19) and (25) we get Combining this with (30) there results P' =J~ (34) Z co Since / is (at least theoretically) the pressure per unit of surface of the gases of one pound of powder at temperature of combustion, occupying unit volume, it follows from (34) that COMBUSTION OF A CHARGE OF POWDER IN A GUN 87 P' is the pressure per unit of surface of the gases of co pounds of powder (the entire charge), occupying a volume equal to the initial air space z % as has already been shown by equation (32). Equation (31) is, therefore, the equation of the pressure curve upon the supposition that the charge is all converted into gas before the projectile has moved from its seat. From equation (30'), Chapter III, we have, 27.68 z u = r- a co cubic feet. Therefore, from (34) 27.68 a Values of the X Functions. These values may be most easily and simply expressed by means of auxiliary circular functions. Thus let (i -f x)* = sec0 (36) Then, by trigonometry, sin 2 = i TT = X 2 . (from (22)) and tan = V(i -f- #)i i Also d x = 6 sec 6 tan d Substituting these values in the expression for X we have X = 6J o sec 3 0- But from (41) dX sin* cos* From (9) and (36) we deduce dX . -7 sm = sin cos 0. d x Also, from what precedes, d sin cos cos 8 2 sin cos -j = ^ = . djc 3sec 8 0tan0 3 Therefore i A 3 = sin cos + X cos 0. o Let X = - i + i X cos 4 cosec COMBUSTION OF A CHARGE OF POWDER IN A GUN 89 Then we have 'v/cuiQ , sin cos 4 X a = Y~ (^ From the foregoing equations we find X 3 dx by means of which are easily deduced from the definitions of X 4 and X*, the following simple equations: X* = X (i + X) (43) and V V 2 ( I ~ V\ f . .\ X^= X 2 (i + 2 A) .... (44) By means of equations (39), (40), (41), (42), (43) and (44), the table of the logarithms of the X-functions given at the end of the volume was computed. Some Special Formulas. Dividing (21) by (15) and reducing by (25), we have, since yl&= k, 7 .2 _ z, yi _ z, y 2 v- / \ V - K V - K V i vV 2 . . . . V45/ That is, the velocity of the projectile at any travel before the charge is all burned is equal to what the velocity would have been at the same travel had all the charge been converted into gas before the projectile moved, multiplied into the square root of the fraction of charge burned. For spherical, cubical, and certain other forms of grain, we have a = 3, \ = i and n = . Substituting these in (15), o we have by obvious reductions, k = i ( i -=2 j (46) and therefore x = x { i -(i -*)M .... (47) QO INTERIOR BALLISTICS For cordite and similar grains we have a= 2, X = - - and ^ = o. Substituting these in (15), gives *-'-('-=;)' .... (48) and X =X { i-(i -*)} .... (49) Equations (46) and (48) give the fraction of the charge consumed for any given travel of the projectile, and, conversely, (47) and (49) enable us to determine the travel of projectile for any given fraction of charge burned. For any other forms of grain the solution of a complete cubic equation is necessary to determine X when k is given. See equation (15). Expressions for Maximum Pressure. It is well known that the maximum pressure in a gun occurs when the projectile has moved but a comparatively short distance from its seat, or when u and x are relatively small. The position of maximum pressure is not fixed but varies with the resistance encountered. As a rule it will be found that the less the resistance to be over- come by the expanding gases the sooner will they exert their maximum pressure, and the less will the maximum pressure be. The differentiation of (29) gives an analytical expression for the maximum value of p; but it is too complicated to be of any practical use. A reference to the table of the X functions shows that Xz is approximately a maximum when x = 0.64, while X and X 5 increase indefinitely. When X is negative it is evident that p is a maximum when x is less than 0.64; and when X is positive, when x is greater than 0.64. Therefore there will be two cases depending upon whether the grains burn with an in- creasing or a decreasing surface. These will be considered separately. (a) When the grains burn with a decreasing surface; or what is the same thing, when X is negative. A function at, or near, its maximum changes its value slowly. Therefore a moderate COMBUSTION OF A CHARGE OF POWDER IX A GUN 91 variation of the position of maximum pressure will have no practical effect upon its computed value. It has been found by trial in numerous cases that x = 0.45 gives the position of maximum pressure when X is negative with great precision. For this value of x the table gives, log X 3 = 9.85640 10 log X 4 = 0.48444. logX b = 0.93587. Substituting these in (29) and designating the maximum pressure by p ml we have approximately, when X is negative, p m = [9.85640 -io]M'{i- [0.48444] ^V + [0.93587] N' } (50) or, , p m = 0.71846 M'{i - 3.0510^ + 8.6273^'] . (50') (b) When the grains burn with an increasing surface. When the grains burn with an increasing surface X is generally positive, and it will not be far wrong to assume that the maximum pressure occurs when x = 0.8. For this value of x the table gives: log X 3 = 9.86027-10. log X = 0.60479 Substituting these in (29), we have, p m = [9.86027 -io]M'{ i + [0.60479] #-[1.17352] #') (51) or, p m = 0.72489^(1 +4-0252 N- 14.911 #'} . (51') Expressions for Computing r and the Velocity of Combustion. From (n) and (14) we have ( *o \*= , . :=l<^t/* ..... (52) If v c is the velocity of combustion under atmospheric press- ure we shall have /, *<=-> 92 INTERIOR BALLISTICS and therefore (6g<*Po\*lo , . V < = -(-^ Z -)Y O - - - (53) Let v' c be the velocity of combustion at any instant under the varying pressure p. Then from (2) we have Working Formulas. English Units. It is customary in our service, following the English practice, to express the volumes of the powder chamber and bore in cubic inches; the various pressures in pounds per square inch; the caliber, reduced length of initial air space, and travel of the projectile in the bore, in inches; while the velocity of the projectile is expressed in foot-seconds and its weight in pounds and ounces. These units are apt to cause confusion and error in the applications of ballistic formulas; and to avoid this as much as possible it will be well to reproduce the most important of the formulas deduced in the preceding pages with all the reductions made and the mathematical and physical constants introduced and combined into one numerical coefficient. The physical constants adopted for English units (foot-pound), are the following: g = 32.16 f.s. (mean for the United States) p = 14.6967 Ibs. per in. 2 V = 27.68 cubic inches. / is taken in pounds per square inch. The formulas are re- numbered for convenience. A = 27.68^- = [1.44217]^- ...... (54) ' c v c I i 8 - A a = T~i = TF ...... (S5) 4X27.68 a Si r oi' 7 "/- u \ / *\ Z = - ~ -^ = I* -54708] js- (inches) . (56) COMBUSTION OF A CHARGE OF POWDER IN A GUN Q3 w-OLCl (57) Ff = 144 X 6 g f -^ = [4.44383]^ (foot-seconds) . . (58) _r_ V* = MX = \M = sMP' _v> kX 2 X 2 a aN ' M f X 2 av- a V, 2 M = -rr=-=- ............ ( ) AI A 6 iv M r , w M M'= - -= 7.82867 - 10 - . . . . . . (61) 27.68^ as, as, aP' 1728 T = ^68 ............. (65) X 27.68 X Vaws, r X Va w a ,, , ' ^ =[8-56006-10] --- - 7 J /72 v c = - = [i.43994 ] -^ /- = (inches per second) . . (67) r X Va w a, v e ' = v c 2 = [941639 - 10] v c V7 .... (68) v fi -43994] r^ 3 [1.43994] lo d 2 XQ= - -- -j==-- ....... (69) V aw w v c \/ aw & It must be remembered that v and /> refer to the period when the powder is burning and V and P to the period after the powder is all burned. 94 INTERIOR BALLISTICS FRENCH UNITS In metric units we shall take V c in cubic decimetres, p in kilogrammes per square centimetre, d in centimetres and z , u and v in metres. Also g = 9.80896 m.s. With these units- our formulas become, A = - (71) 40 a co r , a s, Z = -jT = 11.10491] -# (72) f ~ Vf = [2.76977] -^ (f in kilos, per cm. 2 ) . . . (73) w M M'= L7.70735 - 10] T (tilos. per cm. 2 ) . . (74) d CO wV* P' '= [7.23023 10] - (kilos, per cm. 2 ) . . (75) a co wV 2 f = [7.23023 10] - - (kilos, per cm. 2 ) . . (76) CO v c = [0.63128] (cm. per. sec.) . . (77) X V a w co / d 2 T d 2 X = [0.63128] = [0.631-28]=. . (78) V aw & T = [9.36872 - 10] -^ (79) Characteristics of a Powder. The quantities /, T, a, X and M were called by Sarrau the characteristics of the powder be- cause they determine its physical qualities. Of these quan- tities / depends principally upon the composition of the powder, and, with the same gun, for service charges, is practically constant for all powders having the same temperature of combustion. The value of r depends generally upon the density and least COMBUSTION OF A CHARGE OF POWDER IN A GUN 95 dimension of the grain. The factors a, X and (JL called "form characteristics" depend upon the form of the grain, and for the carefully moulded powders now employed their values may be determined with great precision. They are constant so long as the grain in burning retains its original form. Expressions for M, M', N and N' in Terms of the Charac- teristics of the Powder. When / and v c are known from ex- perimental firings or otherwise, for any gun and powder, the quantities V? and X can be determined either from (58) and (69), or (73) and (78). Substituting these in the proper expres- sions for M, M', N and N' they become For English Units .... (80) If- [0.83356] -- -... (81) N = [8.56006- IO]-KJ v 7 aw n, . . . (82) AT'=A" (83) For Metric Units lrfx * (84) -, <-*. / u c I w w \ * tf'=[o.igoo3]-jM--) . . . (85) N = [9.71291 - io]Vawu . . . (86) A T '= %N* (87) If we substitute the value of M' from (81) in (50) or (51), and reject the terms within the brackets, we have in effect Sarrau's 96 INTERIOR BALLISTICS monomial formula for maximum pressure. But it is evident there can be no monomial formula for velocity or pressure unless X and fj, are approximately zero. Equations (80) to (87) are useful for determining the values of M and N (upon which all the other constants depend), when the charge varies or when there are variations in the weight of the projectile. In these formulas a-, X, n and 1 are independent of & and d and are strictly grain constants. v c is a powder constant, varying only with the composition and density of the powder. / is approximately constant for full service charges of the same kind of powder, in guns of all calibers. For example the magazine rifle, caliber 0.3 inches, and the 1 6-inch B. L. R. give approximately the same value to/ when computed by equation (64) or (65). This factor, however, varies with the charge in the same gun, for it is evident that its effective value as measured by projectile energy must decrease with the charge. Indeed if the charge be sufficiently reduced it is obvious that / becomes zero since we have omitted from our formulas all consideration of the force necessary to start the projectile. The law of variation is not known; but we will assume provisionally that/ varies with the charge accord- ing to the law expressed by the equation /-/. ..... (88) where co is the service charge by means of which M and N were determined and f the corresponding value of / computed by (64) or (76). If the weight of the projectile also varies we will assume that / may be determined by the equation The exponents n and n f must be determined from experi- mental data. If we make K -~-r> (90) co W COMBUSTION OF A CHARGE OF POWDER IN A GUN 97 (89) becomes / = K " w n> (90') Substituting this expression for / in (80) and (84) gives, for English units: ^b^^-ff^)*. . . (91) and for metric units: M = [2.48268] K -^yV^^r^J' - (91') In the applications of these equations f must be computed by (64) or (76) and v c by (67) or (77). 7 CHAPTER V APPLICATIONS THE principal formulas deduced in Chapter IV are here re- produced for convenience of reference. They are the following: (a) Formulas which Apply Only While Powder is Burning. v 2 = MX, [i + NX +N'X 2 } . . . (i) p = M'X 3 {i+NXt+N'Xs} ... (2) It will be observed that these formulas for velocity and pressure are identical in form, and that the constants within the brackets are common to both. Also that M' is a simple multi- ple of M. Moreover, from the manner of deriving p from v 2 , the velocity and pressure deduced from these formulas correspond at every point so that one can be easily and exactly computed from the other without the necessity of laying down velocity curves in order to obtain the pressures. (b) Formulas which Apply Only After the Powder has All Been Burned. ... (3) a p =(iw = p '(<-^ ... - (4) (c) Formulas Which Apply at the Instant of Complete Combustion. ?= MX,{i + NX + N'X 2 }, (from (i)): . . (5) and _ Mlti , ,. v 2 = (from (3)) .... (5 ) Equations (5) and (5') give the same value to zr, since the APPLICATIONS 99 former equation reduces to the latter_at the point w. But (i) and (3) are not tangent at the point w unless the vanishing sur- face (5 r ) of the grain is zero, as with cubes, spheres, solid cylin- ders, etc. From (2) we have at the travel P = M f X*{ i + N Xt + N'X 5 } . . . (6) and from (4) '' Equations (6) and (6') give the same value to p for all grains whose vanishing surface is zero, as may be thus shown : Substituting for M' in (6) its value from (59), Chapter IV, and giving to N and N' their values in terms of X , we have But = i + X and =^ = i + 2 X. For all grains whose vanishing surface is zero we have the relations (Equs. (10) and (12), Chapter III.) a (i + X + M ) = i and a (X + 2 M) = -i which readily reduces to (6'). Therefore for all forms of grain whose vanishing surface is zero the pressure curves (2) and (4) are tangent at u. This is not true for grains for which 5">o. 100 INTERIOR BALLISTICS For these the pressure at travel u given by (2) is greater than that given by (4), and this difference increases with 5". Monomial Formulas for Velocity and Pressure While the Powder is Burning. The expressions for velocity and pressure while the powder is burning (equations (i) and (2)) are generally trinomials because equations (9) and (22), Chapter III, are tri- nomials. And these are so because of the geometrical character- istics a, X and IJL. In order to have a monomial expression for velocity or pressure X and n must both be zero. But upon this supposition (22), Chapter III, would become or, the fraction of grain burned would be directly proportional to the thickness of layer burned; which is impossible, since the grain burns on all sides. This same supposition would also make I.S.-V, which is not true, at least for finite volumes. It has been shown in Chapter III, that all grains which, under the parallel law of burning, retain their original form until wholly consumed and for which 5">o, have one or the other of the following expressions for a, namely, i + x or 2 + x, x being the ratio of the thickness of web to the length (or breadth) of the grain. Only grains for which a = i + x can give approximate monomial expressions for velocity and pressure, and this by making x so small that it may be omitted in comparison with unity, in which case a becomes practically unity and X and /* zero. To this class belong thin, flat grains and long cylindrical grains with axial perforation. When a = i and X and /z are zero, equations (i) and (2) be- come v 2 = M X, . (7) APPLICATIONS 10 1 and p = M'X Z = [7.82867 - 10] M X 3 (8) a & Also, by equation (3), since a = i, we have, after the powder is all burned, It will be seen that the pressures by the monomial formula are directly proportional to X 3j which therefore gives, to the proper scale, the typical pressure curve. Its maximum value, as seen from the table of the X functions, occurs when x = 0.64, and its logarithm is 9.86390 10.* Applying this in equation (8) gives for the maximum pressure, IV p m = [7.69257 - io]M .... (10) If the maximum pressure, assumed' to be the crusher-gauge pressure, is known by experiment, we may compute M from the last equation. Thus we have ,, r i a * P M = [2.30743] ^- .... (n) Substituting this in (7) and (8) we have while the powder is burning, and p = [0.13610] p m X, ..... (13) Since by (12) the velocity is proportional to \/X lt this func- tion represents the typical velocity curve while the powder is * The maximum value of X 3 occurs when x = 0.6336+- But the value of x given above is near enough for all practical purposes. It may be noted here that the curve of X 3 has a point of inflection when x 1.3891. 102 INTERIOR BALLISTICS burning. After the powder is all burned the monomial formulas and _P^_ M'X , } ~ ' are to be employed. Example. As an example of monomial formulas for velocity and pressure take the following data from "Notes on the Con- struction of Ordnance," No. 89, pages 43-47: Gun: 8-inch B. L. R., Model 1888. ^ = 3617 in. 3 ; u m = 205.25 in. Powder: Nitrocellulose composition, single-perforated grains of the following dimensions: length (m) 47.69 in.; outside dia- meter (2 R) 0.4455 in.; diameter of perforation (2 r) 0.1527 in. From these dimensions we find by the formulas of Chapter in., 2 jf. - - (0.4455 - 0-1527) = 0-1464 in. * = ^f = 0.0030525. a = I + x = 1.0030525. x X = ^ ^ = 0.0030433. fj, = O. We may, therefore, in this case, assume a = i and X = o without material error, and employ the monomial formulas (7) and (8), computing M either by (9) or (n), according as we take the observed muzzle velocity or crusher-gauge pressure for this purpose. If the crusher-gauge pressure (assumed to be p m ) is employed equations (12) and (13) may be used. If it is known that the powder is all burned at, or near, the muzzle (9) becomes M = ^ (9') APPLICATIONS 103 in which both symbols in the second member refer to the muzzle. If the charge is not all consumed at the muzzle and we know the value of v c for the powder used, X can be found by (69), Chapter IV., and then ^V can be computed by (9). Finally if v c is not known equations (12) and (13) must be employed. As an example one shot was fired with a charge of 78 Ibs., and a projectile weighing 318 Ibs. The observed muzzle velo- city was 2040 f. s., and crusher-gauge pressure 30450 Ibs. per in. 2 , and it was known that the combustion of the charge was practi- cally complete at the muzzle. From the given data we find (taking 5 = 1.567), A = 0.5969, log a = 0.01584, and z = 44.548 in. Therefore, u m 205.25 and from Table i, for this value of x m , \QgX = 0.77147 logXi = 0.41207. \QgX 2 = 9.64060 10. .'