DU PONT BALLISTIC GRANT STUDIES THE UNIVERSITY OF MICHIGAN, ANN ARBOR, MICHIGAN REPORT # 2 DEVELOPMENT OF BULLET JACKET FACTOR AND RIFLE BARREL FACTORS by Faculty Director Lloyd E. Brownell Du Pont Fellows M. -W. York W. Phillips T. Blackwood Research Assistant K. Jacob Not to be reproduced without special permnlission October 1968

Copyright 1968, Lloyd E. Brownell

REPORT # 2 DEVELOPMENT OF BULLET JACKET AND RIFLE BARREL FACTORS DU PONT BALLISTIC GRANT STUDIES THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN Page List of Tables v List of Figures vii I. RESUME' 1 II. BULLET RESISTANCE, A PROBLEM 1 III. METHODS OF CORRELATION 6 IV. DIMENSIONAL SIMILITUDE 7 V. FOURIER'S EARLY WORK 9 VI. RAYLEIGH'S SERIES 14 VII. LANGHAAR'S PROOF 18 VIII. DIMENSIONLESS GROUPS USED IN BALLISTICS 20 IX EVALUATION OF THE CONSTANTS "K" AND "b" 23 X JACKET FACTORS FOR OTHER BULLETS 29 XI RIFLE BARREL FACTORS 39 REFERENCES 46 APPENDICES 48 iii

REPORT # 2 DEVELOPMENT OF BULLET JACKET AND RIFLE BARREL FACTORS DU PONT BALLISTIC GRANT STUDIES THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN List of Tables Page I. Fourier's Example of Exponents of Dimension 12 II. Experimental Firing Data for Springfield Barrel C 24,25 III. Data Used for Basic Correlation of IMR #3031 Powder in 30-06. J Values for 220 g Rem SPCL Bullets in Springfield Barrel C 32 IVoA J Values for Various Bullets Used in 30-06 (Test Barrel C, 9/16 in. Seat) 33 IV.B J Values for Various 150 Grain Bullets Used in 30-06 (Test Barrel D, 0.340 Seat) 34 V. Data Used for Correlation of 43 R-Values for Five Different Barrels Using IMR # 3031 Powder

REPORT # 2 DEVELOPMENT OF BULLET JACKET AND RIFLE BARREL FACTORS DU PONT BALLISTIC GRANT STUDIES THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN List of Figures Page 1. Dimensions of Bullets of Various Weight Made from 220 g Rem SPCL Bullet 3 2. Test Firing Facilities at The University of Michigan 4 3. Selected Oscilloscope Photo Negatives of 2 ms Display from Firings in 30-06 Test Rifle Using Barrel D, Lot 1229 IMR 3031 and 0.34 in. Seat 5 4. Plot of B' vs S for Bullets of 110 Grain to 476 Grain Weight Data from Table II 26 5. Plots of Data from Table III and Equation 31 27 6. Charts for Calculation of J 36 7. Selected Oscilloscope Negatives with 2 ms Sweep of 3 Rd Firings of Various 150 g Bullets in 30-06 Test Rifle, Barrel D with 51 g IMR 3031 Powder Lot # 1229 37 8. Selected Oscilloscope Negatives with 2 ms Sweep of 3 Rd Firings of Various 150 g Bullets in 30-06 Test Rifle, Barrel D with 51 g IMR 3031 Powder Lot # 1229 38 9. B'/J versus S for Five Different Barrels Plotted on Logarithmic Scales 41 10. B'/J versus S for Five Different Barrels Plotted on Plain Coordinate Scales 42 11. J versus d with Parameters of w/d (pounds/sq. in.) 45 vii

REPORT # 2 - THE DEVELOPMENT OF BULLET JACKET AND RIFLE BARREL FACTORS I. RESUME' For three years, studies in internal ballistics at The University of Michigan have investigated new ballistic methods. The first publication in Ordnance1 described a new method for measuring simultaneously both the bullet travel in the barrel and the chamber pressure. The new technique involved the use of collimated beams of gamma radiation from the radioisotope iridium-192. Since then, two alternate methods have been developed and put into use as described in this report. Technical Report # 1 on studies2 conducted with aid of the DuPont Ballistic Grant described a procedure for measuring the true or absolute chamber pressure without the use of a special test barrel or crusher gages. A calibration curve to convert crusher values to true ones was included. A summary of this study was published in the third edition of the Handloader's Digest.3 In this Report # 2, a new procedure for predicting the maximum chamber pressure is presented. The bullet jacket factor (J) is a new characterization number which, with the sectional density of the bullet and the specific loading density of the powder, permits calculation of the maximum chamber pressure for a given powder, cartridge, bullet, and gun. Thus far the correlation has been limited to IMR No. 3031 powder but the correlation is of a fundamental type, being based upon the principle of dimensional similitude. Based on data for other powders, the correlation will be extended to include powder characterization factors. II. BULLET RESISTANCE, A PROBLEM For many years handloaders have been aware of the influence of bullet type on maximum chamber pressure (p). In the past, however, attempts to correlate the different values of (p) have proved unsuccessful. Waite" investigated such variables as bullet length, bearing length, core hardness, pressure, velocity, and the velocity-to-pressure ratio for bullets of identical weight and caliber but supplied by ten different manufacturers. In these tests, he used the identical seating depths, the same powder load (51 grains of IMR No. 3031 lot No. 157), Western cases, and No. 210 primers by Federal. For all tests, he used a Winchester

-2pressure barrel, 24 inches long with four grooves and a twist of one turn in 10 inches. Ten firings of each bullet were made but no correlation was found to exist between the different bullet factors measured and the magnitude of the pressures and velocities. Waite stated that his most important observation was the presence of a pressure spread of more than 7,000 psi resulting from the use of different bullets. Waite5 reported another series of tests entitled "Loads for the.30-'06." Here, the bullet weight was varied from 110 to 220 grains using over 40 combinations of bullet and powder loads. Again, using identical cases, primers and powder loads but different bullet types, there was a wide difference in pressures obtained. While Waite and other handloaders agree that bullet type does influence the maximum pressure, no serious attempt has been made to explain this phenomenon. This study, based upon research conducted at The University of Michigan, introduces a correlation which successfully predicts the influence of the bullet "jacket factor" on the maximum pressure. The jacket factor (J) is a dimensionless characteristic of the bullet. It is measured in relation to a reference bullet, and is the relative ratio of maximum chamber pressures. We assume a theoretical frictionless reference bullet to have a (J) of 1.0; the bullets to be tested may then have a (J) of 1.10 to 1.40. Since the (J) is proportional to the ratio of maximum pressures, a bullet with a (J) of 1.30 will produce 30 per cent more pressure than a bullet with a (J) of 1.0, other loading conditions being equal. While such a range in pressures may seem surprising, ranges of this magnitude have been measured in test firings at Michigan. The study of the jacket factor (J) developed from the continuation of studies on the maximum chamber pressure in the 30-06 cartridge described in Technical Report # 1. In the earlier studies, the bullet weight and type had been limited in most firings to the 220-grain Remington SPCL bullet. The studies of (J), however, utilized special bullets made from the 220-grain Remington SPCL bullet (see Fig. 1) but having various weights (from 110 to 476 grains), and which were test-fired with varying IMR No. 3031 powder loads. The equipment used to obtain the pressure measurements in the (J) studies (see Fig. 2) was the same as that described in "Absolute Chamber Pressure," Technical Report No. 1. Data were obtained in the form of oscilloscope displays with 2ms time sweep (see Fig. 3).