. log If = 2 log 2O40 0.41207 = 6.20719. Also by (61), Chapter IV, log M' = 4.63036. The equations for the velocity and pressure curves for this shot are, therefore, v = [3-10359] V^i ..... (16) and /> = [4.63036] X 8 . . . . (17). The first of these equations gives, of course, the observed muzzle velocity; and the second gives (by taking x = 0.64) a maximum pressure of 31208 Ibs. per in. 2 , exceeding the crusher- gauge pressure by 758 Ibs. If we determine the value of M by means of the crusher-gauge pressure we shall have by (n), log M = 6.19652; and the equations for velocity and pressure now become v = [3.09826] VX l 104 INTERIOR BALLISTICS and p = [4.61969] X 3 This last equation gives the observed crusher-gauge pressure while the first makes the muzzle velocity 25 f. s. less than the observed. As muzzle velocities can be more accurately measured than maximum pressures, the first set of formulas are probably the more accurate and will be used in what follows in preference to the other set. The expression for fraction of charge burned at any travel of projectile is found from (70), Chapter IV., and is for this example, v 2 k = [3.02134 -- 10] Y" ..... ( l8 ) The travel of projectile is given by the equation u = z x = 44.548 x inches .... (19) The following table computed by means of equations (16), (17), (18), and (19), is represented by the curves v and p in Fig. i. X Travel U inches Velocity V ft. sees. Pressure P Ibs. per in. 2 Fraction of charge burned. k Pressure P Ibs. per in. 2 Velocity ft.- sees. 0.0 O.O 0.0 00 0.0 84084 0.0 O.2 S.QIO 379-7 26II5 0.257 65939 749-3 0.4 17.819 600.2 30260 0-357 53687 1005 . i 0.6 26.729 770-3 3H94 0.430 44931 1175.0 0.8 35.638 910.1 30949 0.489 38402 1301.6 I.O 44-548 1029.2 30214 0-539 33369 1401.5 1.2 53-457 II33-I 29281 0-583 29387 1483-4 i-4 62.367 1225.2 28285 0.623 26167 1552.3 1.6 71.277 1308.0 27288 0.659 23519 1611.5 1.8 80.186 1383-2 26322 0.692 21306 1663.1 2.0 89.096 1452.0 25401 0.722 19434 1708.6 2-5 111.370 1602.2 23322 0.790 15823 1802.8 3-0 I33-644 1729.3 21548 0.849 13242 1877.0 3-5 I55-9I8 1839.1 20034 0.901 II3I8 1937-5 4.0 178.192 1936.0 18733 0.948 9834 1988.2 4-5 200 . 466 2022 . 5 17606 0.991 8661 2031.5 4.6073 205.250 2040 . o 17384 1. 000 8441 2040 . o APPLICATIONS I0 5 The last two columns in the table represented by the curves V and P, Fig. i, show the velocity and pressure upon the supposi- tion that the powder was all converted into gas at the tempera- FlG I. ture of combustion before the projectile had moved. They were computed by the formulas V* = V*X 2 = [6.97866]*, .... (20) and [4.92471] _ ' , . The force of the powder (/) and the velocity of combustion in free air (v c ) for this particular charge and brand of powder can now be computed by equations (64) and (67), Chapter IV. We find/ = 1396.9 Ibs. per in. 2 and v c = 0.13614 in. per sec. If we wish to compute velocities and pressures in this 8-inch gun when the charge varies K must be computed by (90) and M by (91), Chapter IV. Since the weight of projectile is constant 2 n' is zero; and for an 8-inch gun we will assume that n = , O 106 INTERIOR BALLISTICS this assumption to be tested by experiment. With these values of n and n' we have and therefore, / = 76.519 & ..... (22) Substituting this value of K, and the gun and powder con- stants in (91), we have M = [2.09974] a" w v ..... (23) Also, from (61), Chapter IV, M'= [2.43084] ^ .... (24) Therefore from (7), (8), and (10), v = [1.04987] a 1 u^VX'j. .... (25) " X p = [2.43084] r ...... (26) and w = [2.29474]! ..... (27) which are the formulas for velocity and pressure for this gun and brand of powder in terms of the weight of charge. As an example, what would be the maximum pressure with a charge of 79^ I DS ? We first find A = 0.6084 an d then log a = 0.00238. We then have by (27) log p m = 2.29474 + log 79.5 - - log a = 4-5 Io6 5 * Pm = 3 2 4o8 Ibs. per in. 2 This agrees very closely with observation. We have the means of testing the accuracy of these equations to a limited extent, since there were four shots fired with charges of 70, 78, 85, and 88 Ibs. The following table gives -the results of all the necessary preliminary calculations for the four shots APPLICATIONS 107 fired and also for two others " estimated from prolonged empirical curves." The data from the shot fired with a charge of 78 Ibs. have been taken as the basis of the calculations. The gun constants will be found on page 102. AT MUZZLE CO A log a log Z Ibs. X logXo logX t logX 2 60 0.4592 0.18742 .70647 4-0347 0.74978 0.36944 9.61965-10 70 0-5357 0.08940 .67540 4-3339 0.76150 0.39260 9.63110 78 0.5969 0.01584 .64883 4.6073 0.77147 0.41207 9.64060 85 0.6505 9.95382 .62414 4.8769 0.78066 0.42987 9.64919 88 0.6734 9-92777 6I3I5 5.0018 0.78475 0.43770 9.65295 95 0.7370 9.86767 .58629 5-3210 0.79467 0.45663 9.66196 The computed muzzle velocities and maximum pressures in the following table were obtained (witn the exception of the first two muzzle velocities) by equations (25) and (27). The values of / were computed by (22) and X by (69), Chapter IV. CO Ibs. log X f Ibs. per inch 2 MUZZLE VELOCITY, F. S. MAXIMUM PRESSURES, LBS. PER IN. 2 Observed Com- puted O.-C. Observed Com- puted O.-C. 60 70 78 85 88 95 0.74265 0.75819 0.77147 0.78382 0.78931 0.80274 H73 1306 1397 1479 I5H 1593 1600 1839 2040 2200 2275 2454 1600 1844 2040 2205 2276 2441 ~~ 5 - 5 i + 13 18000 24889 30450 35600 39301 47280 18859 25272 31206 37051 39756 46583 -859 -383 -756 -1451 -455 +697 For the first two shots the powder was all burned before the projectile had reached the muzzle, as is shown by the values of log X . For these the muzzle velocities were computed by equation (14). It will be observed that the equations by means of which the muzzle velocities and maximum pressures given in this table were computed depend for their constants upon one io8 INTERIOR BALLISTICS measured velocity only, due to a charge of 78 Ibs. The measured crusher-gauge pressure for this charge has not been made use of at all. The constant M upon which all the other constants depend might have been determined by equation (n) in which the muzzle velocity does not enter. But muzzle velocities can be more accurately measured than maximum pressures and are, therefore, better adapted to the determination of ballistic constants. The computed maximum pressures in the table are probably nearer the actual pressures on the base of the projectile than those given by the crusher gauge. The agreement between the computed and measured muzzle velocities is all that could be expected from any ballistic formulas. To determine the travel of projectile when all the charge was burned we take x by interpolation from the table of the X functions corresponding to the values of log X . We then have : u = .x z . For the travel of projectile when the pressure is a maximum. we have, calling this travel u f , u' = 0.64 z . The following table gives the values of u' and u for all the shots: ft u' u u m u k Remarks Ibs. inches inches inches 60 32.56 196.56 8.69 70 30-3I 201.13 4.12 78 28.51 205.25 0.00 1. 0000 85 26.94 209.30 - 4.05 0.9926 88 26.26 2II.I6 - 5.91 0.9895 95 24.69 215.88 -10.63 0.9813 It will be observed that as the charge increases the sooner it exerts its maximum pressure. The last column gives the fraction of the charge burned at the muzzle and shows that APPLICATIONS approximately the entire charge for the series was consumed at the muzzle, k was computed by the formula v 2 k = [6.I7483-IO] ,1 y s A 2 In order to determine the velocity and pressure curves for any given charge we should compute M and M ' by equations (23) and (24), and then employ (7) and (8) as has already been done for a charge of 78 Ibs. For example, determine the velocity and pressure curves for a charge of 95 Ibs. We have, from (23) and (24), log M = 2.09974 + log a + log co = 6.31864 log M' = 2.43084 + ^ log - - log a = 4.80435 Therefore v = b^spa 2 ! VXi and p = [4-80435] X* are the equations required. Example. Suppose the thickness of web of the grain we have been considering to be increased 10 per cent., all other conditions remaining the same. Deduce the velocity and pressure curves for a charge of 78 Ibs. In this case it is evident that all the charge would not be burned in the gun and that in consequence both the maximum pressure and muzzle velocity would be diminished. It will be seen from (69), Chapter IV, that, other things being equal, the value of X varies directly with the web thickness. Therefore if this is increased by 10 per cent., or, what is the same thing, is multiplied by i . i , X will also be multiplied by i . i ; and from (60) and (61), Chapter IV, M and M f will be divided by i.i. Therefore (16) and (17) will in this case become, v = [3.08290] HO INTERIOR BALLISTICS and P = [4.58897] *8. These equations give v m = 1945 f. s., and p m = 28371 Ibs. This is a loss of 95 f. s. in muzzle velocity and a diminution of 2079 Ibs. in maximum pressure. To determine the fraction of the charge burned at the muzzle, we have from (45), Chapter IV, k- ~ which gives, by employing the muzzle velocity just computed, k = 0.909. Therefore on account of the increased thickness of web, seven pounds of the charge remained unburned when the projectile left the gun. We may next inquire what effect a decrease of 10 per cent, in web thickness would have upon the muzzle velocity and maximum pressure. In this case we must multiply the original value of X by 0.9 and divide M and M ' by the same fraction. We thus get log X = 0.72571 logM = 6.25295 log M' = 4.67612 Therefore, from (9), the muzzle velocity in this case is found to be 2040 f. s.; and, by (10), the maximum pressure, 34675 Ibs. That is, the muzzle velocity remains the same while the maximum pressure is increased by 4225 Ibs. per in. 2 These examples show that for the greatest efficiency (muzzle velocity and maximum pressure both considered), the web thickness for this form of grain should be such that the charge is all consumed at the muzzle. From the value of X given above we find, by interpola- tion, that x = 3.4890; and, therefore, u = 155.43 inches. For this travel the above values of M and M' give v = 1936 f. s., and p = 22294 Ibs. The muzzle pressure comes out 8433 Ibs. APPLICATIONS III Suppose for a hypothetical 7 -inch gun we assume the follow- ing data: d = o". 7 V c = 4,000 c. i. u m = 40 calibers = 280 inches. A = 0.6. 5 = 1.5776 / = 1396.9 Ibs. v c = 0.13614 in. per sec. w = 205 Ibs. What muzzle velocity and maximum pressure would be obtained, supposing the charge to be all consumed at the muzzle; and what must be the thickness of web? The weight of charge due to the given chamber capacity and density of loading is found to be 86.7 Ibs. We next compute the following numbers by formulas given in Chapter IV: log a = 0.0122 1 Iogz = 1.80713 *m = 4-3 6 55 log X= log X om = 0.76269 log Xi= logX lm = 0.39492 log JV- 7-21529 (By (58), Chapter IV), log M = 6.45260 (By (9)) Then by (7) and (10) we find Muzzle velocity = 2653 f. s. and Maximum pressure = 32112 Ibs. per in. 2 The muzzle pressure, by (8), is 18413 Ibs. The necessary thickness of web in order that the charge may all be consumed at the muzzle, is 0.158 inches. The other dimensions of the grains are immaterial. If the volume of the chamber is taken at 3,000 c. i., all the 112 INTERIOR BALLISTICS other data remaining the same, we should have the following results : w = 65.03 Ibs. M . V. = 2413 f. s. p m = 28868 Ibs. per in. 2 M . P. = 14105 Ibs. per in. 2 2l = 0.152 in. If v c = 4500 c. i., we have the following: co = 97.54 Ibs. M. V. = 2753 f. s. Pm= 33574 Ibs. per in. 2 M . P. = 24496 Ibs. per in. 2 2 1 = 0.160 in. Binomial Formulas for Velocity and Pressure. Binomial formulas pertain to grains for which /* is zero or so small that it may be neglected, while X must be retained on account of its magnitude. To this class belong all unperforated, long, slender grains of whatever cross-section, such as strips, ribbons, cyl- inders, etc. The binomial expressions for velocity and pressure for these grains are v 2 = MX, {i - NX } .... (28) and p = M'Xi{i- NX 4 ] .... (29) The second term within the brackets has the negative sign because X is always negative for these forms of grain. Methods for Determining the Constants M and N. The constants M and N can be determined when the given experi- mental data are such that two independent equations can be formed involving M and N. These data may be either two measured velocities of the same shot at different positions in the bore; or a measured muzzle velocity and crusher-gauge pressure, the latter taken as the maximum pressure. In APPLICATIONS 113 addition to these all the elements of loading, as well as the powder and gun constants, are supposed to be known. First Case. Let Vi and v 2 be two measured velocities in the bore at the distances u^ and u 2 from the origin, which is the base of the projectile in its firing position. From the gun and firing constants compute z by (56), Chapter IV, and then Xi and x 2 corresponding to Ui and u 2 by the equation u x = Z With these values of Xi and x 2 as arguments, interpolate from the table of the X functions the corresponding values of log X and log Xi, distinguishing them by accents. We then have the two independent equations r, 2 US V " CT AT V ff \ V 2 = M A! (i J\ X ) from which M and ^V may easily be determined. For simplicity let ^ \ ~ I ' ~v7^ an d &' = ~v7T VV A , A We then have in a form well adapted to logarithmic computa- tion N T ~ b ( \ ~ (i - bV) X" ( $ 0) and ~ X\(i - N X' ) X'\(i - N X" These equations are equally adapted to English or French units. Second Case. When the powder is not all burned in the gun let v m be the observed muzzle velocity and p m the crusher- gauge pressure. We then have the two independent equations U4 INTERIOR BALLISTICS and ((50'), Chapter IV), p m = [9.85640 - 10] M' (i - [0.48444] AO . (32) Substituting for M' its value in terms of M ((61), Chapter IV), and making, for English units, WlPm c == [7-68507 -- 10] ai - ) p m x i we have N " X - 3.051 c " (i -[o. 4 8 444 ] which is a mean of the last four values. Since we find v= 605.1 m. s. N and M can now be computed by (30) and (31), which do not contain the form characteristics. The data are: ^= 543.1 m. s., v 2 = 605.1 m. s., logX' = 0.74183, log *",= 0.80284, log A"! =0.35356, and logX'\ = 0.47205. The results of the calculations are: log N = 8.68636 - 10 1 log M = 5.25171 I While powder is burning. log M' ' = 3.62063 F = [2.94719] VX 2 } _ [3.7^589 I After powder is all burned. (* + *)* J The following table shows the agreement between the ob- served and computed velocities: APPLICATIONS 129 i Travel VELOCITIES of o.-r. Remarks Projectile Observed Computed O.88O m. 543.1 m. s. 543.1 m. s. o.o 1.051 5744 572-8 1.6 1.222 595-0 597-4 -2.4 1-393 612.6 613.1 -0.5 1.564 622.3 623.9 -1.6 1.792 636.5 636.3 0.2 2.O20 648.3 646.5 1.8 The greatest of these differences is less than one-half of one per cent, of the observed velocity and the others are practically nil. The maximum pressure computed by the formula p m = [9.85640 - 10] M'{i - [0.48444] N} is 2555 kilos, per cm. 2 , differing by less than one- third of one per cent, of the mean crusher-gauge pressure. These results show that the assumed value of x= 5.6 is practically correct. Finally, we have, / = 7883 kg. per cm. 2 and v c = 0.438 cm. per sec. - 0.172 in. " " The form characteristics a and X can be computed by the formulas MX. V? and X = N X, From these we find a. = 1.4460 and X = 0.3045. For the B $ 144 powder the equations for the velocity and pressure curves are found, by a process entirely similar to the above, to be, while the powder is burning, v 2 = [5-24187] X, (i - [8.75166 - 10] X ) and p = [3-56310] ^3 (i - [8.75166 - 10] X<) 9 130 INTERIOR BALLISTICS After the powder is burned the equations become V = [2.92007] VX 2 [3-68425] and The following table shows the agreement between the observed and computed velocities: Travel Observed Computed of Projectile Velocity Velocities o.-c. Remarks m. m. s. m. s. 0.880 5037 503.9 O.2 I.05I 534-9 532.2 2.7 1.222 553-1 553-6 -0-5 1-393 565-0 566.0 1.0 1.564 573-5 576.5 -3-o 1.792 591.0 588-4 2.6 2.020 600.7 598.5 2.2 The value of /for B N ni comes out 8001.5 kilos, per cm. 2 , and D C is found to be 0.5268 cm. per sec. This powder is therefore slightly "stronger" than B NI and about 22 per cent, quicker; and this notwithstanding its greater density. Application to the Magazine Rifle, Model of 1903. The following data pertaining to this rifle were obtained partly from a descriptive pamphlet issued by the Ordnance Depart- ment, and partly through the courtesy of officers of the Ordnance Department on duty at the Springfield Armory and Frankford Arsenal, to whom the writer is under special obliga- tions : Caliber, 0.3 inches. Volume of chamber, 0.252 cubic inches. Total travel of bullet in bore, 22.073 inches. Mean weight of powder charge, 44 grains. Weight of bullet, 220 grains. "The standard muzzle velocity of this ammunition is 2300 APPLICATIONS 131 f. s., with an allowed mean variation of 15 f. s. on either side of the standard. The powder pressure in the chamber is about 49,000 pounds per square inch." The powder used with this rifle is composed essentially of 70 per cent, nitrocellulose and 30 per cent, nitro-glycerine. "The grains are tubular, being formed by running the powder colloid through a die 0.09 inch in diameter, with a pin 0.03 inch in diameter; and the string thus made is cut 21 to the inch." There are considerable variations in the length and diameter of the grains "due to the fact that the string is not cut exactly perpendicular to its axis, and to irregularities in shrinking. There are 83,000 to 91,000 grains per pound. The specific gravity is about 1.65, and the gravimetric density is from 0.90 to 0.94." On account of the tubular form of the grains the character- istic M is zero, and therefore the equations for velocity and pressure are binomials. We have reliable measured in- terior velocities for this rifle, obtained at the Springfield Armory in the fall of 1903, by firing with a rifle the barrel of which was successively cut off one inch. Five shots (some- times more) were fired for each length of barrel and the velocities were measured at a distance of 53 ft. from the muzzle, and reduced to muzzle velocity by well-known methods. (See Table A.) It is known that the charge in the magazine rifle is all burned at, or very near, the muzzle. We may, therefore, take the two extreme reduced velocities of the series for Vi and v 2 and thereby minimize the effects of errors in measuring the velocities. The firing data are then, Vi= 1274 f.s.; v 2 = 2277.6 f. s. u i = 3-73 m - ; ^2 = 20.073 m - The weight of charge in these firings was 45.1 grains and weight of bullet 220 grains. The preliminary calculations give 132 INTERIOR BALLISTICS A = 0.7077 log a = 9.90686 10 log z = 0.30878 .'. z = 2.036 in. Xl = 1.5093, loX' = 0.57969, logX\= 0.00146. x 2 = 10.84125, \ogX" = 0.90504, \Q^X'\ = 0.65421. These numbers and the velocities ^ and v z , substituted in (30) and (31), give log N = 8.73379 - 10 and log If = 6.30896. We also find log If' = 4.91902. The formulas for velocity and pressure are, therefore, v* = [6.30896] X, {i - [8.73379 - 10] X ] p = [4.91902] X 3 { i - [8.73379 - 10] X 4 } We have MX, and this substituted in (15), Chapter IV, gives y = > - -X [i - [8.73379 - 10] Xo\ V2 Since in this case v and X 2 refer to the muzzle, we have for the powder burned, in grains, y = [0.99736] X { i - [8.73379 - 10] X } or, in another form more convenient for computation, v 2 y = [4.68840 - 10] TT A 2 Table A gives the measured and computed velocities for the travels of projectile in the first column, and also the weight of powder burned at each travel. APPLICATIONS TABLE A Travel of Projectile, inches Mean Velocity 53 Feet from Muzzle, f. s. Muzzle Velocity Deduced from Measured f. s. Computed Velocity, f. s. o.-c. Powder Burned, grains 3-073 1253 1274 1274 29.99 4-073 1402 1426 1432 - 6 32.61 5-073 1531 1558 1555 3 34-63 6.073 1633 1662 1656 6 36.25 7-073 1742 1772 1740 32 37-59 8.073 1771 1802 1812 10 38.71 9-073 1860 1894 I8 74 20 39-66 10.073 1909 1943 1929 H 40.48 11.073 1957 1992 1976 16 41.19 12.073 1989 2023 2018 5 41.81 13-073 2016 2052 2057 - 5 42.36 I4-073 2050 2086 2091 - 5 42.83 15-073 2069 2105 2122 -17 43-25 16.073 2104 2140 2151 ii 43-63 17.073 2129 2165 2177 12 43-95 18.073 2183 2219 2200 19 44-25 19.073 2163 2200 2222 22 44-5 20.073 2201 2238 2242 - 4 44-73 21.073 2203 2240 226l i 44-93 22.073 2240 ' 2278 2278 o 45-io Table B, on page 134, supplements Table A by giving com- puted velocities and pressures from the origin of motion. The velocity curve in the diagram, Fig. 2, on page 135, shows at a glance the agreement between theory and observation. It will be observed that the computed pressures depend entirely upon two measured velocities. Also that the maximum pressure occurs when x = 0.45, and agrees with the official statement. The muzzle pressure is about 6,000 Ibs. per in. 2 Powder Characteristics. The form characteristics of these grains according to the given dimensions are a= 1.63 and X = 0.3865. But these minute grains, of which there are 560 in the service charge, shrink irregularly and many of them doubtless are more or less abraded and perhaps broken, so that it is impossible to determine the mean values of a and X geometrically with any 134 INTERIOR BALLISTICS TABLE B X u, inches Computed Velocity, f. s. Computed Pressure, Ibs. per inch 2 Powder Burned, grains Pressure on Base of Projectile, pounds O.OOO 0.000 0.000 o.ooooo 0.000 OOO 0.001 0.002 8.590 4501 1.081 318 O.OI 0.020 47.854 13835 3.375 978 O.I 0.204 254-91 37008 10.139 2556 0.2 0.407 408.98 45117 13.842 3189 o-3 0.6II 531.70 48456 16.475 3425 0.4 0.814 635.14 49640 18.554 3509 0.45 0.916 681.48 49769 19.454 3518 0.5 I.OI8 724.89 49695 20.283 0.6 1.222 804.25 49118 21.766 0.8 1.629 939-77 47039 24.221 I.O 2.036 1052.5 44492 26.205 1.2 2-443 1148.8 41882 27.867 '.'.'.'. 1.4 2.850 1232.6 39369 29.291 1.6 3.258 1306.4 37023 30.533 2.O 4.072 32854 2.5 5.090 28^ SO 3 6.108 ^^oo 25061 4 8.144 19812 5 I0.l8o 16086 6 I2.2I6 IT. T.IQ 7 14.252 OO :/ III88 8 16.288 9501 9 18.324 8134 10 20.360 7006 ii 22.396 604. S T^\J certainty. They may, however, be deduced from the values of M and N. From these we find Finally we find and a = 1.7710 x = -4353 / = 1622.5 Ibs. per in. 2 v c = 0.28 in. per second. Formulas for Designing Guns for Cordite. The caliber, of course, is given, and the weight of the projectile of desired length APPLICATIONS and form of head can be computed by known methods.* The grain characteristics and density of cordite and the values of / ID 15 Travel, inches. FIG. 2. 20 22.07 and v c are also known. The necessary formulas for this discussion, given in the order in which they will be used, are the following: & = [8.55783] A V c (a) z = [1.54708] (c) * See the author's "Handbook" (Artillery Circular N), chapter xi. 136 INTERIOR BALLISTICS v:~= [4.44383]^ . . . . . . . to L v u ' ) in which & _ [2.i28 9 o]/ This equation is deduced from (32), eliminating M' and N. 2 V 2 I X ' ~ 2 X ' M'= [7.82867 - io]M^ (g) _ _. __ p ,. _ "i / 7 \ # = M ' ^ 3 (i - N X,) (i) p m = [9.85640 - 10] M' (i - [0.48444] N) . . (j) 1 = [8.56006 - 10] Example. Take the hypothetical y-inch gun already con- sidered on page in, for which d = j ff and w = 205 Ibs. For cordite of o".3 diameter we found / = 2521 Ibs. per in. 2 , and v c = 0.309 in. per sec. Also 5 = 1.56. The only assumptions necessary are the volume of the chamber (V c ) and the density of loading. And this last is not purely arbitrary, since considera- tions of safety to the gun and its efficiency restrict its value to narrow limits, say from 0.4 to 0.6. This latter value is often exceeded, especially in our service; but it is believed that by choosing the proper shape and size of grain this can always be APPLICATIONS 137 avoided. As / and v c are unusually large for cordite, we will take A = 0.4; and for a first assumption will give the chamber a volume of 3,000 c. i.; which is less than the volume of the chamber of the 6-inch wire- wound gun. Finally we will take Pm= 37>oo Ibs. per in. 2 , leaving the muzzle velocity and travel in the bore for later consideration. From the given data we find, by means of the above formulas, w = 43-35 2 Ibs-, lo g# = 0.26928, log z = 1.76317, log Fi 2 = 7.17066, log X = 0.90184, log M = 6.56985, log M f =4.80398 and log TV = 8.79713 10. The equations for velocity and pressure are, therefore, v*= [6.56985] X, (i - [8.79713 - 10] X ) and p = [4-80398] X 9 (i - [8.79713 - 10] X 4 ) This last equation, which is the same as equation (/) when x = 0.45, makes p m = 37000, thus verifying the calculations. The muzzle velocity will, of course, depend upon where we place the muzzle, in other words upon the value adopted for u m . If we regard 40 calibers as a suitable travel in the bore, we shall have u m = 40 X 7" = 28o /r whence x m = ujz = 4-8305 For this value of x, Table I gives log X = 0.77912 log Xi = 0.42689 and these in the above velocity equation give v m = 2487 f. s. We find, from Table I, taking log X as the argument, x = 10.613; 138 INTERIOR BALLISTICS and by (m) u = 615.19 in. Also by (k) 2/ =o".47 and by (/) k m = 0-9394- That is 94 per cent, of the charge was burned at the assumed muzzle. If the maximum pressure is increased to 38,000 Ibs., the density of loading and volume of chamber remaining as before, the velocity for a travel of 280 inches will be increased to 2503 f. s., and the thickness of web, or diameter of the grain, will be reduced to 0^.45. This slight diminution in the diameter of the grain increases the initial surface of combustion of the charge about 3^ per cent., which fully accounts for the increased maximum pressure. If we take A =0.5, V c = 3,000 c. i. and p m = 37,000 Ibs. per in. 2 , there results co = 54.19 Ibs., v 2S o = 2570 f. s., and 2 / =o".66. Trinomial Formulas for Velocity and Pressure. Trinomial formulas occur when the grains of which the charge is composed are of such form and dimensions that the form characteristic /* cannot be regarded as zero. Spherical, cubical, and multi- perforated cylindrical grains are of this kind. For the first two forms mentioned the second term is negative and the third positive; while for m.p. grains (those used in our service), the second term is positive and the third negative. For spherical and cubical grains we may have, before the powder is all burned, the two independent equations, - N X' +- v, 2 = MX," (i - N X" + -N* X'\] APPLICATIONS 139 Put for convenience, X /f o 3 (i a b) 2 c (i a) C = 2(1 -ab*}X", = (i -abYx 7 ^ * Then the quadratic equations give, using the sign applicable to this problem, --^) 5 ) . . ( 4 i) The value of M may now be computed by either of the above expressions for v 2 . Or, if V m is the muzzle velocity, that is, if the powder is all burned in the gun, M may be computed by the formula, derived from (3), M = ~ (42) ^2 If the powder is not all burned in the gun and our data are a muzzle velocity and the crusher- gauge pressure (assumed to be the maximum pressure), N may be computed by the follow- ing process: Compute the auxiliary quantities b, c, and d by the formulas: 5 = [7.68507 - 10] ^r^x','- , bXi 2 X. li -^r d = - 3(I - Then ( / j\ *i .... (43) The functions X 4 and X 5 pertain to the tabular value of x which gives the maximum pressure. If we take this to be 0.45 140 INTERIOR BALLISTICS no material error will ensue. We therefore have log Z 4 = 0.48444 log ^ 5 = 0.93587 The function X pertains to the muzzle. It should be remembered that equations (41), (42), and (43) are applicable to cubical and spherical grains only. Application to Noble's Experiments with Ballistite. The ballistite consisted of equal parts of dinitrocellulose and nitro- glycerine and was in the form of cubes 0.3 of an inch on a side. The gun, powder, and firing data are as follows : d = 6 inches, A = 0.4, 6 = 1.56, w = 20 Ibs., and w = 100 Ibs. From these we find log a = 0.26928 and Iogz = 0.48190. .'. z = 3.033 ft. The following table, which explains itself, is formed for convenient reference: Observed u, X=U/Z Velocity log X log*! logX 2 Remarks ft. f. s. 16.6 5473 2416 0.79917 0.46513 9.66596-10 21.6 7.121 2537 0.84069 0.54I8I 9.70112 34-1 11.242 2713 0.91049 0.66339 9.75289 46.6 15-363 2806 0.95685 0.73940 9.78256 As the powder was not quite all burned in the gun the ex- treme measured velocities are available for determining N and M by means of (41). The data are Vi = 2416 f. s., v 2 = 2806 f. s., logJT = 0.79917, log X" = 0.95685, log X\ = 0.46513 and logXi" = 0.73940. Substituting these in (41), we find log N = 9.03843 - 10 log N' = 7-59974 - 10 log M = 6.62918 log if ~ 4.88754 The equations for velocity and pressure are, therefore, = [6.62918]*! {1-19.03843-10] * and then =[4.88754] * 3 { i -[9.03843-10] ^+[7.59974-10] x 6 APPLICATIONS 141 The equation for velocity will, of course, give the observed velocities for u = 16.6 ft., and u = 46.6 ft. It should also give the observed velocities, if our method is correct, for u = 21.6 ft. and u = 34.1 ft., and indeed for every point in the bore from the firing seat to the muzzle. The velocities computed for these intermediate values of u are 2536.5 f. s., and 2710 f. s., respect- ively, which differ so slightly from the measured velocities as to be negligible. The maximum pressure, which occurs in this case when x = 0.4, is 39,163 Ibs. per in. 2 ; and the corresponding velocity 865.8 f. s., and travel of projectile 1.213 ft. The distance travelled by a projectile to the point where the powder is all burned is determined by means of X and the table of the X functions. We have for cubical grains for which X is unity, X = i/N. Therefore for this example, log X = 0.96157 Corresponding to this value of log X , we find, by interpola- tion from the table, x = 15.8649, logXi 0.74694, and log X 2 = 9.78536 10. To compute #, we have u = 15.8649 X 3.033 = 48.12 ft. The charge was therefore not all burned in the gun, though the fraction of the charge remaining unburned was exceedingly small, practically zero. To get an expression for the fraction of the charge burned for any travel of the projectile, we have, from (45), Chapter IV, v 2 = k VS X 2 But from (3), for cubic grains, M Fi 2 = -jj. .'. log Fi 2 = 7.11363 (for this example). Therefore for cubic grains, . " MX 2 I4 2 INTERIOR BALLISTICS which for this example reduces to k = [2.88637 - 10] -JJT Applying different velocities and the corresponding values of log X 2 in this formula it will be found that the charge was practically consumed long before the projectile reached the muzzle. Indeed nine teen- twentieths of the charge was burned when the projectile had travelled 16.6 ft. We may also determine k in terms of the travel of projectile by means of the equation k = i - ( i - ^) 3 (Eq. (46), Chapter IV). Or, if we wish to know the distance travelled by the projectile when a given fraction of the charge is burned, we have As an example, suppose k = -. Then Y X = Applying the value of N found for this round and completing the calculations it will be found that one-half the charge was burned when the projectile had travelled one foot. The expressions for V and P for this round are v = [3-55681] vx 2 and [4.89487] The last two formulas give the velocity and pressure upon the supposition that all the powder was converted into gas at temperature of combustion before the projectile had started. APPLICATIONS The initial value of P is found by making x zero. Whence p f = 78,500 Ibs. per in. 2 Finally we find ; = 2338 Ibs. and v c = 0.266 in. per sec. The following table was computed by the formulas de- duced for this round for comparison with the deductions from Sir Andrew Noble's velocity and pressure curves. Unfortunately these curves, as published, are drawn to so small a scale and are so mixed up with other curves that it is difficult to get the velocities and pressures from them with much precision. NOTE : The velocities and pressures in the second and third columns were computed by formulas slightly different from those deduced above. But the differences are so small as to be of no account in the discussion. X u ft. Computed Velocities, f. s. Computed Pressures, Ibs. per inch 2 Pounds of Powder Burned V, f. s. P, Ibs. per in. 2 0.000 o.ooo 0.0 0.0 o.o 78500 0.001 0.003 12.449 4196 0.716 65.786 78397 O.OI 0.030 68.889 12672 2.206 207.40 77466 0.05 0.152 221.52 25254 4-683 457.78 73557 O.I 0.303 359-32 32007 6-357 637.37 69132 0.2 0.607 569-36 37680 8.464 875.21 61560 0.4 1.213 869.33 39439 10.967 1174.0 50122 0.6 1.820 IO87.2 37567 12.549 1372.5 41948 0.8 2.427 1257-7 34846 13.686 1520.3 35852 1.0 3-033 1396.6 32050 14-557 1637.0 3H53 2.O 6.066 1842.4 21268 I7-043 1995-9 18143 3.0 9.100 2093.0 15013 18.225 2192.5 12363 4.0 12.133 2257.1 11164 18.892 2322.4 9181 5-o 15.166 2374.2 8645 19.299 2416.9 7200 5-473 16.6 2419.0 7742 19-435 2453.5 6507 7.121 21.6 2538.0 5500 17.742 2554-9 4809 11.242 34-1 27IO.O 2893 19.979 2711.8 2782 I5-363 46.6 2806.0 1890 19-999 2806.0 1890 The computed velocities in the third column of this table, corresponding to the travels of projectile in the second column, 144 INTERIOR BALLISTICS agree very well with those deduced from Sir Andrew Noble's velocity curve, from the origin of motion to the muzzle, a distance of 46.6 ft. As the velocities are thus shown to be correct, the pressures in the fourth column are, from their manner of deriva- tion as given in Chapter IV, necessarily correct also. That is, they correspond to the energy of translation of a hundred-pound projectile. In this respect they are more accurate than the pressures given by Sir Andrew's pressure curve which was de- rived from his velocity curve by graphic methods not sufficiently precise for the great accelerations encountered in ballistic problems. The writer is indebted to Colonel Lissak, formerly Instructor of Ordnance and Gunnery at West Point, for the accompanying diagram (Fig. 3) of the velocity and pressure curves whose co- ordinates are given in this table. Many interesting facts may be gleaned from an examination of these curves, and the formulas by which their coordinates were computed. The two velocity curves v and V are both zero at the origin but immediately separate, attaining their greatest distance apart when the projectile has moved but a short distance. They then approach each other very gradually and become tangent at the point where the powder is all burned, practically at the muzzle. Both curves are tangent to the axis of ordinates at the origin and parallel to the axis of abscissas at infinity. The pressure curve p begins at the origin, attains its maximum when the projectile has traveled about 15 inches, changes direction of curvature when u is about six feet and meets the axis of abscissas at infinity. The pressure curve P is convex toward the axis of abscissas throughout its whole extent. It lies above the curve p from u = o to u = 30 inches (about), then passes below p and the two curves become tangent at the point where the powder is all consumed. Finally the areas under the curves p and P are equal. Example i. Suppose the charge in the example under APPLICATIONS 145 consideration to be increased from 20 to 25 Ibs. Deduce the equations for velocity and pressure. In solving this example, we will compute the new constants M, M f , N and N f by equations (80) to (83), Chapter IV; and as the charge is increased by 25 per cent., a new value of /must be found by (90) and (go'). For a six-inch gun we will take n = t o provisionally; and since the weight of the projectile remains the same, n' must be zero. We therefore have K = 86.12, and the new value of / is 2518 Ibs. The new values of a and z for co = 25 Ibs., are log a = 0.13321 Iogz = 0.44277 .