1.:.536 0 220 Gr. SPCL 1.116 FACTORY BULLET 200 Gr. CUT SPCL li 2H~~~- ~~- -4- ~t~ -Wj-X — Ii3.300 180 Gr. CUT SPCL.816 2.463 150 Gr. CUT SPCL L338 ~- a - ~-Q -X) —..308 128 Gr. 220 Gr. SPCL CARCANO 348 Gr. COMPOUND BULLET 130 Gr. CUT SPCL 3.83C= 1-J25 1.125I 50I3 o.2i - 29 0.3081 +X== t ~ ~~i3- - 9 —- L..308 128 Gr. 128 Gr. 220 Gr. SPCL CARCANO CARCANO 110 Gr. CUT SPCL 476 Gr. COMPOUND BULLET Fig. 1. Dimensions of Bullets of Various Mass's made from 220 gr. Rem. SPCL Bullet

~~~~~~~Oscilloscope ~Strain Gages Removed Section of Barrel and Receiver Showing Location of 0t~~~~~~ ) ~Strain Gages Oscilloscope Input Strain Gage Leads -Strain Gage Rifle Mount Ci rcu i t Note: Power Supplies are not shown Fig. 2. Test Firing Facilities at The University of Michigan

-5- s 2s me 200 s Timeus:r -- 2ms - r 2 ms_ a. Calibration Firing, 3rds, 44gr 3031, 220 gr SPCL b. Herter 150 gr Banana Peel Bullet. 2 rds, p 64, 000 psi 48 gr * 3031 - 51, 200 psi 30 gr' 3031 - 25, 600 psi e.. nps.. ~200P 200 o s Time 200us Times 2 ms 2 ms - ms c. Remington 150gr Bronze Pt, 8 rds, d. Herters 150gr Half Lead 48 gr "3031. 54, 200 psi 44 gr *3031, 35, 5GO psi 44 gr " 36, 000" 42 " 26,600 42 " 34, 500 " 38" " 21, 700 38 " 30, 500 " 30 " 17, 700 36 " 23, 600" 23 " " 6,900" 30.. " 13, 800" Fig. 3. Selected Oscilloscope Photo Negatives of 2 ms Display from Firings in 30-06 Test Rifle Using Barrel D and Lot 1229 IMR 3031 an 0.34" seat

-6III. METHODS OF CORRELATION An understanding of the approach used to solve the problem of the effect of the jacket factor on the maximum pressure requires some discussion of the variables and theoretical concepts involved. When considering possible solution procedures to an internal ballistics problem, one is faced with an extensive number of variables. Among these are: (1) maximum chamber pressure; (2) caliber of the gun; (3) weight of the bullet; (4) weight of the powder load; (5) case and barrel volume; (6) type of powder used; and (7) bore and bullet diameter. Still others are the powder position factor, powder temperature at ignition, case base diameter, primer hole diameter, primer ignition factor (manufacturer and type), bullet engraving factor (manufacturer and type), barrel rifling twist, the barrel-engraving factor, and the barrel-wear factor. The "main" problem of internal ballistics is the solution of the relationships between chamber pressure (p), bullet velocity (v), bullet barrel-travel length (A), and time (t). European ballisticians such as Charbonnier7 of France have usually related these variables to "y," the fraction of powder burnt at any time, "t." American ballisticians for many years related "p," "v," and "t" to bullet travel length "&i" using the empirical methods developed by the French artillery officer Leduc. Both of these procedures have certain disadvantages. The equations of Charbonnier are based on the energy balance. Various modifications of his equations have had worldwide use in solving problems in internal ballistics. A major difficulty in using Charbonnier's equations, however, is that the fraction "y" of powder burnt is not zero at bullet start where v = 0 but "p" is "Po," a definite pressure required to "start" the bullet on its path. The methods of Leduc are not based on the equation of the conservation of energy (First Law of Thermodynamics) and so they lack generality. New equations and constants must be determined for each type of gun. Most correlations in internal ballistics have been based on what is called energy balance. When the energy "stored" in the powder is known, it can be equated to the energy required to: (1) accelerate the bullet down the barrel; (2) accelerate a portion of the powder down

-7the barrel; (3) work on atmosphere by muzzle blast; (4) work on shooter through recoil; (5) while at the same time accounting for heating the chamber, shell case and barrel; (6) frictional losses down the barrel; and (7) initial cold-working (grooving) the bullet into the rifling of the bore. But this method is also complex, and prediction of pressure, velocity or other variables difficult. IV. DIMENSIONAL SIMILITUDE Another approach which allows remarkably simple methods of solution for many problems in internal ballistics utilizes the principle of dimensional similitude. Dimensional similitude allows the correlation of the significant variables by means of compact equations without having to know the individual energy relationships involved. Buckingham's Pi theorem of dimensional similitude8 is a classic in its field. First presented in 1914, the theorem is based upon the Rayleigh9 principle of dimensional similitude and logic. Stated in words the theorem might read as follows: Given any system, natural or artificial, and all of the variables affecting that system, one can, by appropriate grouping, explain the behavior of such a system no matter how complex. The phrase "by appropriate grouping" refers to the grouping of parameters in such a way that each group is dimensionless. These dimensionless forms raised to the appropriate power Buckingham called iT groups. Buckingham based his theorem on logic and made no attempt to prove it mathematically. It wasn't until 1951 that Langhaarll proved it in a rigorous manner. He defines the theorem as follows: If an equation is dimensionally homogenous, it can be reduced to a relationship among a complete set of dimensionless products. A set of dimensionless products is complete if each product in the set is independent of the others and every other product of the variables is a product of powers of dimensionless products in the set. In a later paper,12 Buckingham described a procedure for "dimensional reasoning," which is referred to today as "dimensional

-8analysis." Buckingham gives as an example of dimensional reasoning for the case of the resistance (R) of still air to the motion of a smooth sphere and the influence on velocity, v. This relationship would apply to the external ballistics of spherical shot. Buckingham states that for this system some experience with fluid flow and the fundamentals of physics is necessary. One can proceed by listing the variables believed to influence the resistance (R). Obviously, the total resistance will increase with the diameter (D), one of the significant variables. The properties of the air affecting fluid flow are also important. These are: viscosity (l), density (p), and compressability (C). Density and viscosity change with temperature. The energy of the sphere is partially dissipated by work done on the air and by frictional losses in the form of heat. The transfer of heat to the air will depend upon the specific heat (cp), the thermal conductivity (X), and the emmissivity (a). Thus, a list of nine variables can be compiled: R, V, D, p, iI, Cc, p, a By considering the order of magnitude of each of the nine variables, some may be eliminated. Since for practical problems, the temperature effect will be small, the variables cp, X, and a will have an insignificant effect. If the velocity of the sphere is low, there will be virtually no compression of still air. Thus, compressibility (C) can also be eliminated. This leaves five significant variables which give the following equation: F (R, V, D, p, ) = (1) Thus, by simple analysis and common-sense reasoning together with a general knowledge of the variables affecting the system, an equation including all the more important variables can be deduced. Whether the reasoning is sound and whether or not Eq. (1) is sufficiently complete can be determined by comparing experimental data with the theoretical equation. Although Buckingham's publications are often referred to in discussions of dimensional similitude, the method did not originate with him. As early as 1892, Lord Ra leigh had used the principle in his own publications with great success. However, he did not discuss the procedure until 1915,10 nor did he consider it an original contribution

-9on his part. In fact, he commented that Sir Isaac Newton, probably the greatest natural philosopher of all time, was well acquainted with this basic law of nature and used it extensively. In our review of Newton's Principia13 we found ample evidence of its use. A very simple case is Newton's repeated use of the proportionality of similar sides of geometric figures of different size but similar shape. In a generalization, Newton states that "the ultimate principle is precisely expressible as one of similitude, exact or approximate, to be tested by the rule that mere changes in the magnitudes of the ordered scheme of units of measurement that is employed must not affect sensibly the forms of the equations that are the adequate expression of the underlying relations of the problems." Thus we can trace the origin of the name of similitude to Newton who refers to the principle as the "rule of similitude." V. FOURIER'S EARLY WORK An extensive search of the literature has shown that Jean Fourier, philosopher, mathematician, and friend of Napoleon, was the first to describe the procedure of dimensional analysis and the use of the principle of similitude as we know them today. He comments that the principle is "derived from basic concepts of'quantities' used in geometry and mechanics and is the equivalent of fundamental lemmas which we inherited without proof from the Greeks." He makes this statement in his best known work, Theorie Analytique de la Chaleur, published in Paris in 1822.14 In this remarkable treatise, he presents the development of the mathematical series and the theorem known by his name. However, his contribution to dimensional analysis in this publication has received little or no recognition, and authors of texts and articles on the subject of dimensional analysis omit him from their lists of references, probably because their literature surveys have been limited to more recent publications. There is no question but that he was a master of the use of this versatile tool. In treating the "quantities" of thermal conductivity, heat capacity and the coefficient of heat transfer, he comments as follows: In order to measure these quantities and express them numerically, they must be compared using five different dimensions of: length, time, mass,

-10temperature and, finally, the dimension which serves to measure the quantity of heat (or energy)..... One of the at first surprising facts shown by dimensional analysis is that all purely physical quantities, such as density, velocity, acceleration, work, area, volume, weight, resistance, conductivity, power, and the hundreds of other terms we use to describe physical phenomena, can be reduced to combinations of just five dimensions. Fourier continues to explain this in discussing the problem of heat transfer: In the analysis of the theory of heat any equation (E) which we use expresses a basic relation between the magnitudes: length (x), time (t), Temperature (T), and the coefficients c, h, and k (for heat capacity, heat transfer and thermal conductivity, respectively). The relation we obtain depends in no respect on the choice for the unit of length, because if we took a different unit to measure the linear dimension, the basic equation (E) would still be the same. For example, suppose the unit of length be changed, and its second value equal to the first divided by a number'm.' Regardless of the value of'x' in equation (E), it represents a certain distance'ab' between two points'a' and'b.' This distance, ab, is some multiple of the unit length and thus in the new system of length units becomes'mx.' Although we change the unit of length to mx the values of the time (t), and the Temperature (T) will not be changed. This is not the case with the quantities of h, k and c. The first, h, becomes h/m2 because it expresses the quantity of heat which escapes during the unit of time from the unit of surface at temperature 1. Inspection of the thermal coefficient, k, shows that it becomes k/m because the flow of heat varies directly as the area of the surface, and inversely as the distance between. The heat capacity (c) also depends on the unit of length (as it expresses heat per unit volume) and becomes c/m3.

-11Equation (E) must undergo no change when we write mx instead of x and, at the same time, k/m, h/m2 and c/m3 instead of k, h and c. Thus, the number m must disappear after these substitutions. The exponent on the dimension of length is 1, that of length in k is -1, that of h is -2, and that of c is -3. Now, if we attribute to each quantity its own exponent of dimension, the equation will be homogeneous, since every term will have the same total exponent number. Numbers such as S and V that represent surfaces and volumes have exponents of dimension of 2 and 3, respectively, on the linear measure. Angles, sines and other trigonometrical functions, logarithms or exponents of powers are, according to the principles of analysis, absolute numbers which do not change with the unit of length. Their exponent of dimension must therefore be taken as 0, which is the exponent of dimension of all abstract numbers. If the unit of time is changed from 1.0, so that the new time is 1/n, the number t will become nt and the numbers x and T will not change. The coefficients k, h, and c will become k/n, h/n and c. Thus, the exponents of dimension of x, t and T with respect to the dimension of time are 0, 1, 0; and those of k, h, c are -1, -1, and 0. If the unit of temperature be changed so that temperature 1 becomes a lower temperature in the ratio of 1 to the number p, T will become Tp, x and t will keep their values and the coefficients k, h and c will become k/p, h/p and c/p. The following Table I indicates the exponents of dimension of the three undetermined quantities x, t and T and the three coefficients k, h and c with respect to each kind of dimension.

-12Table I. Quantity of Constant Length Duration Temperature Exponent of dimension of x 1 0 0 Exponent of dimension of t 0 1 0 Exponent of dimension of T 0 0 1 The specific conducibility, K -1 -1 -1 The surface conducibility, h -2 -1 -1 The capacity for heat, c -3 0 -1 To apply this rule to different equations and their transformations, the equation must be homogeneous to each kind of dimensionsand every exponential quantity must cancel to nothing. If this is not the case, some error has been committed or an incomplete expression has been introduced. The following equation for heat transfer may be used as an example: dT k d2T hi T dt c 2 cS dx Fourier's equation above (Eq. 2) which he uses as a demonstration states that the change in temperature (dT) with respect to change in time (dt) is equal to the constant k/c times the rate of change of temperature gradient (d2T) with respect to change of distance per incremental length dx2 minus the heat loss which is directly proportional to the heat transfer coefficient (h), thickness (f) and temperature (T) and inversely proportional to the heat capacity (c) and the surface area (S). If we substitute the values of the exponents from Table I we have: T1 -1 -1 1 -2 -1 -1 1 1 T T1 x T x t T x T (3) 1 -3-1 2 -3 -1 2 (3) t x T x x T x

-13Collecting exponents of dimensions we have T 1 3 T1 2" 7TJt tI[x3 T1 t x The terms in the brackets cancel, leaving the dimensions of unit of temperature per unit of time or: T T T Equation (5) and therefore Eq. (2) are homogeneous. Dividing byy the equation becomes dimensionless. t Returning to more recent times, Rayleigh in 1915 writes: I have often been impressed by the scanty attention paid even by original workers in physics to the great principle of similitude. It happens not infrequently that results in the form of "'laws" are put forward as novelties on the basis of elaborate experiments, which might have been predicted a priori after a few minutes' consideration. However useful verification may be, whether to solve doubts or to exercise students, this seems to be an inversion of the natural order. One reason for the neglect of the principle may be that, at least in its applications to particular cases, it does not much interest mathematicians. On the other hand, engineers, who might make much more use of it than they have done, employ a notation which tends to obscure it. I refer to the manner in which gravity is treated. When the question under consideration depends essentially upon gravity, the symbol of gravity (g) makes no appearance, but when gravity does not enter into the question at all, g obtrudes itself conspicuously. Lord Rayleigh

-14VI. RAYLEIGH'S SERIES A simple statement of the principle is that all physical interrelationships may be expressed as functions and coefficients and are completely independent of any arbitrary system of units used in experimental measurements, provided the system is homogeneous. That is, there exists a general relationship for all physical phenomena: Regardless of the complexity of the functional relationships between the individual physical variables influencing the phenomenon independently or dependently, these variables need be expressed only in the form of a series of dimensionless ratios. For example, consider a system of "n" variables independent of dimension. We will call the variables Q1, Q2, Q3,.. Qn. If we wish to determine the relationship between Q1 and the other variables we can let the power on Q1 be unity and write the following equation: 2 3 4 n 1 1 a2 a3 4 n (6) Where the greatest interest is in "Q2" as a function of the other variables, a similar equation is written: b b b b Q2 = K2 Q1 Q3 Q4 n (7) In both Eqs. (6) and (7), "K1" and "K2" are dimensionless constants and the powers a2, a3, a4 ~ ~* an; and bl, b2, b4... bn are exponents determined by experimental measurements in which all variables but one are held constant. For example, the quantities Q1, Q3, Q4 ~ ~ ~ Qn may be held constant while "Q2" is varied. Experimental data will then show how "Q1" varies with "Q2". If we take logarithms of Eq. (6) with quantities Q3, Q4 ~ ~ ~. Qn held constant, we have a3 a4 a lnfQ1- aa2knQ 2 nK1 n[Q3 4 n (8)

-15or and a2 nQ1 = a2InQ2 + nC2; Q1 C2Q2 (9 By holding quantities Q3', Q4 Q. Qn constant, the product Of "Q3, Q4 4.. Qn an" will also be constant so that Eq. (8) will reduce to Eq. (9) the equation of a straight line. Common practice in evaluating the constants is to plot the logarithm of "Q1" versus the logarithm of "Q2." The slope of the straight line obtained by plotting the experimental data will be equal to "a2" and the intercept at Q2 = 1 is equal to logarithm of C2. The above procedure can be repeated for all the variables Q3, Q4... Qn to establish C2, C3, C4... Cn. Knowing these values for the constants, "C," we see that a substitution can be made for constants K2, K3, K4... Kn, and the exponents a2, a3, a4 ~ ~. an. Expressions for the other variables similar to Eq. (7) for "Q2" can be derived from Eq. (6) by normalizing the exponent for the variable of interest. For example, let us assume that in solving Eq. (8) we find the slope of the line drawn through the data to be 0.50; this means that Q1 is proportional to the square root of "Q2", or: 0.50 a3 a4 a Q=KQ 3 4 n Q K!1Q2 Q3 Q4' Qn Now if we solve for "Q2' we square all other quantities in the equation to obtain power of unity on Q2, or: 0.5( 2 K2 2 2a3 2a(10) (2 Q2 1 1 Q3 Qn (10) 2 Comparison of Eq. (10) with Eq. (7) indicates that K2 = K1; similarly, b1 = 2al, b3 = 2a3... 2an where a2 = 0.50. Thus, once the value of constant "Ki" has been established along with exponents a2, a3... an, Eq. (6) may be rewritten for "Q2" as in the case of Eq. (7) or for Q3 or Q4 by similar equations by the procedure described for normalizing the exponent to unity on the variable of interest.

-16Note that Rayleigh's method involves an infinite series from Q1 to Qn, whereas the methods of Fourier and Buckingham consolidate the dimensionless ratios into a smaller total number of groups. Both methods have their advantages. The variables influencing a particular system are most easily recognized as ratios. Then, by combining ratios according to the magnitude of the exponents, the total number of "Q" quantities may be reduced to shorten the equation. The greatest advantage to such combination is the great reduction in the number of experimental measurements required to determine the constants in the equation. The effort required for experimental determination of a function can be very great. As Langhaar (11) comments, a function with a single variable may be plotted as one curve. If two variables are involved a family of curves may be used with one curve each for selected values of the second variable. With three variables, sets of curves are involved in a chart and so forth. Then we come to sets of charts. If we use five points to establish a curve, 25 points will be required for a chart, 125 for a set of charts, and 625 points for five sets. Thus the effort required increases exponentially with the number of variables. If we have several variables and determine more than the minimum number of values, the effort involved can become excessive. Thus the reduction of the number of variables by combination in dimensionless groupings can greatly reduce the labor of experimental investigations. The proof that an equation of the form of Eq. 6 can be used to solve any physical relation is not self-evident. Let us first consider some forms the equation may take. In its most reduced form, Eq. 6 for an infinite series can be limited to two quantities, Q1 and Q2, with the exponent of aa reduced to unity, or: a2 1.0 1 kQ2 = kQ2 =kQ2 (11) Then as an example let us say that Q is the dimensionless ratio of two units of length, x and y, and that 2 has a value of 1.0. This gives the equation of the simplest of curves, a straight line through the origin, or:

-17() = k(1.0), or, y = kx (1la) y where k is the slope of the line. Then let us allow Q2 to have a value other than 1.0 and be equal to the ratio of two velocities, v1 and v2, with exponent a2 equal to 2.0. To explore the nature of Q2 we will now let Q1 have a value of unity: 1.00 kQ k ( 1) (12) 2 v 2 This could be the equation for the ratio of kinetic energies of two masses traveling at different velocities, v1 and v2, and k, the ratio of the masses. When plotted on plain coordinated paper this equation will give a curve that rises slowly at first and then more sharply. a3 We can Continue in such a manner introducing Q3 with a3 equal to a negative exponent, say -3. This also gives an exponential curve of a reciprocal form of Q2 and will curve downward. The curve of the total products of Qlal, Q2a2, Q3a3 can rise sharply or dip sharply or through various combinations first rise to a maximum, then dip downward, resembling a pressure time curve for a rifle. Inclusion of additional quantities Q4a4, Q5a5... Qnan permits great variation in the shape of the curve of the combined quantities. Herein lies part of the strength of the method. In the experimental studies which are necessary for use of this method we hold all quantities constant except one. Then step by step we can determine the values of "K" and "a" for each "Q" by experimental measurement. When we complete our evaluation of these constants and substitute them into Eq. (6), we can use the combined equation to predict the result when several variables are changed simultaneously. Although these examples demonstrate the power and versatility of the method, we still need proof that the equation must be homogeneous and that the homogeneous equation is adequate for any system.