*. z = 2.772 ft. Applying these numbers in the equations above mentioned we find for a charge of 25 Ibs., logM= 6.73881 log If = 5.03633 log N = 9.01885 logN' = 7.56058 which give the equations required. These constants give p m = 55676 Ibs., and a velocity of 2841 f. s., for a travel of 16.6 ft. That is, an increase of 5 Ibs. in the charge increases the maximum pressure 16,800 Ibs. per in. 2 , and the velocity at 16.6 ft. travel, 425 f. s. Taking the reciprocal of N gives log X = 0.98115 and from the table, x = 18.1425 and u = 18.1425 X 2.772 = 50.29^. 10 146 INTERIOR BALLISTICS The limiting velocity and fraction of charge burned are given by the equations Therefore log Fi 2 = 7.24284 and y = k & = [4-I55 10 ~ IQ ] ~~ rt-a From this last formula we find when u = 16.6 ft., y 24.18 Ibs. The pressure at this point is found to be 10480 Ibs. per in. 2 It is interesting to compare these results with those found with a charge of 20 Ibs. In order to lessen the maximum pressure the grains must be increased in size and thus diminish the initial burning surface. Suppose we increase the size of the cubes from 0^.3 to 0^.5 on a side. Determine the equations of velocity and pressure for a charge of 25 Ibs. An examination of equations (80) to (82), Chapter IV, will show that when the only change in the data is in the thickness of web the new values of M, M' , and N will be found by multiplying the previously determined values of these constants by the ratio of the web thicknesses, in this case by 0.6. We therefore have for 25 Ibs. of 0^.5 cubes log M = 6.51696 log M'= 4.81448 log N = 8.79700 10 log N'= 7.11688 - 10 From these we get p m = 38437 Ibs. per in. 2 and v = 2571 f. s., for u = 16.6 ft. The measured velocity for this travel of projectile was 2416 APPLICATIONS 147 f. s., with a charge of 20 Ibs. of o" '.3 cubes. Therefore by increasing the weight of charge 5 Ibs., and at the same time enlarging the grain from o" '.3 to 0^.5 on a side the velocity is increased 155 f. s., and this without increasing the maximum pressure, though the mean pressure is, of course, considerably increased. The pressure for u = 16.6 ft., with a charge of 20 Ibs. of the smaller grains, was 7741 Ibs.; and with a charge of 25 Ibs. of the larger grains, the pressure for the same travel would be 11181 Ibs. The powder actually burned during this travel of projectile is a little more in this latter case than in the former, and the space in which it has been confined during its expansion is less, both of which facts account for the greater work performed. From equation (19'), Chapter III, it follows that for two equal charges made up of grains of the same form and differing only in their size, the entire initial surfaces of the two charges vary inversely as the thickness of web. Therefore the initial surface of the charge of o. r/ 5 grains is of the initial surface of the same charge of o."3 grains. This accounts for the two charges giving the same maximum pressure. It may be remarked that the same results would have been obtained if the grains had been spherical instead of cubical. Application to Multiperforated Grains. A peculiar diffi- culty arises in the application of any system of interior ballistic formulas to multiperforated grains from the fact that they do not retain their original form until completely con- sumed as do all other forms of grain in use, but each grain breaks up, when the web thickness proper is burned through, into twelve slender rods, or " slivers," which burn according to a different law; and thus two independent sets of formulas become necessary to represent what actually takes place in the gun. It was previously sought to overcome this difficulty by supposing the web thickness to be slightly increased so as to 148 INTERIOR BALLISTICS satisfy the equation of condition a (i + X - M) = i and thus ignoring the slivers.* This method represents quite satisfactorily the actual circumstances of motion so long as the grains retain their original form, but not afterward. It assumes that the slivers are all burned with the fictitious web thickness ; that is, when, in all our guns, the projectile has performed approximately half its travel in the bore ; while it is certain that in most cases with our service powders they are not completely consumed when the projectile leaves the bore. It is necessary, therefore, to divide the entire combustion of the grain into two periods and to deduce formulas that shall represent the law of burning, as well as the circumstances of motion, for each period. From equation (22), Chapter III, we have, for m.p. grains y I ( I k = ~- = a-r-} I +\~j /i- co 1 Q ^ i I which gives the fraction of the charge consumed when any thickness / of the web has been burned, and this without any reference to the law of burning. When / = / Q , that is, when the entire web thickness has been burned, this equation becomes k' = a (i + X - /*) in which k' is the fraction of the charge less the slivers. If we substitute for a, X and n their values for any of our m.p. grains, we shall find for this critical point, k'= 0.85 (about), and therefore the slivers constitute approximately 15 per cent, of the charge. These slivers burn according to another law. We may regard them as slender cylinders whose form character- istics are very approximately a = 2, X = i, n = c. * See Journal U. S. Artillery, vol. 24, p. 196, and vol. 26, pp. 141 and 276. APPLICATIONS 149 We will now deduce formulas for each period of burning. Designate all symbols referring to the point where the grains are converted into slivers by an accent, and those relating to the muzzle including M, M' and N by a subscript m. Equation (n), Chapter IV, becomes, by suitable reductions, , v c \/ a w d> A' = : [8.56006 loj -7j . . . (44) (I IQ in which v c is the velocity of combustion under atmospheric pressure and 1 one-half the web thickness. From (12), Chap- ter IV, we have 1/1 = KX which, when the web thickness is burned through, gives for this critical point, KX' = i Therefore from (44) d*l i) c v 7 a w c = [1.43994] -== . . . (46) A v dWu and / = [8.56006 - I0 ]^^ (460 While the slivers are burning we have from (49) , Chapter IV, K in which X refers to any travel between u' and u m and K = i - (i - k)t If we know the values of X and k for any point, we can 150 INTERIOR BALLISTICS determine N m by the equation Therefore at the point of breaking up into slivers (48) be- comes ~K f T^ f N m = ~r or X' = -- .... (49) . The fraction k f which enters into K' can be computed for a grain of given dimensions by (21), Chapter III; and log K' can be taken from Table II with k' as the argument. Therefore when N m is known X f can be found from (49), and then x' ', taken from Table I with X' as the argument, locates the point where the grains become slivers by the equation u' =x' z ..... (50) In order to determine N m it is necessary to assume a value for k m , or the fraction of the entire charge burned at the muzzle, and check this assumed value by the given maxim am (crusher- gauge) pressure. By (45), Chapter IV, we have, by means of which V l can be determined from muzzle data. The constants M, N and N' to be used in the velocity and pressure formulas from u = o to u' are given by the formulas M = ^, N = and N' = ^T (S 2 ) A o A o X* Finally the value of M m for the travel from u' to u m is given by the formulas, deduced from (3), M m =^N m VS = 4 N m VS . . . (53) As an example of this method we will take the mean crusher- gauge pressure and muzzle velocity of five shots fired March 14, APPLICATIONS 151 1905, with the 6-inch Brown wire gun, by the Board of Ordnance at Sandy Hook. The gun had been previously fired twenty-six times with charges varying in weight from 32 Ibs. to 69 Ibs., and at this time was very little eroded. The gun data are as follows : V c = 3120 c. i. d = 6 inches u m = 252.5 inches (total travel in bore). The firing charge for these five shots was 70 Ibs. of nitro- cellulose powder, with 8 ounces of black rifle powder at each end of the cartridge for a primer. As it is impossible to isolate the action of each kind of powder, we will consider the charge in its entirety and take w = 71 Ibs.* The projectiles varied slightly in weight from 100 Ibs. (about one-quarter of one per cent.) ; but no material error will result if we make w = 100 Ibs. The mean muzzle velocity (v m ) was 3330.4 f. s., and the mean crusher-gauge pressure (p m ) was 42497 Ibs. per in. 2 The charges were made up of m.p. grains designed for an 8-inch rifle, and of the following dimensions: R = 0^.256; r = o".o255; m = i".c29. And, therefore, 1 = 0^.044875; a 0.72667; X = 0.19590; JJL = 0.02378; k f = 0.85174. The granulation of this powder is 89 grains to the pound. The volume of a single grain computed by (15), Chapter III, is 0.197144 c. i.; whence by (23'), Chapter III, d = 1.5776. From these data are found by methods already fully illustrated, A = 0.6299 log a = 9.97940 10 Iogz = 1.82144 .'. z = 66.289 in. \ogX om = 0.74029 * Gossot recommends to increase the weight of charge by one-third that of the igniter. But there is no practical difference in the results by the two methods. 152 INTERIOR BALLISTICS We will assume k m = 0.973. Therefore from Table II, \ogK m = 9.92204 - 10, and log K f = 9.78885 - 10. From (48) we have K m E! ... N = 7JT = Tx r (54) * -^~om * ^ x o whence vf K X om , . .'. \QgX' = 0.60710 We now find from the preceding formulas, log VS= 7.44669 log M = 6.70093 log M ' = 4.69894 log N = 8.68493 ~~ I0 logA 7/ = 7.16201 10 Substituting these values of M', N and N' in equation (51), Chapter IV, gives p m = 42521 Ibs. per in. 2 , differing insensibly from the mean crusher-gauge pressure. The assumed value of k m is therefore correct. We now find from (54) and (53), \ogN m = 8.88072 - 10 logM w = 6.92947 The value of x' taken from Table i, by means of log X' , is x'= 1.757. Therefore u' = 1.757 X 66.289 = 116.47 inches. The two sets of equations for velocity and pressure for a charge of 70 Ibs., and primer of i pound, are: From u = o to u' = 116.47 inches: v*= [6.70093] X t {i + [8.68493 - 10] X - [7.16201 - io]X 2 t p = [4.69894] X 3 {i + [8.68493 - 10] X*- [7.16201 -- 10] X b From u' = 116.47 in. to muzzle: &= [6.92947] X, {i - [8.88072 - 10] X } p = [4-92748] X, {i - [8.88072 - 10] X 4 } } APPLICATIONS P, 1000.J/ 75000 70000- v,V 60000- -3000 55000- -2750 50000- 10000- 30000- 1^000- SWW -250 FIG. 4. 154 INTERIOR BALLISTICS The X functions for the travel u' are log X' = 0.60710 log X\ = 0.06473 log X' 2 = 9-457 6 3 ~ 10 logJT 3 = 9-7933 2 - 10 log ^'4= 0.76498 logX' 5 = 1-48763 Both formulas for velocity give the same velocity for the travel u', namely v f = 2614 f. s. The pressure at this point by the first formula is 38431 Ibs.; and by the second 29324 Ibs. per in. 2 The discontinuity shown by the two curves P and p (see Fig. 4), at the travel u 1 ', where the grains break up into slivers is due to the sudden diminution of the surface of combustion of the grains at this point, whereby the rate of evolution of gas and heat suddenly falls and with it also the pressure. In this particular example the initial burning surface of each grain is 3.2 in. 2 , and goes on increasing until at the point of breaking up the vanishing surface is 4.2 in. 2 It then suddenly falls to about 1.5 in. 2 , which is approximately the surface of the twelve slivers. Of course there is no such absolutely abrupt fall in the pressure as is indicated by the two pressure formulas. Neither can it be supposed that all the grains maintain their original form until the web thickness is completely burned. Nevertheless the two pressure formulas give very approximately the average pressure at or near this point. It might be possible to connect the two pressure curves by another curve of very steep descent; but this is hardly necessary. The characteristics / and v c are / = 1418 and v c = 0.134. These characteristics, computed with the firing data of an 8-inch gun, were found on page 105 to be 1397 and 0.136, respectively. The expression for y (powder burned) is by (45), Chapter IV, y = [4-40457! - APPLICATIONS 155 For the travel u', this formula gives y' = 60.472 Ibs. = 71 k'. At the muzzle, by the above formula, y m = 69.085 Ibs. = 71 k m . If it should be found in any case that the powder was all burned in the gun, it would be necessary to compute X' by the formula X' =K'X, ...... (56) In this case we should assume a value for X (or ~x), and compute the maximum pressure for comparison with the crusher- gauge pressure, following the same steps as before. We will consider a few additional problems illustrative of this method of treating m.p. grains. Problem i. What must be the dimensions of the grains in the example just considered in order that the combustion of the entire charge may be completed at the muzzle? Also what would be the muzzle velocity and maximum pressure? In solving this problem we must first consider the second period of combustion, namely, that of the slivers. It has already been shown that for a charge of 71 Ibs., log VJ = 7.44669. We also have in this case, since k m = i, 2X X being the muzzle value of X ; and by (53), M m = 4 N m VS We thus find for the second period of combustion, v*= [7.00743] X, { i - [8.95868 - 10] X ) \ A P = [5.00544] X, (l - [8.95868 - 10] X,} J The muzzle velocity by the above equation is 3376 f. s., an increase of 46 f. s., due to the combustion of the entire charge in the gun. The muzzle pressure is 11433 ^s. per in. 2 In order to deduce equations for velocity and pressure for the first period of combustion, it will be necessary to determine 156 INTERIOR BALLISTICS the value of k' from which to compute X' and 2 1 . Suppose we adopt grains for which R/r n and m/l = 30. By the method given in Chapter III, we find for grains having 100 48 4 81 these ratios, a = -r- , X = , M = and k = . For 285 ' 199 199 95 this value of k' we find from Table II, log K' = 9.78966; and since K m = i, we have from (55), X' = K f X om which gives \ogX' = 0.52995. By interpolation from Table i, we find x r = 1.15217, and then log X\ = 9.88302 10, log X' 3 = 9.83966 io> log X\ = 0.67980 and \ogX' 6 = 1.32095. Next by equations (52), and equation (61), Chapter IV, we deduce the following equations for velocity and pressure, which apply from u = otow' = 1.1522 X 66.289 = 76-38 inches: v*= [6.76075] ^{1 + 18.85245-10] * - [7.24333 -lo]* 3 .} - p = [4-75876] X 3 { i + [8.85245-10] X,- [7.24333-10] X. } Both sets of equations, A and B, give the same value to z>', namely, 2319 f. s., while the pressures at u' by the two equations are, respectively, 51723 and 39560 Ibs. per in. 2 , a drop of more than 12000 Ibs. when the grains break up into slivers. The maximum pressure (taking x = 0.8) is 52428 Ibs. per in. 2 The dimensions of the grains have yet to be determined. We have found for this powder v c = 0.134 in. per sec. Sub- stituting this and the value of log X' , given above, in (46') gives, 1 = 0.038 in. and then r l /2 = 0.019 m - R = ii r = 0.209 in. = 1-1 in. APPLICATIONS 157 A grain of these dimensions fulfils all the conditions of the problem. These calculations show in a striking manner the great effect which minute variations (scarcely measurable) in the dimensions of m.p. grains have upon the maximum pressure, increasing it in this case by 10,000 Ibs. per in. 2 The cause of this great increase in the maximum pressure is that the initial surface of combustion of the charge of the smaller grains is about 15 per cent, greater than that of the original grains, as is easily shown by equation (26'), Chapter III. Problem 2. What must be the dimensions of the grains of a charge of 71 Ibs., in order that the burning of the web may be .completed at the muzzle? Also determine the circumstances of motion. To solve this problem we obviously have u' = u m ; and therefore x = x m = 3.8091. As all the X functions relate to the muzzle only, we may drop the accents. We have from Table i, log X = 0.74029, log X l = 0.35048, logX 2 = 9.61018, logXz = 9.653 1 1, log X 4 = 0.91582, and log X b = 1.78077. Sub- stituting the value of log X in (46), and making use of the known value of v c , we find that for the new grains, l o = ". 06098 Therefore, as in Problem i, r = = 0^.03049 R = 5-5*0= o"-33539 m = 3 lo^ i"-8294 k' = 0.85263 = k m Since the limiting velocity Vi is independent of the dimen- sions of the grains, we have as before, log V? = 7.44669; and this, with the known values of a, X and /*, substituted in equations (52), gives M, N and N'. We thus derive the following equa- tions for velocity and pressure for a charge of 71 Ibs. of these particular grains. 158 INTERIOR BALLISTICS v*= [6.55041] X,(i + [8.64211 -- 10] X - [6.82262 - 10] X 2 } p = [4.54842] X s { i + [8.64211 - 10] X*- [6.82262 - 10] X 5 } From these formulas we get the following information: Muzzle velocity, 3118 f. s. Maximum pressure, 29897 Ibs. per in. 2 Muzzle pressure, 21014 Ibs. per in. 2 Powder burned in gun, 60.5 Ibs. = 71 k' . The maximum pressure is quite moderate, owing to the thickness of web which gives an initial surface of combustion but 71 per cent, of that of the original grains. The pressure is well sustained to the muzzle, where it would be considered excessive for all except wire-wound guns. If we suppose the length of the grains to be twelve times the web thickness we should have a = 5, X = ^. ju = r-. 