-18VII. LANGHAAR'S PROOF (11) We can prove quite easily that the equation must be homogeneous and either be free of dimensions or, in the case of the equation for a sum, all quantities must have the same dimension. For illustration take the case where "f" is the sum of the Q's, or: Q1 f (Q2' Q3. 3 Qn) Q2 +3 * Qn (13) or, K (Q1 +Q2 + Qn) = KQ1 +K2Q2 +''' K Qn (14) This is an identity in Q's. Therefore: =1 K2 n This can be true only if Q... Q all have the same dimension. This requirement for homogeneity simply means that different dimensions such as inches and pounds are not directly additive. Consider now the case of the product equation, a favorite of Rayleigh, as in Eq. (6) where Q1, Q2... Qn are dimensionless ratios: a2 a3 a Q1 K1Q2 Q3 * n (6) Taking logarithms as in Eq. 8 gives: 1 - K1 + a2 Q2 + a3 hQ3 +. an nQ (8a) Equation (8a) reduces to Eq. 14 because the logarithms of numbers Q1, Q2, and Qn are without dimension, as are the exponents al, a2... an. Therefore, the equation is homogeneous. Equation 8a would not be homogeneous if Q1, Q2... Qn had different dimensions. The equation also is homogeneous if all quantities have the same dimensions as in Fourier's example. In this case, division of the equation by the common dimension leaves the equation dimensionless and demonstrates homogeneity.

-19In a similar manner but introducing the use of matrices, Langhaar (11) proves Buckingham's r theorem: h h h hr 7r22. 1 (16) 1 v2'''2Tn where Tl1, 1T2... 7~n are not simple ratios but instead are groups of quantities arranged in dimensionless forms, such as: a' a' a 1 2 n 1, Q Q2 ***Qn.&(17) al ar art 2 Q1 Q2 Q n (18) a!p a2p anP n Q Q.. Qnn (19) Combining equations gives: Q (a'h +. aPh ) Q (ah + a p) Q (a' + aph) (20) 1_1 p 2 21 2 p n n I n p This is an identity in Q's and the exponents vanish. Langhaar states that "this is contrary to the hypothesis that the rows in the matrix of exponents are linearly independent. Thus it is proved that when the rows in the matrix of exponents are linearly independent, the dimensionless products are independent." This is used to prove the theorem that "a necessary and sufficient condition that the products TI, 7T2... 7p be independent is that the rows in the matrix of exponents be linearly independent." The Buckingham IT theorem basically constitutes a recognition that a truly "infinite" series, such as Rayleigh's Eq. (6), is not required for most problems and usually three to several dimensionless ratios may suffice. In the case of internal ballistics, our studies to date

-2 0indicate that over 20 dimensionless ratios are required but that a number of these may be combined using the methods of Fourier and Buckingham. Buckingham's theorem also introduces an algebraic procedure for manipulation so that these quantities may be combined to produce the minimum number of combined groups required for a given number of dimensions and number of quantities. In our present stage of study we have been able to investigate only a portion of the known variables. We expect to discover new quantities in the course of future study that may have only minor influence but which will permit further refinement of the correlation. Thus, at present we are not in a position to make full use of Buckingham's method and will base the first correlations primarily on the use of the infinite series of Rayleigh. VIII. DIMENSIONLESS GROUPS USED IN BALLISTICS Let us now rationalize a bit about the factors that influence "p" in a rifle. First, there is the bullet, which can be characterized in external ballistics by the sectional density, "w/d2", where "w" is the bullet weight and "d" is the bullet diameter. In external ballistics, a shape coefficient "C" is also used to calculate air friction on the bullet. Before reaching the muzzle, however, the bullet is in contact with the barrel rather than with air, and a different coefficient is necessary. This coefficient, called the jacket factor, (J) must be used for the barrel friction and resistance caused by cold working as the bullet enters the rifling and is "engraved" or grooved. As mentioned earlier, the jacket factor is defined as the ratio of the values of (p) produced by the test bullet to the value of (p) produced by a reference bullet; all other factors (barrel, powder type, powder load, primer, case, seating depth, etc.) are held constant. That is: J (for bullet considered) p (for bullet considered) J (for reference bullet) p for reference bullet) (all other factors constant) (21) Other factors known to affect (p) are the load of powder (i), the case volume (Vs) at a given bullet seating depth, the primer 5

-21and the powder type. For the present, these last two, powder and primer type, will be held constant. The ratio of the powder load (L) to case volume (Vs) is also known as the "loading density (Q/Vs)." In the past, this ratio has been used with success to correlate ballistic information, and, therefore, it will be used here. Loading density has the units, "lbs/cu in," in the English system and "gms/cc," in the metric system. These are the units of the absolute density (a) of the powder. IMR No. 3031 has an absolute density of 1.67 gm/cc in the metric system and 0.0604 lbs/cu in in the English system. The absolute density of the powder can be used to convert the loading density to the specific loading density (S) which is dimensionless and, therefore, independent of any system and of any units used. S = V (22) a V Other factors such as length of free bore, barrel wear and gas leakage also influence "p" but these too shall be considered constants for the present. It is now necessary to determine the number of -T groups required to correlate these variables. This could be done using the mathematical procedure described by Buckingham, or by the intuitive method so often used by Rayleigh. We can see by inspection that one dimensionless rT group is defined by "S," the specific loading density. The remaining factors, "w," "d2," "p" and "gc" can be grouped into another dimensionless quantity as follows: B' = 7T (p) (d) (23) 4(106) (w/gc) Or, for the 220-grain, 30 cal, Rem SPCL bullet: B' - 3.1416(32.17) (7000) (0.308) (p) 4 (106) (220) = 0.762 (10 4) (p) (24)

-22Now, for any bullet for 30-06 B' = 0.01675 (w), or rearranging: (25) p = 59.7 (w' B') (26) where: B' = Brownell number, megagees p - maximum chamber pressure, psi X = 3.1416 gc = 32.17 ft/sec2 w = bullet weight, lb w' = bullet weight, grains d = bullet diameter, in. The reason that the factors of r/4(106) and "gc" were inserted was to give "B"' called the Brownell number for convenience, more significance. The numerator of Eq. (23) divided by 4 defines the maximum force on the bullet because force equals maximum pressure (p) times area (rd2/4) of bullet base. Now, if the bullet mass is divided by the gravitational constant (gc) the result will be the gravitational acceleration, because, by Newton's Law of Motion, F = wa/gc, or a = F/(w/gc). Thus, the numerator of Eq. 23 divided by "4 w/gc" is equal to the ratio of the maximum acceleration in the rifle barrel to the gravitational acceleration. This dimensionless ratio, also found in high velocity aircraft and in centrifugal machines such as the centrifuge, has been termed "gees." The "gee" value for bullet acceleration by the powder is very large (over 1 million). Therefore, the gee value can be divided by 106 (one million) resulting in the more convenient "megagee," equal to a million gees. According to the rules of dimensional similitude, our two groups can be related by: b B' - K - = K (S)b (27) aV

-23Equation (27) is verified by determining the values of "KI" and "b". If these are constants, Eq. (27) is valid. IX. EVALUATION OF THE CONSTANTS "K" AND "b" Experimental data for various firings are given in Table IIa through IIe, and data on "B" versus "S" are plotted in Figs. 4 and 5. Although none of the data produce identical curves, they do have a number of characteristics in common. All bullet masses and powder loads produce data that show a close grouping and a linear relationship for the logarithm of B' versus the logarithm of S, when B' is more than 3 megagees. The line is steep with respect to the logarithm of S scale and gives a slope of 4.0. An expanded scale is used for better analysis. Data for low-loading densities and for values of B' less than 3 megagees show scatter indicating erratic pressures. Bullets of one type and one "J" were produced from the 220grain Remington SPCL (Soft Point Core Lokt) bullets. Lighter bullets were cut from the 220-grain ones, cut precisely on a lathe to a tolerance of ~ 1 grain. The sizes cut were 200, 180, 150, 130, and 110 grains; the lengths are shown in Fig. 1. The objective in cutting bullets of different weights from the 220-grain bullet rather than using commercial bullets of various weights was to eliminate the variation in engraving resistance known to exist in commercial bullets of different weights. Bullets of identical weights but produced by different manufacturers have been shown to produce pressure variations of as much as 10,000 psi when fired in the same rifle with identical cases, primers and charges of the same powder. For the purpose of analysis, data for different bullets but identical powder loads (40 to 52 grains of IMR No. 3031) are assembled in Table III. These powder loads give values for B' of from 3.28 to 9.15. Data in Table III show good agreement in the values of B' for 220, 200, 150, 130, and 110 grain bullets cut from the 220 grain Remington SPCL bullets. Data on these were more closely compared by determining the average values of B' for 40, 42, 44, 49, 50, and 52 grains of load. None of the data for these

TABLE II (a, b, c, d) DATA FOR FIGURE 4 EXPERIMENTAL FIRING DATA FOR SPRINGFIELD 30-06 BARREL C a b c d 110_ Cut (SPCL) 130g Cut (SPCL _ i so RN (SPCL) 150g Cut (SPCL) Load S L.E~gr An/a I ppsi B' ppsi BJ ppsil B' p,psi B' 20.2 0.201 -- --- --- --- 0.67 23.0 0.232 1 ________ 7, 000 0.90 -— ____ --- i 1.23 25.0 0.252 — ___- _ I - 9,000 1.16 -— _ —26.0 0.262 j f ------ --- ____ --- j 000 }2 27.0 0.272 1 -_ _ f _ 11 L92L 1.42 28.0 0.2-Z-8~2~ —]~ I —- I —- ---— _ --- _ --- 1 ____ 3 __ 000 29.0 0.292 --- ______- _____ ____ I _2L01______ 13,500 1,51 38. 0__ -6 10-00 11.A _54 30.0 0.302 _ 12,000 1.5467 20 1_ 21-2 - 1 33.0 0.332 ___10 5 O1, 60 I 13,000 1.67 ____2 20 1 2.2319,0001 36.0 0.363 14,000 093 1 17,400 f 224 -- 21,000 2.35 37. O _I O. 3Z?~~_~?_________~-i-~- 21,000 2.34 1 ____ ______ 37.0 0.373__ _ --- -I _________________ _____ 380 0.383 1 -1- 22,500 2.9011____ 139.0 0.393 f -- Y Ii9 3.24 40.0 0.403 9_28 6,000 3, 2.79 1 _____u0* } 3.46 41.0 0.413 (_ —- --- 3.68 42.0 0.4231~7O00I 3.96 31,000 3.993 96 --- 36.500* 4.08 43.0 0.45-3~3~- j -I --- 34 - 3.80 40,300* 4.48 44.0 0.443 1 2O~ 4.88 j _ --- --- 44,000 4.91 45.0 0.453 36,000 5.48 43000 5.54 1 --- --- 50,000 5.57 46.0 0.463 40,000 6.10 6.18 j 4J09Q 4.58 1 99 6.14 48.0 0.484 4 8 g0 7.32 57,000 7.35 5312._000 5.92 j30 7.10 49.0 0.494 51OOckJ* 7.77 597,800*.70 57-500 L 6.43 j67.800* 7.58 50.0 0.504 54,000 _,.24 62,500 8.05 j 2Pfl 6.93 --- --- 52.0 0.524 60,000 9.15 --- *Interpolated or Extrapolated POWDER DUPONT IMR 3031, LOT 1229; LOT 1229; BARREL "C", SPRINGFIELD 30-06, 03A3, 0.310" GROOVE DIAMETER AT BREECH (FOR 1/4"), BORE WEAR BY US ORDNANCE BREECHBORE GAGE 0.10; SEATING DEPTH 9/16" (TO CANNELURE); PRIMERS AND CASES REMINGTON COMMERCIAL: BULLETS CUT FROM 220 GRAIN REMINGTON SPCL EXCEPT FOR 150 g RN (30-30) SPCL.

TABLE II (e, f, g, h) EXPERIMENTAL FIRING DATA FOR SPRINGFIELD BARREL C e f g h 200g Cut (SPCL) 220gr Rem (SPCL) 348gr Compound (SPCL) 476gr Compound SPCL L,gr S ppsi B p,psi B ppSi B p,psi B 20.0 0.201 11,000 0.86 --- --- 23.0 0.231 13 200 1.11 --- _ _ _ _ _ 25.0 0.251 15,400 1.28 -_1 26.0 0.262 --- -- --- --- --- --- 0000.9 27.0 0.272 17,000 1.43 19,200 1.47 25,000 1.21 1 __ 29.0 0.292 -— ___ _24,000 1.84 29,000 1.40 35__ _ 1.25 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1_ _ _6 8 _ _ _ _ - - - _ _ - - - _ _ _ 1 40..3....._ _ _ _ _ 30.0 0.302 20,000 1.68 --- --- 34,000 1.1-64 - 32.0 0.322 23,200 1.95 -Y ___ 1 47,000 1.65 34.0 0.342 27,500 2.T2 -- ____ I- - _ N) 35.0 0.353 --- --- 30,000 2.30 73,0 2.59 6 2.14 36.0 0.36 29,700 2.50 --- -L0.373 --- --- 1,000~~~~~~~~~~~~~~~~~~~ -- --------— ~~ -~~ULI-M~~37.0 O7 35,000 27 __ 56 70 7 3.06 2.88 --- -— 7 38.0 0.383 36,300 3.06 37 5j00 2.88 _ _ _ -- -- 39.0 0.393 -— _4__ --- _0 64 000 3.09 83o000 2.92 40.0 0.403 j~4j 8 3.51 44,000*_13.35 &j, 000 3,28 86 9p 3.03 41.0 0.413 45 000 47,000 3.57 74,000 3.57 91 000 3.20 42.0 0.423 47,800jQ 3.9 52,000* 3.95 81,000 3.90 102_2 3.59 43.0 0.433 50q600 4.24 57,500 4.37 --- 44,0 0.443 58 500 4.92 64 000 4.88 --- 45.0 0.453 65p000 5.43 --- --- --- 48.0 0.484 93,100 7.11 9.0 0.494 i -- 100000 7_62 -2 __ _ ___ _ _ _ *Iinterpolated POWDER IMR 3031, LOT 1229; BARREL "C", SPRINGFIELD 30-06, 03A3, 0.310" GROOVE DIAMETER AT BREECH (FOR 1/4"), BORE WEAR BY US ORD BREECHBORE GAGE OF 0.10; SEATING DEP1.h1 OF 9/16" (TO CANNELURE); REMINGTON CASES & PRIERS.

-260.0 1' 9.0 8.0 7.0 6.0 5.0.0 - ixz; I o i 1 i c34' Copon 16 cm o I~ ~ ~ ~~~~ DI I z9 o "~ X; OxcY/ + I10 Gr. SPCL Cut 0 03 130" o I 9iv f A 150" "11 2.0 - - 200 II 0 220,,,, Factory ox to 348 Compound 476, 70 x 4 cm 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 S = Specific Loading Density Fig. 4. Plot of B vs S For Bullets of 110 Grain to 476 Grain Mass (Data from Table I)

Modified Brownell Number, B' its 0~~~0 0~~ Its 0'3 PI 1 I - Cf9~~C 00 C ~ ~ ~ __ __ I___ 4-K'_ ___ oct~c 0 - Ub) 0 C3 I~~~~~~~~~~~~ o o 4 ~~~~~~~~J. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ O ~3 N~ -_ CI _ _ _ _ _ I.- C I\. o c Cf9 4!.~~~~~~~~~~~~~~~~~~~~~~~~~~ (D__ ____ _ __ __ I K _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ __ _ _ cn ~ 0 __ _ _ __1 _ _. _ 1 t_ O ~~ U C-~~~~~~~~~~C

-28bullets and loadings deviated more than 3.5% from the respective average value. The data became more erratic at lower loadings of 30-38 grains. In the region of loads as low as 20-30 grains, "pressure excursions"' giving over 100% variation were experienced (see "Pressure Horn" and "Pressure Excursions" in Technical Report No. 1);2 these were not plotted. In Fig. 5, the logarithm scale for S was expanded four times the logarithm scale of B' permitting a more precise determination of the constants for the linear equation of the logarithm of B' versus logarithm of S plot. The scale for the logarithm of S extends only from 0.38 to 0.60 because of the four-fold expansion. Note that five data points (indicated by circles 0) equaling S values of 0.403, 0.423, 0.443, 0.483, and 0.494 fall almost exactly along one straight line (labeled Curve A). These points correspond to powder loads of 40, 42, 44, 48, and 49 grains and represent average values of "B " for the same values of "S" for three to five different bullet weights. The two points to the upper right are based on two bullet weights (130 and 110 grain) for a 50-grain load (S - 0.504) and one weight (110 grain) for a 52-grain load (S = 0.524). Since the other five points represent averages of five different bullet weights (110 to 220 grains), they were considered the most important in locating Curve A. The slope of the curve is 45~ or, 4.3 in./4.3 in. or 1.0. If this is multiplied by the scale expansion of four to one, the product is 4.0. This value is the true slope of the curve and the value of the exponent, "b," in Eq. (27). The next step is to determine the coefficient "K" by substitution of corresponding values of "B"' and "S" into Eq. (27). For convenience, we can let B' = 3.00 and read from Curve A of Fig. 5 the corresponding value of "S." Inspection of Curve A shows that for B' = 3.00, the value of "S" is 0.392. The only unknown in Eq. (27) now is "K." If we substitute the numerical values for "S," "b" and "B"' we can solve for "K": B' = K (S)b (27) 3.00 = K (0.392)4 K223.00 3.00 = 20 4(0.392)4 0. 0236