228' 163' 163' k' = -7, and m = 0.916 in. Then, as before, log M = 6.56065 logM' = 4.55866 log N = 8.60383 - 10 log N f = 6.90929 10 These constants give Muzzle velocity, 3124 f. s. Maximum pressure, 30163 Ibs. per in. 2 Muzzle pressure, 20875 Ibs. per in. 2 Powder burned in gun, 60.72 Ibs. = 71 k f The initial surface of combustion of the shorter grain is about 2.4 per cent, greater than that of the longer grain, which fact is shown in the maximum pressures. Problem 3. Suppose the powder we have been considering to be moulded into cylinders with an axial perforation. If the length of the grain is 50 inches (approximately the length of the cartridge), and the diameter of the axial perforation APPLICATIONS 159 one-twentieth of an inch, what must be the diameter of the grain and thickness of web in order that a charge of 71 Ibs. may all be burned just as the shot leaves the muzzle? Also determine the equations for velocity and pressure. We have already found the thickness of web satisfying the conditions of the problem to be o". 12 196. (See Problem i.) Therefore, by means of the formulas pertaining to this form of grain given in Chapter III, we find the diameter of the grains to be o".294 and a = 1.0024392 X = 0.0024333 fJL = O Since log X = 0.74029 and log Fi 2 = 7.44669, we find v 2 = [6.70746] X l {i - [6.64590 - 10] X } p = [4.70547] X 3 { i - [6.64590 - 10] X 4 } which are the equations required. The muzzle velocity and maximum pressure by these formulas are v m = 3376 f. s. p m = 37040 Ibs. per in. 2 This latter, on account of the smallness of N, occurs when x = 0.64. The muzzle pressure is 22750 Ibs. per in. 2 A comparison of these results with those deduced in Problem i shows the great superiority of the uniperforated grain over the multiperf orated grain so far as maximum pressure is con- cerned. The muzzle velocity is the same in both cases since the same weight of powder was burned in the gun. But the maxi- mum pressure given by the m.p. grains is more than 15,000 Ibs. greater, and the muzzle pressure 11,000 Ibs. less than with the u.p. grains. For these latter grains the pressure is remarkably well sustained from start to finish. The monomial formulas for velocity and pressure for this example are easily found to be l6o INTERIOR BALLISTICS ^ = [3-353 20 ] V% and p = [4-70441] ^3 The first of these gives the same value for the muzzle velocity as the complete formula; while the second gives maximum and muzzle pressures differing about o.i per cent, of their former values. During the test- firing of the 6-inch Brown wire- wound gun at Sandy Hook, shots were fired with charges varying from 32^ Ibs. to 75 Ibs., thus enabling us to determine whether our formulas have any predictive value. Unfortunately the object of the firing was simply to test the endurance of the gun and no special effort was made to give to the results any scientific value. Many of the recorded velocities and pressures are inconsistent with each other as when, more than once, an increase of charge gave a diminished velocity and pressure. Some of the recorded muzzle velocities are so manifestly wrong that they cannot be used in getting averages. They suggest that the chronograph velocities were not always reduced to the muzzle. We will compute the new values of / due to a change in the weight of charge by (88), Chapter IV, taking = 71 Ibs., and Jo = 1418 Ibs. per in. 2 , and for a six-inch gun, n = . We o therefore have / = [2.53451] of To determine X' , we have X'. = [1.43994!- v c V aw a which, by substituting the known values of d, 1 , v c and w, reduces to [1.52243] A = -=- v a u APPLICATIONS 161 N m is given by (54), which easily reduces to (since log K' = 9.78885 - 10) ^V w = [7.96539 - 10] \/ a a, .... (a) Next we have from equation (58), Chapter IV, substituting for/ its value given above and for w its value, 100 Ibs., F, 2 = [4.97834! tf and lastly from (53), Jf - 4 ^Fi ! = [3.54579] } a v ... (6) The following table computed by these formulas shows the agreement between the observed and computed velocities for a range of charges between 75 Ibs. and 33^ Ibs. The differences in the last column follow no apparent law and are unimportant. dj Ibs. x m logM m **"m Observed Velocity Computed Velocity o.-c. 75-o 3-9574 6.95290 8.87242-10 3455 3477 22 74-5 3.9383 6.95008 8.87347 3422 3459 -37 73-5 3-9005 6.94436 8-87557 3402 3423 21 72.5 3-8635 6.93849 8.87764 3380 3385 - 5 71.0 3-8091 6.92947 8.88072 3330 3330 69.0 3-7392 6.91693 8.88474 3254 3257 3 68.0 3-7052 6.91047 8.88672 3236 3220 16 59-0 3.4244 6.84548 8.90384 2879 2888 - 9 49.625 3.1742 6.76170 8.92037 2484 2536 -52 33-25 2.8146 6.55588 8.94643 1913 1896 17 The two sets of equations for velocity and pressure for the charge of 75 Ibs. are: From u = o to u' = 117.43 inches: v 2 = [6.72436] X, {i + [8.67663 - 10] X -[ 7 .i45 4 i - 10] X 2 } p = [4.73M * 8 {i + [8.67663 - 10] X, -[7.14541 - 10] X, } From u 1 ' = 117.43 in. to muzzle: ir= [6.95290] X l ]{i - [8.87242 - 10] X } p = [4.96750] X 3 { i - [8.87242 - 10] X*} ii !62 INTERIOR BALLISTICS By the first equation for pressure we find p m = 46509 Ibs. per in. 2 And by the second, muzzle pressure = 15375 Ibs. per in. 2 Both expressions for velocity give v f = 2744 f. s. For a charge of 62 Ibs., the two sets of equations are From u = o to u' = H4-54 inches: tf= [6.63999] X t {i + [8.70249 - 10} X - [ p = [4.60288] X 3 {i + [8.70249 - 10] X 4 - [ From u' to muzzle: v 2 = [6.86853] Xi (i - [8.89828 - 10] X } p = [4.83142] X s { i - [8.89828 - 10] X 4 } These formulas give a muzzle velocity of 3,000 f. s., with a maximum pressure of 34,263 Ibs., and a muzzle pressure of 11,784 Ibs. per in. 2 It would seem as if these last results are all that could be desired for a 6-inch gun. APPLICATION TO THE FOURTEEN-!NCH RIFLE The i4-inch rifle was designed by the Ordnance Department to give a "muzzle velocity of 2,150 f. s..to a projectile weighing i, 660 Ibs., with a charge of nitrocellulose powder of about 312 Ibs., and with a maximum pressure not to exceed 38,000 Ibs. per square inch." The gun has a powder-chamber capacity of 13,526 cubic inches and a travel of projectile in the bore of 413.85 inches. The type gun has been fired to date 55 times with charges varying from 102^ to 328 Ibs., producing muzzle velocities ranging from 901 to 2,252 f. s., and crusher-gauge pressures from 4,875 to 46,078 Ibs. per in. 2 , this latter with a charge of 326 Ibs. The powder employed was " International Smokeless powder, lot i, 1906, for 1 2 -inch gun." The grains were cylindrical multiperf orated (7 perforations), of the following dimensions: APPLICATIONS 163 Outside diameter, 0.826 in. Diameter of perforations, 0.0815 in. Length, 1.883 m - Thickness of web, 0.145375 in. These dimensions give: a = 0.71584 X = 0.20974 /A = 0.02151 k' = 0.85058 log^' = 9.78778-10. The granulation of the powder is 20.6 grains to the pound, which by (24'), Chapter III, makes the density (5) 1.4291. We will base our calculations on round No. 55, fired January 23, 1911, with a charge of 328 Ibs. of nitrocellulose powder plus an "igniter" of 9 Ibs. of rifle, or saluting, powder. This round affords the following data: co = 337 Ibs. w = 1664 Ibs. v m = 2252 f.s. p m = 43640 Ibs. per in. 2 The preliminary calculations give A = 0.68965 log a = 9.87523-10 Iogz = 1.65768 x m = 9.1025 log X om = 0.87855 log X im = 0.60885 By a few trials it will be found that the observed values of v m and p m are satisfied when k m = 0.953 and therefore from Table II, log K m = 9.89388-10. We also find log X' = 0.77245, log V, 2 = 6.99575, x r = 4-6354 and u' = 210.75 inches. 164 INTERIOR BALLISTICS The equations of the velocity and pressure curves are found to be Fromw = otow' = 210.75 in.: v 2 = [6.07812] Xi {i + [8.54923-10] Xo ~ [6.78775-10] X 2 } p = [4-72511] Xs{i + [8.54923-10] X, - [6.78775-10] X b } From u f = 210.75 in. to muzzle: v 2 = [6.31211] X l {i - [8.71430-10] X ] p = [4.959*0] Xs{i - [8.71430-10] X*\ Both of the velocity formulas give v' = 1921 f. s. The first formula for pressure gives p' = 27457 and the second 19,772 Ibs. per in. 2 The muzzle pressure comes out 9,485 Ibs. per in. 2 This round makes the powder characteristics, by (64) and (67), Chapter IV, / = 1759-7 v c = 0.10214 For computing the velocity and pressure constants when the charge varies, we will consider v c constant and assume / to vary directly as the weight of charge. That is, we will compute / by the formula Equation (69), Chapter IV, becomes, by substituting the values of J 2 , 1 and v c , [3.58446] ...... (b) Also (58), Chapter IV, becomes, by employing the expression for /given above, \ APPLICATIONS We then have ad> \? - f a& \ < VT^/ GO and ~X N' = -~ N* = [9.68929-10] A"" By (49), we have K' Combining this with the expression for N we have N m = -~ = (0.16507) N . . . Finally we have from (53) M m = 4 N m Ff = = (0.23399) M (g) (K) The following table gives the computed muzzle velocities and maximum pressures for certain charges, computed by these formulas, together with the observed velocities and crusher- gauge pressures for comparison: M Iha w It.- Observed Velocity, Computed Velocity, O.-C. f c Observed Pressure, Computed Pressure, O.-C. f.s. f.s. Ibs. per in.' J Ibs. per in. 2 337 1664 2252 2252 43640 43628 12 335 1660 2238 2240 2 42811 42944 -133 334 1660 2232 2232 42877 42637 240 284 1662 y z 1857 1871 -14 25530 29142 3612 263 1660 1738 1724 14 21190 24431 -3241 239 1660 1567 1556 II 16795 19704 -2909 The greatest difference between the observed and computed muzzle velocities is considerably less than one per cent, and 1 66 INTERIOR BALLISTICS may be disregarded. The same is true of the differences of the observed and computed maximum pressures of the first three charges. Then, as the charges are greatly reduced, these differ- ences are largely increased. This may be accounted for if the same kind of copper cylinders were employed for all the charges. For a charge of 314 Ibs. of service powder and an igniter of 9 Ibs. of rifle powder, making o> = 323 Ibs., and density of loading 0.66 1, the equations for velocity and pressure are as follows: From u = o to u' = 208.75 inches. ir = [6.05003] Xi!i + [8.55696-10] x -[6.80321 -- 10] x 2 .} p = [4-67944] X 3 [i + [8.55696-10] X< - [6.80321 -- 10] X,} From u' = 208.75 inches to muzzle. v = [6.28402] X, {i - [8.72203-10] X } p = [4.91343] ^3 {i ~ [8.72203-10] X 4 } These formulas give / - 1686.6 Muzzle velocity = 2152 f. s. Maximum pressure = 39351 Ibs. per in. 2 This muzzle velocity is that for which the gun was designed, but the maximum pressure is about 3^ per cent, greater. The muzzle pressure comes out 8714 Ibs. per in. 2 Example. Suppose the volume of the powder chamber to be increased (as is proposed by the Ordnance Department) to 15,000 cubic inches, by lengthening the chamber 6.65 inches, thereby reducing the travel of the projectile to 407.2 inches. If the density of loading remain 0.66 1, what would be the charge, the muzzle velocity, and maximum pressure? Answers : = 349-2 + 9 = 358.2 Ibs. p m = 41683 Ibs. per in. 2 M . V. = 2233.5 f - s. APPLICATIONS 167 With a charge of 337 Ibs. of service powder and an igniter of 9 Ibs. of black powder, we should get, with the lengthened chamber, a muzzle velocity of 2150 f. s., with a maximum pressure of about 38,400 Ibs. per in. 2 These results are prac- tically those sought for in designing the present 1 4-inch gun. Example 2. Suppose, instead of enlarging the powder chamber of the 1 4-inch gun, we lengthen the grains of powder, and employ the ratios R/r = n and m/l = 200. These ratios give, as is shown in Chapter III, 1210 a = - - = 0.64158 1900 = . 3l829 1219 fJL = = 0.00328 1219 . . 84368 1900 logtf' = 9.78150 - 10. (By Table II.) Employing these grains, what muzzle velocity and maximum pressure may be expected with a charge of 314 Ibs. of service powder and an igniter of 9 Ibs. of black powder, in the gun as it is now, where V c = 13526 c. i., and u m = 413.85 in. ? The preliminary calculations give: A = 0.661 log a = 9.91017 Iogz = 1.67419 x m = 8.7630 log X om = 0.87273 log X lm = 0.59873 By equations (a) to (ti), inclusive, we find, the web thickness remaining as before, 1 68 INTERIOR BALLISTICS / = 1686.6 log Fi 2 = 6.95993 logX f = 0.76472 .'. x' = 4-4201 and u' = 208.75 in - The equations for velocity and pressure are From u = o to u' = 208.75 m - : v 2 = 6.00247 X, { i + [8.73812-10] X - [5.98664 - 10] X 2 p = 4.63188 X 3 { i + [8.73812-10] X, - [5.98664 - 10] X 6 From u' = 208.75 in. to muzzle: v 2 = [6.27775] X l {i - [8.71576 - 10] X } p = [4.90716] *i'fi - [8.71576 ~ 10] X,} From these equations we find, Maximum pressure = 37851 Ibs. per in. 2 Muzzle velocity = 2146 f. s. Muzzle pressure = 8789 Ibs. per in. 2 v' = 1820.3 f. s. . , 26299 The dimensions of these grains are found from the ratios given above, and are as follows: Diameter of perforations = 1 = 0.0727 in. Diameter of grain = 1 1 1 = 0.8 in. Length of grain = 200 1 = 14.54 in. The following table gives the pressures (p') at different points of the bore for a charge of 314 Ibs. of service powder plus an igniter of 9 Ibs., making a> = 123 Ibs., and also the pressures (p") of the same charge made up of the grains whose dimen- sions are given above. It will be seen from this table, and the previous calculations, that increasing the length of the powder grains relieves the maximum pressure more than is accomplished by lengthening the powder chamber, for the same muzzle energy: APPLICATIONS 169 X u Inches. P' Ibs. per in. 2 P" bs. per in. 2 P'-P" Remarks. 0. I 4.72 24325 22388 1937 0.2 9-45 31329 29138 2191 0-3 14.17 35084 32888 2196 0.4 18.89 37234 35I3I 2103 05 23.61 38453 36483 1970 0.6 28.34 39090 37275 1815 0.7 33-o6 39351 37691 1660 0.8 37-78 39351 37851 1500 Maximum pressure. 0.9 42.50 39I8I 37833 1348 I.O . 47-23 38892 37690 1202 i-5 70.84 36650 36065 585 2.0 94 45 34150 34019 131 2-5 118.07 31849 32060 211 3-0 141.68 29820 30293 - 473 3-5 165.29 28049 28726 - 677 4.0 188.91 26500 27341 841 4.4201 208.75 25344 26299 - 955 The web thickness is 5-0 236.13 16308 16295 + 13 burned at this point. 6.0 283.36 13566 13585 - 19 7.0 330-59 H45i 11496 45 8.0 377.82 9775 9838 - 63 8-763 4I3-85 8714 8789 - 75 Muzzle. CHAPTER VI ON THE RIFLING OF CANNON Advantages of Rifling. The greater efficiency of oblong over spherical projectiles is twofold. In the first place they have greater ballistic efficiency, that is, for the same caliber, muzzle velocity and range, an oblong projectile has a higher average velocity during its flight than a spherical projectile. This gives to the former a flatter trajectory which increases the probability of hitting the target. Experimental firing has demonstrated that the mean deviation of the shots from a rifled gun at medium ranges, when all known and controllable causes of deviation have been eliminated, is only one-third that from a smooth bore. This advantage results both from the greater sectional density of the oblong projectile whereby it is enabled the better to overcome the resistance of the air, and also because this resistance is diminished by the more pointed head. In the second place the penetration of oblong projectiles, other things being equal, is much greater than can be realized with spherical shot, while the bursting charge of oblong shells is as great or even greater than that of spherical shells on account of their greater length. These are very substantial advantages; but to secure them it is essential that the oblong projectile should keep point foremost in its flight, otherwise it would have neither range, accuracy nor penetration, but would waste its energy beating the air. The only way to secure steadiness of flight to an oblong projectile is to keep its geometrical axis in the tangent to the trajectory it describes by giving it a high rotary velocity about this axis. This is accomplished by rifling, as it is called, that 170 ON THE RIFLING OF CANNON iyi is, by cutting spiral grooves in the surface of the bore into which a projecting copper band, securely encircling the projectile near its base, is forced as soon as motion of translation begins, thus giving to the projectile a rotary motion in addition to its translation as it moves down the bore. The rifling may be such that the grooves (or rifles) have a constant pitch, that is, make a constant angle with the axis of the bore; or, this angle may increase. In the first case the gun is said to be rifled with a constant twist, and in the second case with an increasing twist. In all cases the twist at any point of the bore is measured by the linear distance the projectile would advance while making one revolution supposing the twist at that point to remain constant. This linear distance is always expressed in calibers, and is therefore independent of the unit of length employed. The Developed Groove. Uniform Twist. The element of a groove of uniform twist developed upon a plane is evidently a right line A C making, with the longitudinal element of the surface of the bore A B, the constant angle B A C, whose tangent is B C/A B* Suppose A B to be the longitudinal ele- ment passing through the beginning of the groove at A, which is near the base of the projectile when in its firing seat and directly in front of the rotating band. Make A B = nd, n being the number of calibers the projectile travels while making one revolution. Then B C will be equal to the circumference of the projectile; or, B C = TT d. If we designate the angle of inclination of the groove, B A C by /?, we shall have BC ird TT tan )8 = -7-5 = i = - . . . . (i) A B nd n Increasing Twist. With a uniform twist the maximum pressure produced on the lands (or sides of the grooves) occurs (as will be shown presently) at the point of maximum pressure * The simple diagrams required in this Chapter can easily be constructed by the reader. 172 INTERIOR BALLISTICS on the base of the projectile, which point, as we know, is near the beginning of motion. From this point the pressure on the* lands decreases to the muzzle where it is not generally more than one- fourth of its maximum value. It is considered by gun-designers a desideratum to have the pressure on the lands as uniform as possible, and to this end recourse is had to an increasing twist- that is, the angle which a groove makes with the axis of the bore, instead of being constant as with a uniform twist, increases from the beginning of rifling toward the muzzle. If this variable angle be represented by 0, we shall still have, as before, tan = - (i) in which n is now a variable, decreasing in value as the distance from the origin of rifling increases. If, as before, we take the origin of rectangular coordinates at A, the beginning of rifling, and suppose A B to be a longitudinal element of the bore and B C the length of arc revolved through by a point on the surface of the projectile while it travels from A to B, then the developed groove, A C, is a curve convex toward A B, the axis of abscissas, for the reason that by definition tan increases from A toward B. The two forms of increasing twist that have been most generally adopted are the parabolic and circular. General Expression for Pressure on the Lands. Before attempting to decide upon the best system of rifling it will be necessary to deduce an expression for the pressure upon the lands. Take a cross-section of the bore and suppose for sim- plicity that there are only two grooves opposite to each other, and let the prolongation of the bearing surface pass through the axis of the bore, as is practically realized in the latest systems. Let M represent the point of application of the bearing surface of the upper groove. We will take three coordinate axes: one axis (x) is coincident with the axis of the bore, while the others (y and z) are in the plane of the cross-section of the bore and ON THE RIFLING OF CANNON 173 perpendicular to each other. Let the axis y be directed along O M, and z in a direction perpendicular to M. The pressure at M , whatever may be its direction, can be replaced by the three following mutually perpendicular com- ponents: The first, perpendicular to the axis of the bore and consequently acting along the radius through M; the second lying in the plane tangent to the surface of the bore (normal to the radius O M) and acting along the normal to the groove; the third lying in this same plane and tangent to the groove. The first of these components will be destroyed by the similar component of the opposite groove and does not enter into the equation of motion of the projectile. The second component, which is the normal pressure against the bearing surface of the groove, we will designate by R. The third component, being in the tangent to the groove, represents the friction on the guiding side of the groove, and may be designated by f R, in which / is the coefficient of friction. The forces R and / R give the following components along the axes x and z: Axis of x. Axis of z. Force R . . - R sin 7? cos Force fR . - f R cos -fRsm0 The positive direction of the axis of x is toward the muzzle; that of z in the direction of the force R, and is the angle which the groove makes with the axis of x. The full component for the upper groove is : On axis of x . . - R (sin + / R cos 0) On axis of z . R (cos f R sin 0) For the lower groove the component along the axis of x has the same value and sign as the upper one; while the component along the axis of z has the same value but the opposite sign. Besides these forces the projectile is also subjected to the variable 174 INTERIOR BALLISTICS pressure of the powder gases on its base acting along the axis of x in the positive direction, which force call P. If we replace the grooves by the forces enumerated above, we may consider the projectile a free body and apply to it Euler's equations. These equations are six in number; but, as is readily seen, they reduce to two in the problem under consideration, namely: an equation of translation along the axis of x, and of rotation about the same axis, or, what is the same thing, the axis of the projectile. The first equation is M-^= P - 2 R (sin 6 + /cos0) ... (2) and the second -fsmd) ... (3) in which r is the radius of the projectile, co its angular velocity about its axis and k its radius of gyration. The angular velocity co of a projectile about its geometrical axis for an increasing twist, continually increases as it moves along the bore from zero to its muzzle value, which is IT v/n r, v being the muzzle velocity of translation and r the radius of the projectile. Its magnitude at any instant is given by the equation where

n^). For these two points we have, by (i), tan 61 = - - and tan 2 = n, n 2 0! and 6 2 being the inclinations of the grooves with the axis of x at the points considered. Differentiating (10), we have, ^ = tan0=|/>(* + <*) 2 . . . (n) At the origin x = o, which gives 3 p \/a TT At 2^2 calibers from the muzzle where the increasing twist ends, x u 2 , and we have at this point 2 n% From these two last equations we find " . () and 1 78 INTERIOR BALLISTICS Since at the origin x and y are both zero, we find from (10) and (13), 2-n-a Thus all the constants in the equation of the developed groove are given in terms of HI, n 2 and u 2 . Lastly, differentiating (n) gives /'(*)= ^L= ..... (15) 4 V x + a If the vertex of the semi-cubical parabola is at the origin, or beginning of rifling, a and b are zero, and (10) becomes y = P x* ....... (16) In this case the twist is zero at the origin and increases to one turn in n 2 calibers near the muzzle. The values of tan 0, p and j" (x) for this particular form of rifling are deduced from (n), (13), and (15), by making a zero. This form of groove is that adopted by the navy for all their heavy guns of recent construction. Common parabola. The equation of the common parabola is y + b = p (x + a) 2 ..... (17) where a and b are the coordinates of the vertex. The constants are determined as for the semi-cubical parabola, and are as follows: 2 n (19) > Trr i 7T t 2n 2 (u 2 +a) f" (*}=2 P (21) ON THE RIFLING OF CANNON 179 Relative Width of Grooves and Lands. In our service siege and sea-coast guns the number, N, of grooves (or lands) is given by the equation N = 6 d in which d is the diameter of the bore in inches, and is a whole number for each of these guns. If w g is the width of a groove and Wi the width of a land, we have the relation inches - The best authorities lay down the rule that the width of a groove should be at least double that of a land. In our guns the lands are made 0.15 in. wide, and the grooves are therefore 0.5236 0.15 = 0.3736 in. wide. Application to the lo-inch B. L. R. Model 1888. This gun has 60 grooves which, beginning at 20.1 inches from the bottom of the bore with a twist of one turn in 50 calibers, increase to one in 25 at 20 inches from the muzzle, and from thence continue uniform. We therefore have HI = 50 and n 2 = 25. The bore is 22.925 ft. long, and therefore u 2 = 19.583 ft. The developed groove is a semi-cubical parabola whose equation is (10). The constants are a = 19^3 = 6 . 52g ft . 6.S287T b = -- = 0.27344 ft. .75 P = " =5 = - l6 395- .528 The equation of the developed groove (changing x to u to indicate travel of projectile) is therefore, y + 0.27344 = 0.16395 ( u + 6.528)* - (22) in which y will be given in feet. l8o INTERIOR BALLISTICS From (n), we have tan = 0.024592 \/ u + 6.528 Making u zero in this last equation gives i=335'4*" which is the inclination at which the groove starts. At 20 inches from the muzzle where the twist becomes uniform (and which is therefore a point of discontinuity on the developed groove) we have u-i = 19.583; and at this point *>=79'45" This value of is retained to the muzzle. From (15), we have /" f x \ = o-oiffg^ Vu +6.528 This function decreases from the origin to the point of discontinuity. From this point to the muzzle /" (x) is zero. If we put K = _ Msec " i - tan 0{ f - n (f + tan 0) } ' equation (8) becomes 2 R = K [P tan + M ir f" (x) } . . (24) Captain (now Sir Andrew) Noble, as the result of very careful experiments made by him with i2-cm. quick-firing guns, found / = 0.2, and this value will be adopted in what follows. We also have for cored shot /* = 0.5, nearly. Substituting these values of / and /* in (23), it will be found that K increases very slowly as increases. The values of K for u = oandw = 19.583 are, respectively, 0.5032 and 0.5064. We might therefore take for K the arithmetical mean of these two values and write (24) 2 R = 0.5048 { P tan + M v~ f" (x) } (25) ON THE RIFLING OF CANNON l8l without any material error. This formula may be employed for all our sea-coast guns. If the lo-inch gun were rifled with the kind of groove given by (16), we should find 2 7T P = - X - -- = O.OI893I tan = 0.028397 V u In this form of rifling the initial inclination of the grooves is zero and increases to 7 9' 45" at 20 inches from the muzzle, where the twist becomes uniform. Between this point and the muzzle, j" (x) is zero. Uniform Twist. If we suppose the lo-inch gun to be rifled throughout with a uniform twist of one turn in 25 calibers, we have p = 7 9' 45". Employing the values of ju and / already given, (9) reduces to 2R = 0.063624 P (26) Working Expressions. If the equation of the developed groove is (10), we have ( Mv 2 > 2 R = K tan B -( P + , r L . . (27) L 2(u + a)} and TT / u + a \ \ tan 6 = - 25\w 2 + a / If (16) is the equation of the developed groove, we have and tan 9 = -I- l82 INTERIOR BALLISTICS Pressure on the Lands of the lo-Inch B. L. R. The equations for velocity and pressure for this gun are the following: ir= [6.20536]*! {i - [8.59381 -- io]X } . . (29) and p = [4.720601*8 1 1 ~ [8-593 81 " io]* 4 } . . (30) The gun and firing data were V c = 7064 c. i., u m = 22.925 ft., co = 250 Ibs. of brown cocoa powder, w = 575 Ibs., muzzle velocity 1975 f. s., maximum pressure on base of projectile, 33300 Ibs. per in. 2 , A = 0.98, and z = 3.461 ft. It will be convenient to change (30) so that it will give the entire pressure (P) on the base of the projectile; and to avoid large numbers we will adopt the ton as the unit of weight. We then have - ~ 8960 and (30) becomes P = [3.26544] M 1 - [8-5938i -- 10] X<\. . (30') Finally, the mass of the projectile expressed in tons is We have now all the formulas and data necessary for comput- ing the pressures on the lands of the lo-inch B. L. R., by means of (26), (27), and (28), for the three principal systems of rifling adopted in our service. These calculations are given in the table on page 183. The last three columns of this table show that the maximum pressure on the lands is greater for uniform twist than for either form of increasing twist; but the difference between these max- ima is not very great. Moreover, the maximum pressure for uniform twist occurs at the trunnions where its torsional effect upon the gun so far as deranging the aim is concerned is a ON THE RIFLING OF CANNON Pressures on lands required to produce rotation of shot in the lo-inch B. L. R. for different systems of rifling. Charge 250 Ibs. Projectile 575 Ibs. Muzzle velocity 1975 f. s. Maximum pressure on base of projectile 33300 Ibs. per square inch. Travel of Velocity of Pressure on Base of PRESSL RE ON LANDS. TONS X Shot, feet Shot, f. s. Shot, tons Uniform Twist Increasing Twist Eq. (28) Increasing Twist, Eq. (27) 0.0 0. 0.0 0.0 0.0 0.0 0.0 .1 0.3461 227.7 841.1 53-5 I2.O 27.0 .2 0.6922 366.7 1036.4 65.9 21.5 36.9 o-3 1.0382 478.1 1122.4 71.4 29.0 42.3 4 1.3843 572.4 1158.3 73-7 35-3 46.1 5 1.7304 654.7 1167.4 74-3 40.9 48.9 0.6 2.0765 727.8 1161.1 73-9 44.8 5LI .7 2.4226 793-6 1145.6 72.9 48.5 52.9 .8 2.7686 853-4 1124.7 71.6 5L6 54-3 0.9 3-II47 908.2 1100.7 70.0 54-3 55-5 I.O 3.4608 958.7 1075.0 68.4 56.7 56.5 i.i 3.8069 1005.6 1048.5 66.7 58.7 57-3 .2 4.I530 1049.3 1021.9 65-0 60.5 58.1 3 44991 1090.2 995-5 63-3 62.1 58.7 4 4.8452 1128.6 969.7 61.7 63-5 59-2 5 5.I9I2 1164.7 944-5 60. i 64.7 59-7 .6 5-5372 1198.9 920.1 58.5 65.8 60. i 7 5.8833 1231.4 896.4 57-0 66.7 60.5 .8 6.2294 1262.1 873.6 55-6 67-5 60.8 9 6-5755 1291.4 851.7 54-2 68.2 61.1 2.0 6.9216 I3I94 830.5 52-8 69.0 61.3 3-0 10.3824 15434 6594 42.0 72.7 62.4 4.0 13.8432 1702.8 541-4 34-4 73.5 62.3 5-o 17.3040 1824.9 456.4 29.0 73-2 61.6 5.6586 19.5833 1891.6 412.4 26.2 72.6 61.0 6.0000 20.7648 1922.8 392.4 25.0 25.0 25.0 6.6242 22.9250 1975.0 359-9 22.9 22.9 22.9 1 84 INTERIOR BALLISTICS minimum; while the position of the maximum pressure upon the lands for either form of increasing twist is well down the chase. It is difficult to see any superiority of an increasing twist over a uniform twist, especially in view of the fact demonstrated by Captain Noble's experiments, that the energy expended in giving rotation to the projectile with rifling having an increasing twist is nearly twice as great as with a uniform twist. Application to the 1 4-inch Rifle. This gun has 126 grooves and the same number of lands, in this respect differing from the rule followed with the other seacoast guns. The values of n\ 9 HZ, 61 and 2 are the same as those found for the lo-inch rifle. The rifling begins 7.05 inches from the base of the projectile when in its firing seat and becomes uniform 22.8 inches from the muzzle. Therefore u 2 = 4i3- 8 5 ~ (7-5 + 22 -8) = 384 inches. From (12), (13), and (14), we now find a = 128 p = 0.0037024 b = 5.36165 Therefore the equation of the developed groove is y + 5-36165 = 0.0037024 (u + 128)' From (n) and (15), we have, finally, tan = 0.0055536^ u + 128 VU + I2S M = - = c 2240 and in which P is the entire pressure on the base of the projectile in ON THE RIFLING OF CANNON tons while p is the pressure in pounds per square inch given by the equation on page 166, for a charge of 314 Ibs. Substituting these expressions for tan 0,/" (x), and M in (25) and reducing, we have the working expression I 128 2R = [7.44769-10] V I* -f 128 jP|4- [8.06145-10] in which 2 R is the normal pressure on all the lands in tons and u the travel of the projectile in inches. To determine the normal pressure on each land 2 R must be divided by 126. For a uniform twist 2 R is given by (26). In the following table v and p were computed by the formulas on page 166 for a charge of 314 Ibs., and P and 2 R by the formulas given above: M inches. V L s. p tons. 2 R Increasing T 2 R Uniform T O.I 4.72 198.7 1671.7 54-i 106.4 0.2 0-3 9-45 14.17 325.2 429.1 2153-0 2411 . I 71.1 81.1 137.0 153-4 0.4 0-5 0.6 18.89 23-61 28.34 518-9 598.5 670.3 2558.8 2642 . 6 2686.3 87-7 92.2 95-3 162.8 168.2 I7I.3 0.7 0.8 0.9 33-06 37-78 42.50 735-8 796.1 852.1 2704.3 2704.3 2692 . 6 97.6 99-2 100.4 I72.I I72.I I7L3 I.O 2.O 47-23 70.84 94-45 904-3 1123-3 1295.1 2672 . 8 2518.7 2346.9 IOI.2 102.5 101.8 I70.I 160.3 149 < 3 2-5 3-0 3-5 118.07 141.68 165.29 1436.7 1557-5 1662.9 2188.7 2049.3 1927.6 100.5 99.1 97-8 139-3 130-4 122.7 4.0 5-0 6.0 188.91 236.13 283.36 1756.5 1891.0 1981.2 1821.1 1120.7 932-3 96.5 66.0 59.3 II5-9 7i-3 59.3 7.0 8.0 8.763 330-59 377-82 413-85 Muzzle 2053-7 2H3-3 2152.0 787.0 671.8 598.8 53-6 48.8 50.1 42.7 1 86 INTERIOR BALLISTICS Influence of the Rifling for a Uniform Twist. For a uniform twist we have where n is constant, and r is the radius of the projectile. Differ- entiating with respect to / we have do) TT dv TT d z jx ~dt " "nr dt~ nr dt z Substituting this value of d u/d t in (3), it becomes *-^*J = 2 IZ(cos/J-/sm0) . . . (31) Eliminating 2 R between equations (2) and (31), gives P M ,. ~-~dP\ M n i-/tan/3j' If the bore were smooth the equation of translation of the projectile would be from which it appears that the effect of the grooves upon the motion of the projectile for a constant twist is equivalent to increasing the mass of the projectile by the quantity TT_MM / + tan |8 n i tan By making / = 0.2, n = 25, ^ = 0.5 and /3 = 7 9' 45", the value of this supplemental term is found to be 0.021 M. That is, the retarding effect of a constant twist of one turn in 25 calibers is equivalent to increasing the weight of the projectile 2 per cent. * Artillery Circular, N, p. 201. TABLES PAGE I. X Functions l8 9 II. K = 1- (l-fc)i 2I1 III. Work of Fired Gunpowder 2I2 TABLES 189 HH co ONOO oo t^OO CO . -vO ON ON CO ON N co HH to r^ r^ to vO t^vO vO to iO CO HH O O O 00 ^00 rt- M HH ON O HH Ol CO ^t" tOVO t^* t^OO ON O HH HH CS ON ON ON ON ON ONONONONON ONOOOO ONONOOO OOOOO OOOOO OOOOO O HH HH HH HH IO COOO HH oo r>> c< t~> oo ON r-^ HH ON . . IO IO CO CO ONOO t^vO to rh co *t- IO CO *^- co N ON Ol ON vO to O 00 ONOO I-I ON O ON CO lO^ CO CO TfTl-^- OOOOO OOOOO OOOOO o o o o o vO CO O to iO t^ -3- cooo to N (M vO co O) CO HH r>. o) c t^ ON 1-1 HH O O * ^t- CO CO (S O ON cooo fO ON iO ON ON to N O rf r^vo rOO 00 ^tOO ci iovo r^ i^. VO r) HH HH t>. O \O ON 1OOO H ON >-i O 1-1 1-4 CS rj- IO IO co oo oo oo oo CO O ON O n vO ^t- tOOO iO HH co CO rj- co HH ONVO ( CO TJ- lOvO t^. 00 00 ON to to to to to tototov ONOO oo oo oo co oo oo ON ON ON ON ON ON ON ON ON ON w _) PQ VO * O M Tf O Tt-00 00 00 v> h-i 00 C< 00 ON ON Tt- i/>vO O to co t^ IO 1^00 00 \O to Th HI ON t^VO t^. M 00 O Tj- (S oo r>. r^ t^t^ ON cOvO ON CO O ON t^ O rj- n I-H o oo \O vO vO iO O ON iO ON ON O< O ON to t->- to CS Tf-vO f^ HH COHH O ON to cs oo to Tf IO O) t^ M CO 00 ONOO O O CO ^ co O vO vO t~>-vO co t^ Ol O OO vO co o o o 5 o o co n to HH O O HH O 00 Tf HH 00 IOHH 00 100 ON O O HH CO HH VO t^ Th 00 HH C) CJ HH T*- HH t^ co ON HH ^ CS CO CO t>.oo oo oo oo OO OO ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ONOO rC 1-1 01 M2 00 t^ (N IO IOOO O) 1-1 I-H O O t^ O I-H CS O) IO OJ O O) IO i ior>.