-29Therefore: B220 =127 S4 (28) To check Eq. (28), let S = 0.50 and solve for "B,"' or: 4 B' = 127 (0.50) - 7.95 Reading from Curve A we obtain B - 8.00, for S = 0.50. The error is 5/800 or 0.6 per cent. Thus we have established the validity of Eq. (27) for the 220 grain Rem SPCL bullet. Note that Eqs. (27) and (28) are identical in form to the exponential version of Eq. (9). X. JACKET FACTORS FOR OTHER BULLETS In compiling Table II, only data for bullets of a similar type, fired in the identical rifle barrel were used. These data were for 220grain, Rem SPCL bullets and for the special bullets made from them: 110, 130, 150, 200, 348, and 476 grains, respectively. The value of "J" for all of these bullets is the same, 1.38. Table I lists one set of data for the Rem 150-grain, SPCL (RN 30-30) bullet which has a different value of J. The 220-grain Rem SPCL bullet has a thicker jacket than does the 150 grain Rem factory SPCL bullet. This is because the former is designed for use with large game where penetration is important. The heaviner iMack__ et keeps the bullet from mushrooming as quickly as lighter bullets do and allows more penetration before expending its energy. Therefore, bullets cut to 150 grains from the 220-grain Rem SPCL bullet will have thicker jackets than the factory made 150 grain bullet. The bullet with the tougher jacket will require more work from the powder gas to force it into the rifling of the barrel. The result is an increase in pressure for the more penetrating bullet, although the mass is the same for both. In Fig. 5, Curve B (indicated by crosses +) representing 150grain Rem SPCL bullets (low penetration) can be compared to Curve A representing the 220-grain Rem SPCL bullet (high penetration) and the specially cut 200, 150, 130 and 110 grain bullets. Curve B is parallel to

-30Curve A, but lower, corresponding to a lower pressure. The equation for Curve B as for Curve A can be determined using a slope of 4.0. Again, if we select a value of B' = 3.00 for convenience, we read the value of S to be 0.410. Therefore: B 150 = K150 (0.410)4 = 3.00 3.00 3.00 K10 -'4 - 106 (0.410) 0.0282 B'0 106 S4 (29) 150 To show the significance of Eq. (29), we can take the ratio of Eq. (28) to Eq. (29) and obtain the ratio of "p" for 220 grain to that for 150 grain Rem SPCL bullets. B220 P220 K220 220 BP150 P150 K150 150 or: K150 _ 1.38(106) 83(1.38) = 1.15 150 220 K220 127 Thus, the "J" for 150-grain, Rem SPCL factory bullets (RN 30-30) is 83% of the values for the 220-grain Rem SPCL bullets. To demonstrate, consider a value of "S" of 0.433 which corresponds to a load of 43.0 grains. From Eq. (28) we have: B' = 127 (0.433) = 4.48 and, from Eq. (26) for J - 1.38 p = 59.7 w'B' = 59.7(150)(4.48) - 40,200 psi

-31(This checks with the value of 40,300 psi in Table Id for 150-grain, specially cut SPCL) Multiply by 0.83 to correct for the lower jacket factor of the 150 grain factory bullet: p = 0.83(40,200) - 33,400 psi (for 150 grain RN) (This checks with the value of 34,000 psi in Table Ic for 150-grain RN) As demonstrated, we can switch from Curve A to Curve B by using the ratio of J values. Another procedure is to divide "B"' by "J" and plot the curve of "B'/J vs S." All bullets having values of "b = 4.0" would then be represented by the single line of Curve C (see Fig. 5). The equation for Curve C may be written using Eqs. (28) or (29). Using the first: B'/J = 127/1.38 (S)4 = 92 S4 = U (30) or, the second: B'/J = 106/115 (S)4 92 S = U (30) Curve C of Fig. 5 will be useful for graphical solutions of Eq. (30) and, therefore, is not cluttered with data points. However, all of the data from Tables II and III in the range of S = 0.38 to S = 0.57 will fall on, or very close to the line of Curve C. Handloaders interested in chamber pressures will have an interest in J values and the relative pressures for commercial bullets supplied by various manufacturers. More measurements need to be made to permit prediction of "J" from basic physical characteristics. Table III summarizes the data obtained as of June, 1966. A few comments should be made about Table IV. In Part A, some data are included from Tables II and III, along with new data on additional firings using the same barrel, C, and a range of loads with each bullet type. These are presented with their corresponding J values. These

-32TABLE III DATA USED FOR BASIC CORRELATION OF IMR #3031 POWDER IN 30-06 J VALUES FOR 220 GRAIIN REMINGTON SPCL BULLETS IN SPRINGFIELD BARREL C Powder Load, Grains 40,0 42.0 44.0 48,0 49.0 50.0 52.0 Specific Loading Density S, Dimensionless Bullet 0.403 0,423 0.443 0.484 0.494 0.504 0.524 Weigh'iJ, w' Gra ins B = Brownell Number, Megagees 220 B' 3.35 3.95 4.88 7.26 7.62 % Dev -1.2% -1.2% 0.2% 0.0% -0.6% 200 B/ 3.51 4.01 4.92 % Dev 3.5% 0.3% 1.0% 150 B' 3.46 4.08 4.91 7.10 7.58 % Dev 2.1% 2.0% 0.8% -2.0% -1.2% 130 B' 3.35 3.99 4.77 7.35 7.70 8.05 % Dev -1.2% -0.2% -2.2% 1.2% 0.4% -1.2% 110 B/ 3.28 3.96 4.88 7.32 7.77 8.24 9o15 % Dev -3,2% -1.0% 0.2% 0.8% 1.3% 1.2% 0.0% Avg B' 3.39 4.00 4,87 7.26 7.67 8.15 9.15?OWDE'R DUPONT IMR 3031, LOT 1229; BARREL "C", SPRINGFIELD 30-06, 03A3, 0.310" GROOVE DIAMETER AT BREECH (FOR 1/4"), BORE WEAR BY US ORDNANCE BREECHBORE GAGE OF 0.10; SEATING DEPTH OF 9/16" (TO CANNELURE). CASES AND PRIMERS REMINGTON COMMERCIAL; BULLETS CUT FROM REMINGTON 220g SPCL.

-33TABLE IV A J VALUES FOR VARIOUS BULLETS USED IN 30-06 (Determined by Range of Firings in Test Barrel C with Groove Diameter of 0.310" at Breech) IMR 3031, Lot 1229, Seating Depth 9/16", Rem. Cases and Primers Bullet Bullet Supplier or Jacket We ~, ht Type Manufacturer Factor w', gr. 220 SPCL Remington 1.38 110,130 Cut from 150,200 220 g SPCL Remington (Modified) 1.38 348 Compound Remington + FMJ 1.38 220 SPCL (Italian) Military + 128 Carcano 476 Compound Remington + FMJ 1.38 220 SPCL (Ital san)Military + 2x128 Car. 150 SPCL Remington. 20 RN(30-30) 150 Partition Nosier 1.10 150 Spire Pt. Hornady m.i8 150 Soft Pt. Speer 1.22 Spitzer

-34TABLE IV B J VALUES FOR VARIOUS 150 GRAIN 30 CAL. BULLETS Based on a Load of 51 Grains of IMR 3031 Powder (Lot 1229) and Triplicate Firings in 30-06 New Test Barrel D (Springfield) with Seating Depth of 0.34 Inches, Remington Cases and Primers Bullet Supplier or Maximum *B=Brownell Jacket Type Manufacturer Pressure Number, Factor pmax, psi Megagees J** Partition Nosier 61,100 6.82 1.17 SOnic Herter 61,100 6.82 1.17 Spire Pt Herter 69,000 7.69 1.30 Half Lead Herter 66,000 7.38 1.25 Banana Peel Herter 63,000 7,04 1.20 FMJ-M2 US Army 63,000 7.04 1.20 Pt SPCL Remington 63,000 7.04 1.20 S.P, Spitzer Speer 64,000 7.16 1.21 Spire Pt Hornady 65,000 7.26 1.23 pmax pmax * by Eq 12: B = 1 - ** J150 = 150 59.7m 896 (9/16) 6.4 by Eq 17: B = 92S4 = U150 = 92(00695)4 = 6.4 J150 B Ji~~~~.(0.34") 5.9 (9/16) J150(0.34") = 1.08 J150(9/16") Note: The quantity 1.08 is the "seating depth factor" and gives the pressure ratio for the same load of IMR 3031 when seating depth is decreased from 9/16" to 0.34" as determined from Fig 20 of Ref 2 (Technical Report No. 1).