<3-co co O 00 1^ i HH ^- ON COO co O t^ iO co HH HH O O O oo vO ON ON ON ON a g - n oJ I ONTfO toco HH O to t^vO ^O ON HH co to OJ co t^vO ON c< to to cooo r^oo ON o O CO rt- iO t^.00 HH (S HH l^ Cl HH HH HH O O ONONONONON ONONONONON "1" "? "? ' ON ON ON ON ON O CO \O C< 00 t^OO to O ^O n- ^-oo vo i*- ON (S HH 00 iO 10 1-1 ONOO CO CO C ON HH ThvO O t>- t^-vO iO iO CO CO CO CO co HH t^ ONt^HH COOO CO M 00 1000 O HH HH ON ONO 6 6 s ^8"o O cOOO (S \O ON W CJ n 0) O) co co tO CO ON TJ-OO cOOO M r- HH CO CO ^h ?* iO cO co co CO co t>. ON N to ON HH CO tO IO Tj- vO O ^t-00 C iovo vo vo r^ co co co co co OOOOO OOOOO OOOOO vO tOvO O vO COHH 00 IOO vO O co t^ HH l^OO 00 00 ON co co CO co co 66666 HH IO O IO O O HH HH (S 66666 iO O iO O >-i fN CO co ^f * OOOOO OOOOO t^QO ON O HH rt- rf- ^4- tO tO 66666 cs co TftOvO to to to to to 00000 igo INTERIOR BALLISTICS ^ M O ONOO t~* JS t^vO vO vO O ON ONOO ON O O ONOO f^ vO iO ^h ^ co 10 10 10 10 CN CN HH O O IO iO IO IO iO J2 HH ro ^ CO CN rf HH 00 IO CN co rh Th iOvO O O O O O CN CN HH O 00 flO CN 00 rf- r^oo oo ON O O O O O OO ON ONOO t^ O VO CN 00 -5J- HH HH CN T+~ O t^* IO t^^ ^)- M VO HH IO O VO HH t^ CN CN ON ON CN 00 ON HH co lO iO t^. 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ON t~>- HH co HH rl- t^ O CS HH CO IOOO O CO co co co Tt- 00 00 00 00 00 HH \O t>- IO HH d- lOvO t^.OO Cl T)-VO 00 O ^J- rh rt- ^ IO OO OO 00 00 00 * co ON co iO oo oo r^ t^o CS TfvO 00 O IO iO to lOvO oo oo oo oo oo 00000 O O O O O O O O O O O O O O O o o o o o IO O iO O 00 ON ON O 1-1 CN) rO^HOO t^OO ON O HH CS COThiOvO t^OO ON O HH IOO IOO \O vo NO O O O vO vO vO t^ t^ t^ r^ t^ t^ t^ t^ t^ t^oo oo "8 TABLES 2OI w CQ Q ON cor-- cooo hH HH O O ON COOO CO ONTt- ONOO OO t->- t^ cO cO cO co CO ON NO CM t^ CO NO NO NO IO O CO CO co CO co O to CM oo to iO ^ rt-coco i-i t^rt-CM 00 CO CM CM CM I-H CO CO co CO co * oc t^o r^o O CM rt rt o rtOO CM NO O 10 1C NO NO t> O O O O O 00 HH ONCS I-H ^ rt- 0) i-t ON ^OO CN O ON i^>. t^oo oo oo O O O O O O rj- O CM ON NO CO O NO l-< co r^ I-H Tt-oo g'g s 2 2 2 CM CM l^ ON t^ t>. CM NO O rt- I-H lOOO CM IO HH HH HH CM CM CM COO rf-NO 00 HH rt-NO 00 00 CM lOOO I-H CM CO CO CO rf CM cs CM CM' CM CNJ M CM CN CNJ CM CM CM CM CM CM CM CM CM CM CM CM CM CM 0 CO O t^NO Tt- ON ONOO 00 00 CM ONOO vO rt- CM i-i ON t^ iO t^. t~>.NO NO NO >< M O 00 IO O CM l-l rtNO OO ON O *" ON | - 1 CONO iO IONO NO NO O O O O O 00 CM rf CO ON OO O CM "3-vO NO l>. t^. I s - t^. O O O O O ^- r> i^>* ^d~ O O ONOO t^NO OO ON HH co >O S'S-'SS'S rhNO iO CO ON rt- CM O 00 IO t^. ON I-H CM rf OO OO ON ON ON O O O O O to ONO to N CO O t>. rh HH NO 00 ON I-H CO ON ON ON O O q 00 rt <-> 00 rt Cs CM CM i-i HH fO co co co co 1 2^g>8 cO tO cO cO cO t^ IO CM ON t^ ON ON ONOO OO CM CM CM CM CM rt- CM ON t^ to oo oo r^ t^ t^. CM CM CM CM CM CM O 00 IO CO t^. l^.VO NO NO CM CM CM CM CM * 9 I-H tO ONOO O rt I-H OC O IO CO O NO co O NO NO iO iO iO NO ri-0 ONVO CO CM 1-1 O O t^ Tf I-H 00 IO rj- rj- rf co CO NO ON rl- CM CO O O >-H CM CO CM ONNO co O CO CM CM CM CM Tl- ^ rt- -* rj- NO CM O "H rh rt-NO 00 O CM t^ rj- I-H ONNO HH I-H hH O O rt- t^ O CO t^ CO O 00 iO CM O O ON ON ON rt rt-cO COCO ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON 0-0, ON ON ON r> CM O 00 lOcO rt rtcOCOCO O 00 t^. n-co CO CM M CM CM HH ON\O 10 CO CM I-H 00 NO iO 3-808% i * M O O CM O) O IO co r^ I-H 1000 00 ON I-H CM CO HH i-i CM fs CNJ oo oo vo cor^ -i TJ- t^ O CM IOVO t^ ON O CM CM CM CM CO O I-H O NO M IO t^ O\ O CM HH CM CO IOVO CO CO CO CO CO rhNO t> to >-H CO ri- lONO r^ t^.00 ON O I-H CO CO co rl- rt- t>- t~>. t" t^- 1^ NO O CM CO co t^.00 00 00 00 CM to rt-iONO rj- rt- rf rf- rt- ON ON O ON ON ON ON ON ON ON ON ON ON ON ON ONONONO^ON ON ON ON ON ON Q ONTj-ONiOO 04 (^ i_ _c HH CO CO co CO CO IO HH 00 CO ON O O ON ONOO CO CO CM CM CM IO CM 00 rt- HH oo oo r^ r>> t>> CM CM CM CM CM tN. rt- HH t^ rh NO NO NO IO IO CM CM CM CM CM I-H ONNO CM ON iO rt rj- rf CO CM CM CM CM CM * g rj- ro t^vO 1-1 00 HH co ID f>- O TJ- t^ O co OO OO OO ON ON IO IO UO IO IO M \o r>. ooo 00 00 00 00 t^ >O ON CM lOOO ON ON O O O iO lO^O ^O NO t>. CM rj- CM NO NO iO CO HH 00 HH Tj- l^-. O CM HH I-H P-H CM CM NO NO O NO NO l^. rt-OO ONNO O CM 00 rl- O tOOO O CONO CM CM CO CO CO NO NO NO NO NO O M O NO 00 NO i-i NO O rt- OO I-H CONO 00 CO rt- rt rt rt NO Nfi VO NO NO O O O O O O O O O O O O O O O 00000 O O O O O Q t^ TJ- fS ON t>. oo oo oo r^ t^ 10 CO -i ONO r^ i>. I^NO NO O CO >-i ON t~~- NO NO NO IO IO NO rh CM I-H ON IO IO IO IO rf t^O IO CO CM c >.^o IO rj- CN O OO O TJ-vO OO ON 00 00 00 00 00 COOO M CM HH NO CO M oo >O I-H tO iO\O OO oo oo oo oo oo t^ CM >ONO IO I-H 00 rh O vO O HH CO O^O 00 00 00 00 00 oo oo oo oo oo CM 00 CM rt- IO CM f^ COOO CO 00 ON I-H CM rt- 00 00 ON ON O OO OO OO OO OO ^j- HH J> CM tO 00 CO t^ CM NO 10 r^oo O I-H ON ON ON O O OO 00 00 ON ON d d d 6 d O O O O O O O O O O O O O O O O O O O O CNJ CO Tj- IOVO t^OO ON O I-H CM co rt- >ONO 1-^00 ON O I-H ON O\ ON d C CM CO ^-"P^O d o o o o 06 od 06 06 06 00 00 00 ON ON ON ON ON ON ON 2O2 INTERIOR BALLISTICS ^- HH 00 *O <*O O I s "* ^^ M 00 ^^ CO O NH HH O O O O ON ON O^oO oo oo oo CO CO CO CO rO CO Ol M Cl 0* (N M . t- t>O vO \ CM W CN tN CN O l^. tO CO vO vO iO iO iO CN CN CN CN CN I O O tN CM ON 'd- ON NO 04 OO cO ON 3 w ^hri. 6 covd ONCN tooo 6 covo ONhnrft^ON CNIOI^OCN ^t-iOtOtO VOvOvOvOt^ l> t>-OO COOO OOONONONON OOOhHhH OOONr>-CN lOtOCNVOt^ lOhHTt-Tj-HH VOON HHNCO^ ^"t^-cocN hHOogvort- HHOO CN CN CN CN CN CN CN CN CN CN VO vO vO to iO to iO iO iO iO T)- rj- rf Tt- -^ ^ rj- * CO CO CO CO CO CO CO I cOCN ONIOONCNCO cOCNONiOON CN.t->.iO cOONcOt^O vO CN ^ COOO ^t" ON <* ON cOOO CN t^nnioONcO 1^-OTl-t^.HH O HH ro -^-vo t^ ON O hH co -^-vO 1^-00 O CNCNCNCNCN CNCNCOcOcO cOcOcOcOTt- CN OM^ to co yft to IO iO IO CN CN C- CN CN I-H O CN CN CN CN CN CN CN co CN O t^vO to co hH O ON COCOCOCNCN CNCNCNCNhH CNCNCNCNCN CNCNCNCNCN ^ S M ON O COCO O t>- ^t" ONOO oo co co co IO rf IO t- 0) (N 1^ C^ t^ CO t> fO 10 i~i 00 \O \Q O^ cOOO iO T}~ ^d" t"> rO O ^O ^*O O t^ 1 * *-O C^ O 00 ^O t"> 10 to o oo *o co *^ cr\ i^* T^ cs CO rO co co co to ro to ro rO rO rO co to ro co CO ro to to ro CO ON ON ON ON ON ON ON s - ON ON ON ON ON ON ON \O ^s IO ID CO O) I-H ONOO t^vO ONOO 00 00 00 O Tf- Ti- CO CN CQ $ o CN to rt- CN ON ON ON ON O^ ON ON ON O\ ON ON O^ ON ON O*^ ON |{ t^ ^ (N ONVO rt-CNONt^.iO CNOOOtOrJ- CNOOOtO'^r CNOONt^O COCOCOCNCN CNCNi-iMHH hnnnOOO OOONONON ON ONOO 00 00 CNCNCNCNCN CNCNCNCNCN CNCNCNCNCN t^ rj-00 O ON to ON M O OO CN IO ON hH O CO IO IO 10 O\ Tj-\o ON M CJ W Tj-vO ON I-H HHCOCOhHVO OCNCNhHOO oooooooot^- r^.voio-Ti-cN CO iO t^ ON hH CO iO t^ ON hH CO iO t^ ON >-< ' -00 OOOOOOOOON ONONONONO 00000 00000 00000 00000 00000 I-H o ON C4 CN O ON ONOO t^iOiOrJ-cO ON ON ON ON 60660 fO t^ ON O O ON (N iO ON ON 1-1 (S CO iO 66666 ^t- r^ O o) 10 ^O t^ ON O I-H hH M M (^ CN ON ON ON ON ON 66666 rj- O O hH ON CO lOr}- CN t>. ON M CO O t>-C CN CO tOvO l^ 00 ON HH CN CO CNCNCNCNCN CNCNCOcOcO ON ON ON ON ON ON ON ON ON ON 66666 66666 r^oo ONO M t^OO ON O I-H t^oo ON O M OOOhHHH HHhHHHHHhH HHHHHHCNCN CNCNCNCNCN CNCNCNfOrO TABLES 20 3 O4 OJ 04 04 04 M ONOO O cotO CN) M iO ON cONO t-> 10 Tf rl-cOcO rh Tf rj- rfrf ' I-H I-H 04 04 O4 O4 04 O4 04 O4 04 04 04 04 04 N CO 0< O O oo M vo O r- t^ O M iO ON C< fO fO CO ro tOOO t^O NO cooo cooo t-i ^-00 CO t^ Cl <3- t^. CO iO co ON M oo O NO 00 I-H O O OO O t^ i-i IOOO O4 l^ ON O O O -* ON co M 04 04 04 04 O O ONOO t^ CO CO 04 O4 Ol I-H 00 co i-i l^. Tt- i-HOOTJ-cO <* COCOCOO4O4 C4i-i>-ii-H 04 04O404O4O4 04 O4 04 Ol o"S ON ON" 04 04 HH HH VO "H 10 ON rO OO i-i ro iO O t> ON O M ^f r^- ri- IO IO IO NO 00 HH CO iO t^OO NOOcO 0100O4COO4 ON I-H ro lONO > t^OO 00 00 t>. O 01 ^t-O oo O 04 ThNO NO i^* t^ t^ IN* i>*oo oo oo oo t^ O TJ- co HH ONOO 00 O O O O I-H ON iO O iO i-i CO ON ONOO 00 COCO CO CO CO iN-i-it^coON lOt-ir^coO r^t^NOOlO IO IO Tl- rf rf cococococo cococococo ^ 8 O 00 NO co M CO M M 04 O4 cOfO fO CD to \D\O 00 O O ON t"** iO ro C^ CO coco coco Q\ &\ QN Qs Q\ coc iO HH coco C^ 1 ON ON O\ O^ looo i^ o 00 CO Ol O O4 Tj- ON 10 r^ co o o co IOCNOO-^-'-H r^roo CS CJ CS CNJ OJ CJ CN] CJ CN | Cl Os ON ON ON ON ON ON s - O^ ON l^^O O 00 NO NO co (N 04 O oo O O ON W CQ < H ON ON ON O ON ON ON ON ON ON ON ON ON ON ON 00 COOO O4 00 Ol Ol HH >-* O cO co co co co r^-00 CO ON IO O ON ONOO OO CO O4 M O4 04 >-H t^ CO ON O O4 O4 04 04 O4 00 NO ON f^ ON O co 10 r^oo ON O ON HH ON 04 HH VO O ON ONOO NO Tj- O4 ON\ CO IOOO HH Tht^O 04 IO lONO NO dodoo ooooo ooooo ooooo ooooo ON NO 00 ON ONOO NO ^t i-i t^ t^ cONO t^ ^- ON ^- lONO t^-OO ONO>-ii-HO4 COCOCOCOO4 rt- lONO t->.OO ONi-iOlcOiO I>.ON>-icOiO COCOCOCOCO CO -^- * rj- rj- T^ rt- IO iO iO do odd ddodd o'dddd HH O t^- 04 CO 04 ^ ONOO NO t^ ON O 04 * IO ONO NO NO ON ON ON ON ON co O IO TJ- O4 ON 00 OO 00 O cO >-> .00 ON O 04 CO coco TJ- 4" O OJ q o; iO iO lOO O 204 INTERIOR BALLISTICS w CQ Q n 00 ICOO IO t^^O sO 1C i-O rO fO cO cO co >- t^ CO ON 1 ^- co CO CO CO CO cOOO O \5 cO CO 04 0) 04 04 coco CO CO CO 00 10 O NO co "I -. O O 00 IO >-i t^ CO ON ON ONOO OO Ot O4 04 O4 04 J to t^ tOO 00 ri- I-H oo to O \O O co t^* I-H co co fO co CO CO ^t- "-< rt- CO vO -> vO O TJ- TfOO i- UOOO O CNJ CO CO CO CO COCO CO CO t^ O 00 ^t- O t^. HI crjvo ON W 1000 HH Tj- Tt- ^- ^IO IO CO co co CO CO co I-H vO ^C 04 I-H CO "sf tOO 00 M * t^ O iO\O \O vO l>> CO co co co co to CO X O 1 O NO O iO r)- CO COO ON O4 to t^ !>. l^QO 00 CO CO co co CO 04 04 O4 04 04 (N CN CN M CN 04 04 04 O4 04 04 O4 04 04 04 O4 04 04 04 0) q co 04 o o o ON ON ONOO 00 CO O ONVQ r}- oo oo t^* t~^* t^ Th i-< oo o r^ !>. I>-O 1>NO CO HH QN Q\ QN NO ^O to iO to IO IO O4 ON to * ON04 ri- Tt-O i^. r^vo 10 ^j- 00 O O4 rJ-vO IOOO 00 t^ CO CM O oo ^o ^r 00 O I-H CO iO t^ i-< 04 O IO h-i ON^O CO O t^oo O 04 Th 04 tOvO uo ^t- t^. co ON to HI IO t^OO O O) COOO CO iO rt- r^ 01 co coco rO iOO oc ON '3 M CS O) 04 04 O4 04 O4 O4 O4 04 Ol 04 O4 04 0) 0) O) O4 0) q O 04 ONO 04 CO CO 04 O4 O4 <0 ro rococo \ ONVD CO O O _, _, _ _ o coco coco CO rt- HI oo 1000 O O ON ONOO CO CO 04 O) 04 IO H-I OO l^-O ON ONOO OO OO 04 O4 04 04 O4 O COOO iO CO r- t^-o o O 04 04 04 O4 O4 * 9 04 vO ^- iO ON M (^ ^J- |_ 00 O NO CO O NO O -d- -^- rj- ro Ol Ol 04 04 04 1^00 04 ON ON \O Thco w O CO O t^ ^ i-" CO CO O) 04 O4 04 04 04 04 04 cO ONOO O iO O QN ^s O O oo "^ HH o\^o 04 O4 O4 04 04 t^. 04 tn CO\O HH O4 CO ^t tO CO O t^ rj- I-H %%222 O rj- I-H rOOO t^ ON O4 IOOO oo o co O r^ oo oo oo oo r^ ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON q ON ON ON ONOO t^. l^. IO <* O4 oo oo oo oo oo 04 O ON r^sO OO OO J>^ t^* t^* iO iO Tf CO I-H I-H O ONOO 00 r- r^o NO o * &0 .0 Tt- ON fO lONO 04 I-H I-H O ON fO^iOvO vO ON ON ON ON ON IO O4 ON Tt-00 00 *> IO -^t- 04 1N.OO s O "-" ON ON ON O O t^ l^ t^OO 00 O 04 04 HH 00 I-H C\t^ IO O4 04 O4 CO -3-IO O O O O O oo oo oo oo oo Tf-ON^OO I-H O t^ to O4 O O4 CO CO 04 O t-~ -4- oo to ON C I-H HH O4 00 00 00 00 00 ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON 0,0, ON ON ON ON ON Q fS OO IO O OO vO iO ir; 10 T|- CS CS fN CS C4 IO 04 00 IO 04 Tf r}- CO CO CO 04 04 04 04 04 O 00 iO 04 O CO O4 O4 O4 O4 04 04 04 04 04 ONOO t^ to ON |_ _ HH (-H O O4 04 04 Ol 04 3^8:8; O) 04 O4 HH M * *- cO HH vO vO CO ONIO O IO 00 O COO 00 Tj-ON w ON^ %% vO vO OM-< O\ 04 IO t^ O O4 lOl^. ON 04 HH O 00 tOO Ol Tj-lO t^ ON 0^0 2 2 ON r^ co ONOO ON O I- <~> I-H xh t^ ON "H CO | t^^ t^. r t^. t^t^t^t^t^ t^ t^ !>* t^OO OO OO OO OO OO CO 00 00 00 00 O O O O O o o o o o O O ooooo ooooo q \O vO vO vO >O t^O CO O4 ON IOIO iO iO Tt- ON r^ ioioo Ti- CO CO 04 00 "* Tt- ^f rj- co t^.VO t^ HH h-C CO co co co co * t^COOO w M O 1^ co O vO lOvO OO O 1-1 r t^ t^oo oo ON ON ON ON ON O t^ O4 iO t^ 04 t^. COOO CO CO ^NO t">- ON oo oo oo oo oo tlO O4 l^- 04 COOO 04 t^ O4 CO lONO ON ON ON ON ON l^ hH rf- f^ ON HI vO O Tt-00 00 ON I-H OJ co ON ON O O O ON ON O O O t^ * O t>-QO 04 O O cOO tONO OO ON O OOOOO O O O O O 00000 00000 ThvO 00 O CM -d-v> 00 O 04 -3-NO 00 O 04 rh\O 00 O 04 ^J-O 00 C oi t^. t^ 1^.00 oo 00 00 OO ON ON ON ON ON O O O O O "H i-< |_! |_l HH O) O4 TABLES 205 n d ON t>- lO \O vO lO lO 10 C4 W M CN M ON >-i ON >O ON (M co co O lOt^ ONOO lO i-iot^-'O oiOf^tt-'-i r^. co ON 10 I-H OO M cO'O ON CM ^}- l^- O CM to l^- O co ... OOONONON ON O O O - I-H 1-1 hi CM CM o c ? f ? r ? c ? lOiO^ft-i I-H O O ON ON OOOOOOt^-t^ ^j- ^- *^- ~j- r^- rf" ^h CO CO co CO CO CO CO ON ^1- ON CO rt- 10 10 10 ri- ^^^^T^ i 4 CS Tt" ^O t'^OO ON "- ' CN ^t" lOMD CO ON (VJ^^M vOCMOOiOrj- CNOONi-<00 ^ xOvONO^O lOiOTh-^-Tt- COCOO4COOJ Q CM04C4O< O4C^CSO ^O CM O "t^ * O^ONO^ON ONONONONON ONONONONON 3-3--tf- cOCMi-O 1 O^O t^ 1 * t^*OO OO ON O O '^ _ i- _< 1-1 H-II-II-IMI-I i-i>-iCM(MCM g 1 oooqoooq ocsoqoqoooo oqoqoqcooo G\G\O\O\ ONONONONON ON ON CN ON ON CO -* ^O 1 O O OO t^^^O ^O ^" ^" co i"^ ON ONOO OO OO Q t>.O'-'l 1> " cMCMOt^cO ONCor^O'-' i-ti-nOoO t^iOcOOOO lOcOOooiO lO t- ON O CM rJ-vO 00 ON t-i CO lO^O OO CMCMCMCO COCOCOCOCO ThTj-rf-rJ-Th o oqoooqoq oqoqoqoqoo oooooqoooq dodo ddddd do* odd t^ -rt- CO CM r>.v> vOvO^O lOiOlO^J-rt- d ONNO OcO lOdOO'^-O ^Oi-iNO>-iiO K. ONMior^. ONI-ICM ^^o t^. ON o CM co r\ H- t co ^" 1 O ^O OO ON O hH 01 cO ^O^ g 1 qqqq qqqqq qqqqq _ _ O CM ^i-\O OO O CM rJ-\O OO O H - : - : CM'CO cocococo'^- >..-, ^. - -. CMCJ CMCMCMMCM CMCMCMCMCM 206 INTERIOR BALLISTICS Cj CMOOO & 10 CO O O O 00 oo r^ t^ IOVO * o rh co * vO iO O Tj-Q I-I CN CNJ 00 I-I HH oo CN 10 ci coco 00 COVO i^. O (N CO 4- rj- oo oo oo rl-\O 00 4-^-^ vO ^ M O CN Tj- lO lO lO 00 co ON 10 r^oo 10 10 10 COOO co co HH 00 00 10 10 O t^- iO iO l^> ON vO d t^ r^t>.vo 1^^ ^8 CN vO CO ~+ 1 * I-I t^ IO Tj-l-l 10 oo i-i r^ VO vO 666 OO O^ t N * 00 Tj-O O CN CO r^ t^ r^ 666 OOv CN ON ^t- 1^ (N CN i-i r^ t^. r^ 666 t>. HH ON r^ t^ 10 O ONOO t^vO vO 666 CN lOOO ^t-oi O f^vO iO vO vO vO 666 o r>- 10 ONt^VO CO CN i-i vO VO vo 666 ^^ O ONOO vO iO iO 666 q t^t^ rt- r>. rf CN covo -i 8* ^ oo t^ IN. 10 Tt-co OO O CO (N CM i-i r^ i-i vo O O ON *-< vO co ONOO 00 OO vO (N >< CO O t ro lOOO i-i TJ- O \O vO iO O CN Tj- w ONVO t> 0) W CN CN i-i t^t^ t^ O ONOO r^vo vo ONVO iO VO 0^ vO vO vO ^0 I-I CO < G\ lO M ro ON ^h COOO iO vO ^J- tN coiooo < 00 Tj-cvj ro t> HH O "000 CO Tj- N t^vo ON O coco I-H CO IO fN . COCO co 00 O CN (N (N t>. ON ONOO CO ON O cocOTi- ON cOvO oo r-^ M r^vo 10 w CN CO 4- -si-4- ti ,"-< M CO O O O rhicvo O O O r^oo ON O i-i CN CO rj- iO vO t^oo ONO HH w PQ W w TABLES 207 I a w 5 W Q CO COM COHH N hH hH hH O ' O O ON ONO oo r^* r^* * .000 00 ON C ON O HH HH CO rf 01 co CO 00 ON O co COM HH 01 CO ss NO NO NO vO O t>. t^t^t^ t^t^l>. t^ i^oo 00 00 OO OO 00 00 ON ON ON 00 00 00 o oo 10 oo t^ r^ CO HH ON r-ri-co SS5ft 2?g; I m vO oo ri- r^-vo to SSS8 10 10 t^ CN HH O M ONOO r^ ON 01 ONOO 00 ON M 00 >O ON M to rh 10 00 00 T|- oo r^ t>. CO Ol HH IO IO IO IO O >O IO IO IO ^J" "^ '^t - ^J" ^ ^* ^ ^ ^ 000 coo 000 O O O O O O O O O O O O Q ONt^. <* hH O fx 10 COM IO IO IO O oo t^ vO rj- CO rt* * * HH oo o CO co CO rj- 10 IO M t^O ONTl-O lOvO I s - CO coco M M O IO O IO r^oo oo co co co r^ co t>. CO co co III M 00 O iO M O 000 O O O O O O O O O O O O O O O O q C HH t"NO IO IO IO IO O) co TJ- rf co O) IO IO IO zn ONOO 00 M NO O t^NO NO COOO M OO OO 00 10 o) oo IO t^. O CO l^ HH rfh- o -go O M 00 0) ON iO 00 HH os hH t^M vO d r^* X 04 00 CO iO O ^O t^oo oo to o ON O ON COOO M HH ON Cj ON t^ 10 HH CO ^ Hi at o oo r^ ON I^NO IO IO IO t^ IO IO lOrtCO IO IO IO CO ON >O ^- o r>. ON hH HH M hH t^. HHO.IO ow O ro Tf O O 00 IO Tt" HH IOM r^ M 10 Tj- O vO t^** K, * CO HH ON U"} ^^ HH 00 Tf HH o o o "st- IO IO NO t^OO M co CO TJ- rJ-iO VO t^oo *"** 44^- ^^-10 101010 IO IO IO IO IO iO IO iO IO 01 C0- ^vot^ OO ON O hH 04 CO ^100 t>.oo ON O M "3- M M M M M M M M CO CO CO CO CO CO CO cocOfO *** 208 INTERIOR BALLISTICS q t^vovo 10 10 Tj- iO tf CO rt- co CO f^ co CN 04 N K. CM HH NH $ ON HH N cOOO CO ON ON ON 00 O " ONO 6 ONCOVO fN rh IO 666 ON 1-1 "* vO 00 ON 666 vO 00 O o -; co CO^J- q * HH \O CO oo t^ r^ O t^ co t^vO vO C4 00 VO vO iO iO IO IO IO 00 l^rj- ^- I-H HH 1 * iO CM CO t^OO ON O ONOO <* CO CO fOO M O) IO CO COCO ON ON 0) IO IO rf CO CO CO ON t^ ON COCO IN CO co co COOOVO O rhON CN HH O CO co co O s * ON CO d CNI t^ co H O vO (N ONOO 00 . T)-OO co iO CO CN| ON ^t- CO O ONOO f^vO vO ON t^. t^ VO IO T^ vO vO vO CO (N) 1-1 vO vO vO vO IO to r^oc o t-N^o vO A t*** C>J O O "?J- CS vo I-H 10 00 OO ON ill vO cOOO" C4 CO CO cOvO ON ^" O vO Tt- IO IO oo n t^ O '"I O co ON iO *o ^ t^* CO O 00 ON !