-35data are plotted in Fig. 6. Basically, Fig. 6 is a graph of "J" times Eq. (30) with "B"' plotted versus 92 S4. Because 92 S4 is inconvenient we will let it equal U, University number (see Eq. 31). Note that in Fig. 6, "J" is not included with "B!" Therefore, only firings with a "'J" of 1.0 will fall on the 45~ line running from 1,1 to 10,10. Parallel lines of J = 1.10, 1.20, 1.30, 1.40, and 1.50 are drawn which permit interpolation for J. Equation (30) holds only for B' values of 3 to 10 or higher because of erratic burning and pressure excursions at low loading BJ = U, (where U = 92 S (31) J densities (such as those giving pressures much under 30,000 psi). Note the single Curve X at the upper left corner of Fig. 6. This line represents a cross plot of "J" versus "B"' for a specific value of U (or 92 S4) equal to 6.40 (this is equal to 51.0 grains of IMR No. 3031 powder in a 30-06 case with bullet seated to 0.34 in.). Curve X was used to determine the J values for the firings in Table III, Part B, in which all of the loads were 51 grains of IMR 3031. The data in Part B were taken from Figs. 3, 7, and 8 after the test barrel, C, had been replaced with a new one, D. Barrel C became slightly oversized (0.310 in. groove diameter for about the first 1/4 in.) as a result of earlier tests. Barrel D was a new Springfield 03-A3 which met government specifications when installed. For comparison with NRA data, several 150-grain, 30 caliber bullets from different suppliers were tested with 51 grains of IMR No. 3031 powder. In order to use data in the literature for comparison with the data of Parts A and B, it is necessary to convert crusher values to absolute pressure. Curve Y in Fig. 6 is a plot which corresponds approximately to the relationship between crusher value and absolute pressure, psi (see Refs. 3 and 6). To use Fig. 6, enter with the crusher value, move to Curve Y and read absolute pressure. Note that the relation for Curve Y is not linear but exponential.

-36For Use With "Y" U = 6.4 J(For 92S4 =6.4) psi, Absolute Pressure x 104 1.0 1.10 1.2.31.4 1.5 2 3 4 5 6 7 8 9 1 91 j===::1 "i== —''=~ —I ta ~-~ ~:. ~vr.:: 77'::: ~:;, +:~~~~~~~: 6 6 0 _i-_i~~~i';= -~~~- I- - - ~~~~~ ~~~LU -!::t;' ii il i":;I!:i" 0 ~~ ~~ ~~~~~~~~~~~~~~~;t11,110 ~~~~~~~~~~~~~~~~~~~ ~o -~~~~~~ Q)~~~~~~ 3 3 2 3 4 5 6 7 8 9 10 92S =U:= (University No.) Fig. 6. Working Charts for Calculation of J + 4i f ue~tl 1 - jt~i i~it f " t Ifi4ut( -r~~~~~~~i itt11 i 1 c Z i ~~~~-4 tf i1 t- t!.t - 1: 1 5~~~~~~~~~I I il 11il IH IJ I 1. III111111,111h~~~~~~~~~~~~~~ri I I ci-( -t —Fig.6. Workino- harts for Caculation of

- - I -X.I a. 150 gr Nosier Partition Bullet mean-61, 100 psi, b. 150 gr Herter Sonic Bullet mean 61, 100 psi, 3 rd5s, 0 g er at31, i34o seat 3 rds, 51 gr r3031,.34" seat d. 5IOgr US-M2 FMJ mean-63,000, c. 150gr Rem. SPCL Ptd. mean-63,00, 3 rds, 51 gr "3031,.34" seat 3 rds, 51 gr "3031,.34 " seat Fig..7. Selected Oscilloscope Negatives with 2 ms Sweep of 3 Rd Firings of Various 150 gr. Bullets in 30-06 Test Rif le, Barrel D with 51 gr. I'fR 3031 Powder Lot #1229.

a. 150 gr Herter Banana - Peel Bullet b. 150 gr Speer Spitzer S. P. Bullet mean 63, 000 psi 3 rds 51yr 3031 34 seat mean - 64,000 PSI, 3 rds, 51 yr 3031,.34' seat c. 150 yr Herter's Half Lead d. 150gr Herter's Spire Pt. mean - 66, 000 psi, 3 rds, 51 yr 3031,.34"seat mean 69.000 psi, 3 rds, 51 gr 34" sa Fig. 8. Selected Oscilloscope Negatives with 2 ms Sweep of 3 Rd Firings of Various 150 gr. Bullets in 30-06 Test Rifle, Barrel D with 51 gr. IMR 3031 Powder Lot #1229.

-39XI. RIFLE BARREL FACTOR A comparison of the data of Tables IIIA and IIIB shows that different barrels do not give the same values for J for certain bullets. This is particularly apparent with the 150 g Nosler bullets which had a J of 1.10 in barrel C and 1.17 in barrel D. This can also be observed with Hornady bullets and to a smaller degree with Speer bullets. Remington SPCL bullets gave about the same values for J in all barrels and continued to be used as a reference. In the tests with barrel C the original calibration load of 44 g of IMR No. 3031 powder and the 220 g Rem SPCL bullets were used. It was found that barrel C, which had been fired in a large number of high pressure tests, had changed its barrel characteristics. It now gave lower pressures than measured previously. The groove diameter at the breech was determined by "slugging" with lead "00"' Buck shot and was found to have increased its diameter from.308" to.310". The throat bore diameter was checked with a U. S. Ordnance breechbore gage and showed 0.10 units of wear. A new 30-06 military (Springfield) barrel identified as barrel "D" was installed in the test gun for the next series of firings, using bullets of various manufacture. This brings us to the need for "R' the rifle barrel factor which is defined by Eq. (31) in a manner similar to J. The value of R is: R (for barrel considered) p (for barrel considered) R (for reference barrel) p (for reference barrel) (with all other factors held constant) (31) R is of primary importance to the handloader in finding whether his rifle is "typical", and if not, whether his rifle will give higher or lower pressures than those reported in the loading data that he is using as a guide. Many experienced riflemen testing a new rifle with standard ammunition may find that the rifle shoots "hot", possibly blowing primers and giving difficult case extraction, while other rifles apparently identical and of the same caliber and manufacture will give normal pressures with the same ammunition. These "hot" rifles have rifle barrel factors, R significantly greater

-40than unity. On the other hand, either erosion as a result of firing many rounds or deliberate "freeboring" of a barrel will reduce the pressure and this reduces the rifle barrel factor to less than unity. The magnitude of the variation in R from rifle to rifle is rather surprising. We have tested a number of different barrels in 30-06 caliber in which R varied from about 0.6 for worn barrels to 1.3 for new "tight" barrels. This would mean that for R - 1.0 with p = 60,000 psi as a standard of reference, the barrel with R = 0.6 would give a p of only 36,000 psi, and the new tight barrel would give a p of 78,000 psi. To determine J values for commercial bullets and R values for different rifle barrels, a series of firings was conducted in which pressures were measured using the strain gage technique. Some of the test data are shown in Figs. 7 and 8 and are summarized in Table III B. The next problem was the choice of the most suitable reference for R, the rifle barrel factor. The first procedure considered was to use Michigan military barrel "B" as the reference. This barrel was considered to be a "normal"barrel-neither new nor old, with some, but not excessive wear. However, on comparing our pressure data with those of the DuPont loading tables(15) it became obvious that our barrels were giving significantly lower pressure than the DuPont test barrels in 30-06 caliber. The choice of reference was changed in favor of using R = 1.0 for the loading data published by E. I. du Pont de Nemours and Co., Inc. (15). These data represent the major accumulation of recent ballistic information on rifle pressures available to the handloaders, other than the excellent data provided by NRA (16). Unfortunately, the duPont data tables give no description of the barrel characteristics. The NRA tables do provide data on barrel groove diameter, bore diameter and length. An inquiry to duPont provided the information that, in general, the data for each cartridge were the averge of results from numerous firings sometimes in several barrels. Therefore, as we were unable to characterize the duPont barrels, we concluded that the procedure for the best use of all data was to assign R = 1.0 to the rifle barrels used in the duPont tests. Figures 9 and 10 show plots of data obtained using the Michigan military barrels B, C and D as well as duPont data and some of Waite's NRA data for the 30-06. Note that in the upper (high pressure) portion of the log-log plot of Fig. 9, the curves for all firings in all barrels are straight lines with the same slope but with different intercepts. Figure 10 shows this region on an expanded plain coordinate scale so that the intercepts can be read more easily.

-4110.0 9.0 8.0 7.0 rn O 6.0 5.0 <: 4.0 /i D ~30 7 Du Pont I Id F U of M (D) NRA m 2.0 U of M(B) Uof M (C).2.3.4.5.6.7.8.9.10 SPECIFIC LOADING DENSITY, S Fig. 9. B'/J versus S for five different barrels.