> CO O vO 01 00 00 ON l~>- 00 O CO Tf CO fH as o 10 1->. VO Tf CO iO CO CO (N >-> O vO vO vO ON COO O O\ O*^ O^ 0^ O (N) CO <-< f CO cO CO >-< CO >-^ ON rj- IO IO vO Tt~ ^* v"So? r^oo TJ- vO vo iO OO IO iO IOO IOVOVO vO vO vO r^r^t^ i^ r^oo 00 00 00 u OQ u OH PL, CXI TABLES 20Q Q n t^lO tOTi-CO ff) f^ _ HH M O ON ONOO- OM^OO t-^vo , tOvO H HH N N N N coco CO COCO<* Tj"^-^ Thtoto to to to * ON N iO CO ONOO 1 SR * vo vO to to to ^t- tO CO 1-1 oo t^ to CO CO CO * N O CO CO CO m -" -i ^J* t^^ ^* vO O to HH VO CO O ON ON 00 ^O* oo oo oo N 00 ^O t^cOO odd odd odd odd odd odd odd q RRR vO N ON V^Q ff) HH OO ^O IO fN O ON rj- rf CO CO coco CO cow CO CO CO * t^.00 CO IO IO IO odd * O N iO lOvO to o to odd odd odd O N N ON cO t^ ON O O tO^O vO odd M oo rh d d d ON CN iO HH lOOO N N N odd q CO t^ 't' tO rf O tOfO CO vO N 00 CN N M IO IO ^- OC M VO tO N N ON r>- to ON ON ON t-< CO to CO 1-1 ON ON ONOO oo oo oo oo t-H oo r^ rO *-O iO ^^ ^O vO w CO t> t> ^t" tOvO ON ON CN CO ^ CN N CO - l> ON O 1-1 CO "^- iO vo t^oo o S n TJ- ^J" ^ IO IO iO to to to lO lO to 00 ONO q sH CO ON iO cN cO cO O O N -vO oo oo oo 21^2. OO ON O t^ t^OO t^. ON fN vo to lO 1-1 CN O ^t- tO CO O CM ^ IOVO t^> N CO TJ- tOvO t> 00 00 00 00 00 00 t^OO 00 00 ON O 00 00 ON vO O 00 ON lOOO "-i N CO ^N O O O 00 vO iO cO H oo O to o too d d M to o to o to o I-" N N CO CO Tj- iO O iO o too ^O ^O I s * too to 2IO INTERIOR BALLISTICS w PQ w w PL, PL, q vC vO o 10 o Ti-Ti-Tj- rj-rj-rt-rf * 00 Tf-O ONM CO lOvO vO M3 M t^ ^-vo r^ vO O O CN VO O O O . Q O 00 t>- coc CM 1 vO MD * O CS CM CO C< f> < vO vO 00 r TJ- HH vO vO vO M, 10 O ONO CO lO lO IO 10 M O M Ot^ O rf Tj- 00 00 ON O rt-n o ON rt-Tf rj- CO odd odd odd d d d d Q O OOO CO N CJ 00 f^vO CN c< M 10 Th rf 01 CN CN CO CO CN tN tN 0) W * \O vO 10 HH Tj- t^ fO fO CO v> >O vO CO" 00 O CO lO rh Tt-^J- v} vO vO ^ ON co 00 O CO Tj-lO O \O vO vO t^ O CO IO lOOO O . t^ t>. W hH O iO "3- CO t>- 1^ t^ ON ONOO 00 R3 H t^OO rj- HH ^-XO rl-CN O >-< C< CO \o o ^o vO iO CN vO lO CO OO >O ^ CO *^" >O vO vO vO vO t^.00 ?S:s vO vO t>. vO vO vO 00 t^vO rf * CO rf TJ- Tf M oo 10 00 ON ON O VO vO vO t^ Q 100 vo ^ CO HH 00 0000 O OM^. oo r^ t>. oo oo oo r^. o co r^ t^. t^ oo oo oo C0 HH O t^ t>. t^ t^ 00 00 00 00 X w r^t^ OJ O ON i^oo oo ONO\O\ CO CO CO CN ^0^ c->o d O\vO 1-1 O 00 vO CN CN CO odd -TJ- r^oo ON Tt- 10 too d d d d H o >oo OvONO HI M M 10 o 10 d i-I -! CM M OJ o 10 o M M CO IN O) CS iO O VO O CO ^ rt- iO C< CN M IN TABLES TABLE II 211 k logK D k logK D k logK D 0.60 9.56531 930 0.908 9.84304 2O6 0.979 9.93201 177 0.61 9.57461 922 0.910 9.84510 207 0.980 9.93378 181 0.62 9-58383 915 O.9I2 9.84717 209 0.981 9-93559 185 0.63 9.59298 9 08 0.914 9.84926 2IO 0.982 9.93744 189 0.64 9.60206 902 O.9I6 9.85136 212 0.983 9-93933 194 0.65 9.61108 896 0.918 9.85348 213 0.984 9.94127 199 0.66 9.62004 890 0.920 9-8556I 215 0.985 9-94326 205 0.67 9.62894 886 0.922 9.85776 217 0.986 9-94531 211 0.68 9.63780 882 0.924 9-85993 218 0.987 9.94742 219 0.69 9.64662 879 0.926 9.86211 221 0.988 9.94961 227 0.70 9-6554I 875 0.928 9-86432 222 0.989 9.95188 236 0.71 9.66416 872 0.930 9.86654 22 4 0.990 9-95424 122 0.72 9.67288 871 0.932 9.86878 226 0.9905 9^5546 125 0-73 9-68159 869 0-934 9.87104 229 0.9910 9.95671 128 0.74 9.69028 869 0.936 9.87333 231 0.9915 9-95799 132 0-75 9.69897 869 0.938 9.87564 234 0.9920 9-95931 135 0.76 9.70766 869 0.940 9.87798 236 0.9925 9.96066 139 0.77 971635 871 0.942 9.88034 239 0.9930 9-96205 144 0.78 9.72506 873 0.944 9-88273 242 | 0.9935 9.96349 150 0.79 9-73379 877 0.946 9-885I5 245 0.9940 9.96499 154 0.80 9.74256 880 0.948 9.88760 248 0-9945 9.96653 162 0.81 9-75I36 886 0.950 9.89008 252 0.9950 9-96815 I6 9 0.82 9.76022 893 0.952 9.89260 256 0-9955 9.96984 I 7 8 0.83 9.76915 900 0-954 9.89516 260 0.9960 9.97162 I8 9 0.84 977815 454 0.956 9.89776 264 0.9965 9-97351 203 0.845 9.78269 456 0.958 9.90040 269 0.9970 9-97554 218 0.850 9.78725 459 0.960 9.90309 274 3.9975 9.97772 241 0-855 9.79184 462 0.962 9.90583 28O 0.9980 9.98013 272 0.860 9.79646 465 0.964 9.90863 285 0.9985 9.98285 319 0.865 9.80111 469 0.966 9.91148 292 0.9990 9.98604 414 0.870 9.80580 473 0.968 9.91440 300 0.9995 9.99018 982 0.875 9.81053 478 0.970 9.91740 152 I.OOOO o.ooooo . . . 0.880 9.81530 482 0.971 9.91892 155 0.885 9.82012 488 0.972 9.92047 157 0.890 9.82500 492 0-973 9.92204 1 60 0.895 9.82992 500 0.974 9.92364 161 0.900 9.83492 2OI 0-975 9-92525 165 0.902 9-83693 2O2 0.976 9.92690 167 0.904 9.83895 204 0.977 9.92857 170 0.906 9.84099 205 0.978 9.93027 174 14 212 INTERIOR BALLISTICS TABLE III. Giving the total work that dry gunpowder of the W. A. standard is capable of performing in the bore of a gun, in foot-tons per Ib. of powder burned. 1 Number of volumes of expansion. Corresponding density of products of combustion. 1* Ms !<2J H S.g Difference. Number of volumes of expansion. Corresponding density of products of combustion. Total work per Ib. burned in foot-tons. Difference. .OO I .OOO .56 .641 T.A !?OO 8lQ .01 .990 .980 . 9 80 58 633 OT* O 35-30^ ul y .801 .02 .980 1.936 -956 .60 .625 36.086 785 03 .971 2.870 -934 .62 .617 36.855 769 .04 .962 3.782 .912 .64 .610 37.608 753 05 952 4-674 .892 .66 .602 38.346 -738 .06 943 5-547 873 .68 595 39.069 723 .07 935 6-399 .852 .70 .588 39-778 .709 .08 .926 7-234 -835 .72 581 40.474 .696 .09 .917 8.051 .817 74 575 41.1.56 .682 .IO 909 8.852 .810 76 568 41.827 .67'! .11 .901 9-637 -785 78 .562 42.486 659 .12 893 10.406 .769 .80 555 43-133 -647 13 .885 i i . 160 754 .82 549 43-769 .636 .14 877 11.899 739 .84 543 44-394 -625 15 .870 12.625 .726 .86 537 45.009 .615 .16 .862 13-338 713 .88 532 45-6I4 -605 17 855 14.038 .700 .90 526 46 . 209 595 .18 .847 I4-725 .687 .92 521 46.795 -586 19 .840 15.400 -675 94 515 47-372 577 .20 833 16.063 .663 .96 .510 47.940 .568 .21 .826 16.716 -653 .98 505 48.499 559 .22 .820 17-359 643 2.00 .500 49.050 55 1 23 .813 17.992 -633 2.05 .488 50-383 1 -333 .24 .806 18.614 .622 2.10 .476 5I-673 .290 s .800 794 19.226 19.828 .612 .602 2.15 2.20 465 454 52-922 54-I32 .249 .,210 2 l .787 20.420 -592 2.25 444 55-304 .172 .28 .781 21.001 -581 2.30 435 56.439 135 29 775 21.572 571 2-35 425 57-539 . 100 30 .769 22.133 -56i 2.40 4 T 7 58.605 .066 32 758 23-246 1.113 2-45 . .408 59-639 034 34 .746 24-324 i .078 2.50 .400 60 . 642 i .003 36 735 25-37I 1.047 2-55 392 61.616 974 38 725 26.389 1.018 2.60 384 62 . 563 947 .40 42 7H .704 27-380 28.348 .991 .968 2.65 2.70 377 370 63-486 64-385 923 899 44 .694 29.291 943 2-75 363 65 . 262 .877 .46 .685 30.211 .920 2.80 357 66.119 -857 .48 50 .676 .667 31-109 31 986 .898 .877 2.85 2.90 351 345 66.955 67.771 .836 .816 52 54 .658 .649 32.843 33-681 857 -838 2-95 3-oo 339 333 68.568 69-347 797 779 1 From Noble and Abel's " Researches on Fired Gunpowder TABLES 2I 3 Number of volumes of expansion. Corresponding density of products of combustion. Total work per Ib. burned in foot-tons. Difference. Number of volumes of expansion. Corresponding density of products of combustion. "% C iis 321 3- Difference. 3-05 .328 70.109 .762 7.10 .141 105.125 539 3.10 .322 70.854 745 7.20 -139 105-655 530 3-15 317 7L584 731 7-30 137 106.176 521 3.20 .312 72.301 .716 7.40 135 106.688 512 3-25 .308 73.002 .701 7-50 133 107. 192 504 3-30 .303 73.690 .688 7.60 131 107.688 .496 3-35 .298 74-365 675 7.70 .130 108.177 .489 3-40 .294 75.027 .662 7-80 .128 108.659 .482 3-45 .290 75.677 650 7.90 .126 109.133 474 3-50 .286 76.315 .638 8.00 125 .109.600 467 3-55 .282 76.940 .625 ! 8.10 .123 110.060 .460 3.60 .278 77-553 .613 8.20 . 122 110.514 454 3-65 .274 78.156 .603 8.30 . 1 2O 110.962 .448 3-70 .270 78.749 593 8.40 .119 III. 404 .442 3-75 .266 79-332 583 8.50 .117 III. 840 436 3.80 .263 79-905 573 8.60 .116 112.270 430 3-85 .260 80 . 469 564 8.70 US 112.695 425 3-90 .256 81.024 555 8.80 .114 113.114 .419 3-95 -253 81.570 546 8.90 . 112 113.528 .414 4.00 .250 82.107 537 9.00 . Ill 113-937 .409 4.10 .244 83.157 1.050 9.10 . no 114 341 .404 4.20 .238 84.176 i .019 9.20 .109 IH 739 398 4-30 .232 85.166 990 9-30 .108 II5-I33 394 4.40 .227 86.128 .962 9.40 . I0o 115-521 .388 4-50 .222 87.064 .936 9-50 .105 115-905 384 4.60 .217 87.975 .911 9.60 .104 i 16.284 379 4.70 .213 88.861 .886 ! 9.70 .103 116.659 375 4.80 .208 89.724 863 9.80 . 102 117.029 370 4.90 .204 90-565 .841 9.90 . 101 H7-395 366 5.00 .200 91-385 .820 10 . 100 H7-757 362 5.10 .196 92.186 .801 II .091 121.165 3.408 5.20 .192 92.968 .782 12 .083 124.239 3-074 5-30 .188 93-732 -764 13 .077 127.036 2.797 5-40 .185 94-479 -747 H .071 129.602 2.566 5-50 .182 95-2io 731 15 .066 131.970 2.368 5.60 .178 95-925 -715 16 .062 134-168 2.198 5-70 175 96.625 .700 17 059 136.218 2.050 5.80 .172 97.310 -685 i l8 .055 138.138 .920 5-90 .169 97.981 .671 ! 19 .052 139-944 .806 6.00 .165 98-638 -657 20 .050 141.647 703 6.10 .154 99.282 .644 21 .047 143-258 .611 6.20 .161 99.9I5 633 22 045 144.788 530 6.30 159 100.536 .621 23 043 146.242 454 6.p .I 5 5 ! 101.145 .609 24 .042 147.629 -387 6.50 -154 101.744 599 25 .040 148.953 ! 1-324 6.60 151 102.333 589 30 033 154.800 1 5-847 6.70 .149 102.912 579 35 .028 159.667 4.867 6.80 .147 103.480 .=68 40 .025 163.828 4. 161 6.90 145 104.038 558 45 .022 167.456 3.628 7.00 143 104.586 548 50 .020 170.671 3-215 WORKS CONSULTED HUTTON: "Mathematical Tracts," London, 1812. RUMFORD: "Experiments to Determine the Force of Fired Gunpowder," London, 1797. RODMAN: "Experiments on Metal and Cannon and Qualities of Cannon Powder," Boston, 1861. NOBLE AND ABEL: "Researches on Explosives," London, 1874, 1879. NOBLE: "On the Energy Absorbed by Friction in the Bores of Rifled Guns." Reprinted as "Ordnance Construction Note," No. 60. "On Methods that have been Adopted for Measuring Pressures in the Bores of Guns," London, 1894. "Researches on Explosives." Preliminary Note, London, 1894. OFFICIAL: "English Text-book of Gunnery." Editions of 1897 and 1902. SARRAU: "Recherches sur les effets de la poudre dans les Armes," and "For- mules pratiques des vitesses et des pressions dans les Armes." A translation of these memoirs into English by Lieutenants Meigs and Ingersoll is given in Vol. X of the Proceedings U. S. Naval Institute. "Recherches theoriques sur le chargement des bouches a feu." Transla- tion by Lieutenant Howard, O. D., in "Ordnance Construction Note," No. 42. "War Powders and Interior Ballistics." A translation by Lieutenant Charles B. Wheeler, O. D., as "Notes on the Construction of Ordnance," No. 67. Washington, 1895. Gossox AND LIOUVILLE: "The Ballistic Effects of Smokeless Powders in Guns." Translated by Major Charles B. Wheeler, O. D. "Notes on the Construction of Ordnance," No. 88. Washington, 1906. DUNN: "Interior Ballistics." Part I. "Notes on the Construction of Ordnance," No. 89. Washington, 1906. SOUICH: "Poudres de Guerre. Balistique Interieur," Paris, 1882. BAILLS: "Traite de Balistique Rationnelle," Paris, 1883. LONGRIDGE: "Internal Ballistics," London, 1889. MEIGS AND INGERSOLL: "Interior Ballistics," Annapolis, 1887. PASHKIEVITSCH: "Interior Ballistics." Translated from the Russian by Captain Tasker H. Bliss, A. D. C. Washington, 1892. BERGMAN: "Larobok i Artilleriteknik." Del I. Krutlara. Stockholm, 1908. A part of this work was translated for the author by Colonel Lundeen, Coast Artillery Corps, U. S. Army. GLENNON: "Velocities and Pressures in Guns," Annapolis, 1889. CROZIER: "On the Rifling of Guns." Ordnance Construction. Note No. 49. A. W.: "Des Armes de guerre Modernes et de leurs Munitions." Revue Militaire Beige, Vol. II, 1888. McCuLLOCH: "Mechanical Theory of Heat," New York, 1876. PEABODY: "Thermodynamics of the Steam Engine," New York, 1889. RONTGEN: "The Principles of Thermodynamics." Translated from the German by Professor A. Jay Du Boise. New York, 1889. LISSAK: "Ordnance and Gunnery," New York, 1907. Encyclopaedia Britannica, eleventh edition, 1911. 215 INDEX ABSOLUTE temperature, definition of, 17; of fired gunpowder, 40, 47. Adiabatic expansion, definition of, 26. Air space, initial, definition of, 76; expressions for reduced length of, 76, 77, 92, 94- Angular acceleration, 174. Applications of velocity and pressure formulas: To magazine rifle, 130 to 134; to Hotchkiss 57 mm. rapid-firing gun, 125 to 130; to 6-inch English gun, 115 to 125 and 140 to 147; to 6-inch Brown wire gun, 150 to 162; to 8-inch rifle, 102 to no; to lo-inch rifle, 179; to i^-inch rifle, 162 to 169 and 184; to hypothetical 7 -inch gun, in, 136. Artillery circulars M and N, references to, 88, 135, 186. Atmospheric pressure, value of, 21. Axite, form-characteristics of, 61. BALLISTIC pendulum, 2, 3. Ballistite, 61, 140. Binomial formulas for velocity and pressure, 112. Bliss, Captain Tasker H., 53. Board of Ordnance, reference to, 151. Boyle, Robert, 15. B N powders, form-characteristics of, 61 ; computation of by velocity for- mula, 129. CAVALLI, reference to, 8. Centervall, law of combustion, 79. Chamber, reduced length of, 31 ; alignment of grains in, 71; effect cf varying volume of, in, 112, 166. Characteristic equation of gaseous state, 17. Characteristics of a powder, 94. Charge of powder, behavior of when ignited in a gun, 12, 13; in a close vessel, 12, 35; initial surface of, 73, 77. 217 21 8 INDEX Chase, excessive pressure in, 55, 158. Chevreul, reference to, 8. Chronograph, Noble's, 116; Boulenge'-Breger, 126. Coefficient of expansion of a perfect gas, 16. Combustion of a grain of powder, 11; under constant pressure, 55, 79; under variable pressure, 79, 80. Composition: of gunpowder, i; of cordite, n, 117, 124; magazine rifle powder, 131; ballistite, 140. Constants, physical, adopted, 92, 94. Cordite, composition of, n, 117, 124; form-characteristics of, 63. Cube, form-characteristics of, 61. Cylindrical grains: solid, 63; with axial perforation, 65; with seven perfora- tions (m.p. grains), 66 to 72. D'ARCY'S method of experimenting, 4. Density: of powder, 1 1 ; of a gas, 21 ; of loading, 37, 75, 77. Dulong and Petit, law of, 21, 23. ELSWICK works, mention of, 115. Encyclopaedia Britannica, eleventh edition, reference to, 115. Energies neglected in deducing equation for velocity, 121. Energy of translation of projectile, 32, 51, 52, 53, 80, 144. English Text-Book of Gunnery, reference to, 115. Euler, mention of, 88; equations of, 174. Examples: of expansion of gases, 28; of the formulas of Chapter III, 77; relating to 8-inch rifle, 109; to 6-inch gun, 122, 124, 144, 155; to 14-inch rifle, 1 66. Expansion, work of: isothermal, 25; adiabatic, 26; in the bore of a gun, 30, 47. FACTOR of effect, 49, 52, 54. Force of the powder, 33, 36. Formulas: Characteristic equation of gaseous state, 17. For specific heat under constant volume, 22. For work: of an isothermal expansion, 25; adia- batic expansion, 26, 27, 32; of gases of fired gunpowder, 49. For tempera- ture: of an adiabatic expansion of a perfect gas, 26, 27; of gases of fired gunpowder, 47. For pressure: isothermal, 15, 17; adiabatic, 27; gases of fired gunpowder in close vessels, 6, 36, 39; in guns, 45. For pressure in guns with smokeless powders: While powder is burning, 85, 86, 101, 106, INDEX 219 112, 140, 152; after powder is all burned, 86, 102. Maximum pressure, 91, 101, 1 06. Initial pressure when powder is all burned before projectile moves, 86, 87, 93, 94, 99. For velocity of projectile in guns with smokeless powders: while powder is burning, 83, 84, 89, 100, 101, 103, 112; after powder is all burned, 84, 85, 102. For limiting velocity, 85, 93, 94, 98, 101, 127, 141, 146, 150, 164. For computing f, 32, 36, 84, 93, 94, 96, 97, 106, 164. For v c , 91, 92, 93, 94. For k and k' ', 58, 59, 61, 62, 63, 65, 68, 69, 72, 89, 90, 109, 136, 141, 148. For M, M' , N and N', 84, 85, 93, 94, 95, 97, 101, 102, 106, 112, 113, 114, 115, 118, 136, 139, 150, 155, 161, 165. For y, 39, 83, 93, 132, 146. For a, A, fJ., 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 84, 129, 134. For P', 86, 87, 93, 94, 98. For the X functions, 87, 88, 89, For X r ^o' 84, 93, 94, 119, 136, 149, 150, 152, 164. Working formulas, 77, 93, 94, 95, 97. For inclination of groove, 171, 172, 177. For pressure on lands, 175, 176, 180, 181, 185. For semi-cubical parabola, 176. Common parabola, 178. Frankford arsenal, mentioned, 130. GAS, perfect, 17. Gay-Lussac, law of, 16; mentioned, 8. Gossot, Colonel F., law of combustion, 79; igniter, 151. Graham, mentioned, 8. Grains of powder, combustion of under constant pressure, 55 ; vanishing-sur- face, 56; volume burned, 57; form-characteristics, 58; their relation to each other, 58, 59. Granulation, 151, 163. Groove, developed, 171; width of, 179. Gun-cotton, 10, n. Gunpowder, i, 2. HAMILTON, Captain Alston, length of m.p. grains, 71. Heat: mechanical equivalent of, 18; specific heats, 18, 19, 21, 22. Hugoniot, law of combustion, 79. Hutton, Dr. Charles, experiments with gunpowder, 3, 4. INFLAMMATION of a grain and charge of powder, n, 12, 13. Isothermal expansion, 25. JOURNAL U. S. Artillery, references to, 71, 79, 125, 148. 22O INDEX LANDS, width of, 179. Lenk, General von, experiments with gun-cotton, 10. Liouville, R., law of combustion, 79. Lissak, Colonel O. M., ordnance and gunnery, 29; construction of velocity and pressure curves, 144. Longridge, Atkinson, loss of energy in gun, 53. MAGAZINE rifle, description of, 130. Marriotte, law of, 15. Maximum pressure in a gun, 90, 91 Maximum value of X 3 , 90, 101. Mayevski, mention of, 8. Monomial formulas, 100. Muzzle velocities and pressures, computed, 107, 120, 123, 124, 129, 130, 133, 134, 143, 161, 169. NATURE, reference to, 115. Neumann, mentioned, 8. Nobel, N. K. powder, law of combustion for, 79. Noble and Abel, experiments with fired gunpowder in close vessels, and deductions therefrom, 33 to 54. Noble, Sir Andrew, experiments with 6-inch gun, 115; coefficient of friction, 1 80. Notation, 15, 17, 19, 23, 24, 31, 51, 56, 58, 60, 67, 72, 74, 75, 76, 79, 80, 81, 82, 83, 84, 85, 86, 91, 92, 108, 149, 171, 172, 174, 176. Notes on the construction of ordnance, reference to, 102. ORDNANCE Department, reference to, 130, 162, 166. Otto, mentioned, 8. PARALLELOPIPEDON, form-characteristics of, 60. Pashkievitsch, Colonel, lost work in a gun, 53. Piobert, mentioned, 8. Point of inflection of X 3 , 101. Powder grains. See Grains of powder. Powder, smokeless. See Composition. Pressure: of fired gunpowder in close vessels, 6, 7, 9, 35, 37; in guns, 41. RADIUS of gyration of projectile, 174, 180, 186. Retarding effect of uniform twist, 186. INDEX 221 Rifling of cannon, advantages of, 170. Robins, Benjamin, experiments with fired gunpowder, 2. Rodman, Captain T. J., experiments with fired gunpowder, 8; perforated grains, 9; cutter gauge, 9. Rumford, Count, experiments with fired gunpowder, 4; comparison of re- sults with those of Noble and Abel, 6. SAINTE-ROBERT, Count de, law of combustion, 79. Sandy Hook, mention of, 151, 160. Sarrau, ET., law of combustion, 80; monomial formula for pressure in a gun, 95. Schonbein of Basel, discoverer of gun-cotton, 10. Sebert, law of combustion, 79. Spherical grains, form-characteristics of, 59. Springfield Armory, mentioned, 130, 131. TABLES, in text: of specific heats of certain gases, 22; of pressures in guns of fired gunpowder, 46; of velocities and pressures in guns, 104. 107, 129, 130, 133, 134, 143, 161, 165, 169, 183, 185; of pressure on lands, 183, 185. Temperature of fired gunpowder, 45. Trinomial formulas, 138. Twist, uniform, 171; increasing, 171. VIEILLE, law of combustion, 79. WEAVER, General E. M.. Notes on explosives, referred to, n. Work of fired gunpowder, 47. Working formulas, 77, 92, 181, 185. SHORT-TITLE CATALOGUE OF THE PUBLICATIONS OF JOHN WILEY & SONS NEW YORK LONDON: CHAPMAN & HALL, LIMITED ARRANGED UNDER SUBJECTS Descriptive circulars sent on application. 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