~s{ ISE~q;uGIt;44p GATJ otO S sns2a[ f/,A / *'0 T ~ S'A/ISN3Q 9NIlaVO I 31J133dS Zg' 0cj' 8lz 91'' 9S Z9' 0'~2 0 z u1 rC( I) I Io n, I I, L9' _= P 1_ Q'9 C-:0= 0'8 OO( -= 00'6 - t

-43Table V DATA USED FOR CORRELATION OF R VALUES FOR FIVE DIFFERENT BARRELS USING IMR #3031 POWDER S B'/J S B'/J Dupont.442 6.12.480 8.70.445 6.25.493 9.65.447 6.40.497 9.85 NRA.417 3.85.454 5.75.424 4.25.465 6.35.432 4.60.478 7.15.438 4.90.482 7.35.439 4.95 Univ. of Mich. (B).430 3.60.450 4.30.440 4.15.515 7.65 Univ. of Mich. (C).441 3.05.500 5.75.460 3.85.519 6.65.480 4.95 Univ. of Mich. (D).431 4.70 4.60 4.95.431 5.10 4.79 7.50.460 5.85

-44For convenience in use of the duPont data as reference for R = 1 we can locate the intercepts at the value of S where B'/J is equal to 10.0. Figure 10 shows this to be at S = 0.498 for the duPont data. The other intercepts at the same value of S = to 0.498 are 8.60, 8.30, 6.50 and 5.70 for Michigan barrel D, NRA barrel for tests A, and Michigan barrels B and C respectively. Since these intercepts are directly proportional to p, and p to R, the corresponding values for R for the 5 barrels are obtained by dividing the values of B'/J by 10.0 to give 1.0,.86,.83,.68 and.57 respectively for the five barrels. With the values of R determined for the different barrels involved, a further study of J the bullet jacket factor, now is possible. If the bullets are of the same design, material, shape, hardness, yield point, etc., we can predict the value of J from the bullet diameter and sectional density. This relationship between J, sectional density and diameter exists because: (1) the basic characteristic of sectional density is that all bullets with a given sectional density require the same pressure for the same value of acceleration (or deceleration as by air resistance); and (2) diameter, d, is involved because as the size increases the ratio of engagement area to volume decreases. The work of bullet engraving by the rifling and the bullet friction in the barrel is a function of the surface area of engagement between the barrel and the bullet. Thus, as the bullet diameter increases and the ratio of surface area to bullet volume decreases the bullet becomes more efficient and J decreases. This relationship for J is shown in Fig. 11 which plots J versus bullet diameter d in inches with parameters of sectional density. As used here, the sectional density, w/d2, is the weight of the bullet in pounds (grains/7000) divided by the bullet diameter squared (inches squared). The Speer Reloading Manual lists sectional density for each Speer bullet, and most other handloading books give tables of sectional density for bullets of various weights and caliber. To determine J, simply enter Fig. 11 on the horizontal scale for diameter and move up to the estimated distance between the curved lines for constant sectional density. After locating the intersection of d and w/d2, move across to the vertical scale and read J.

-451.7 1.6 2 1.5 0 LL 1.4 1.3 t- -too LrJ 1 w._J So CD 1.2 4?00 1.1 tloo~~~~~~~~~~~~~~~~~~ fO0 1.0.20.25.30.35.40.45,50 BULLET DIAMETER d(inches) Fig. 11. "J" versus "D" with Parameters of "w/d2" (pounds/ square inch)

REFERENCES 1. York, Michael W., Gyorey, Geza L., Brownell, Lloyd E., "Ballistic Breakthrough," Ordnance, Washington, D.C., July-August, 1964. 2. Brownell, et al., "Technical Report No. 1, "Absolute Chamber Pressure," Du Pont Ballistic Grant, The University of Michigan, Ann Arbor, Michigan, June, 1965. 3. Brownell, L. E., and York, M., "Ballistic Breakthrough-A New Method of Determining True Pressure," Handloading Digest, 3rd Ed., p. 53, Gun Digest, Inc., Chicago, 1966. 4. Waite, M. D., "Bullet Types and Pressure," in NRA Illustrated Reloading Handbook, p. 122, The National Rifle Association, Washington, D.C., 1961. 5. Waite, M. D., "Loads for the 30-06," in NRA Illustrated Reloading Handbook, p. 168, The National Rifle Association, Washington, D.C., 1961. 6. Charbonnier, P., "Ballistique interieure," Droin, Paris, 1908. 7. Olger, P. R., "The Le Duc Velocity Formula," U. S. Naval Inst. Proc., 37, No.138, p. 535, 1911. 8. Buckingham, E., "On Physically Similar Systems; Illustrations of the Use of Dimensional Equations," Phys. Rev., 4, p. 345, 1914. 9. Rayleigh, Lord, "On the Stability of the Flow of Fluids," Phil. Mag., 34, p. 59, 1892. 10. Rayleigh, Lord, "The Principle of Similitude," Nature, 95, p. 66, 1915. 11. Langhaar, H. L., Dimensional Analysis and Theory of Models John Wiley and Sons, Inc., New York, N. Y., 1951. 12. Buckingham, E. "Notes on Methods of Dimensions," Proc. Roy. Soc. Lond.,, p. 696, 1921. -46

-4713. Newton, J. Isaac, Principia Mathematics, 1686. 14. Fourier, J. B., "Theorie Analytique de la Chaleur," Paris, 1822.

THE APPENDICES It is hoped that the Appendices of the University of Michigan Technical Reports in internal ballistics will serve as a source of information on ballistic research in the United States and abroad. When Michigan investigators looked for unclassified U.S. literature on ballistics they found it to be almost non-existent. The reasons for this seem to be that 1) the United States Army classifies nearly all of its ballistic research; and that 2) there is no journal in the United States which is suitable for printing technical articles on ballistic research. Therefore, the Appendices are an attempt to alleviate this problem in part by presenting technical articles from the University of Michigan (see Appendix I), pertinent articles not generally available (see Appendix II), and translations of significant articles published abroad (see Appendix III). Readers of the Technical Reports are encouraged to contact us about any articles and/or books they feel ought to be included in future appendices. A second purpose behind the Appendices is to present data which will enable us to make comparisons between the conventional methods for internal ballistic calculations as used in Europe and the United States with those developed at the University of Michigan. The Michigan methods are based on the principle of dimensional similitude in contrast to the convention methods which involve the use of the energy balance, sometimes referred to as the First Law of Thermodynamics. By comparing the calculated results obtained from both conventional and the Michigan methods with experimental measurements, the Michigan correlation can be evaluated. Now a word about the Appendices of Technical Report No. 2. Appendix I, "Ballistic Breakthrough: A New Means of True Pressures," reprinted from the Third Anniversary Edition of the Handloader's Digest, was chosen because it includes the only known curves of pressure versus distance of bullet travel in which a curve for absolute pressure is compared with the corresponding curve for crusher pressure. The article also contains a very significant discussion of the two procedures for describing chamber pressure by Dr. Wendall F. Jackson, Associate Director of the Explosives Division, Du Pont Laboratories, Wilmington, Delaware. -48

-49Appendix II contains "Chamber Pressures in Small Arms" by Berwyn J. Andrus, a thesis describing his ballistic research and written as part of the Bachelor of Science Degree requirements at the University of Utah. The work by Andrus was done independently ten years before the studies at the University of Michigan began. Both Andrus and Michigan investigators have used the same type of devices for measuring pressure: the strain gage and the oscilloscope. The Michigan work was described in detail in Technical Report No. 1; "Absolute Chamber Pressure in Center-Fire Rifles." Andrus discusses points that were not mentioned in Technical Report 1. Andrus's thesis is well written and represents an important contribution to American internal ballistics literature. It is reprinted here with the special permission of Professor R. L. Sanks and the Library of the University of Utah, Salt Lake City, Utah. Appendix III is a translation of Chapters 1 through 5 of an excellent and quite recent (1961) East German text on internal ballistics, Innere Ballistik by Professor Waldemor Wolff. Chapter 5 constitutes the major portion of the translation to date and covers the burning of smokeless powders at constant volume. This discussion provides the basis for the introduction of Abel's equation and the various powder constants. The translation of Chapter 6 and Chapter 7 is not quite complete and it is hoped it will be included in Technical Report No. 3.

APPENDIX I BALLISTIC BREAKTHROUGH A New Means of Determining True Pressures by L. E. Brownell and M. W. York Department of Chemical and Metallurgical Engineering The University of Michigan Ann Arbor, Michigan Reprinted by special permission from the Third Anniversary Edition of The Handloader's Digest, John T. Amber, Editor, 1966.

BALLISTIC BREAKTHROUGH A New Means of Determining True Pressures by L. E. Brownell and M. W. York Absolute chamber pressures have been completely in unaltered rifle barrels-not in pressure barrels. unknown to most reloaders until now! Here is the As an added bonus, we can look forward to a reasonfirst popular account of a completely original method ably priced, reliable instrument for recording chamfor recording true, accurate breech pressures, taken ber pressures in our own rifles ~~~60,000 _ -Pmax (AB SOLUTEI PRESSURE) 30 s 0,000 3000 50,000 r-, 40000 2 000 03 QI~ 30,000 2 4 6 8 10 1 1 20 22 U. q) 20000 travel in inches. Data from several firings of 4.8 r Hi Ve,0,000 0 2 4 6 8 10 12 14 Is Is 20 22 BULLET TRAVEL, INCHES Fig. 1. Composite curves of chamber pressure and bullet velocity vs. bullet travel in inches. Data from several firings of 45.8 grs. Hi Vel 2, 173-gr. FJ bullet in 30-06 rifle. FOR YEARS many reloaders have the limits of current ballistic informa- Pressure is of primary concern to relied upon the powder companies and tion. Rather, it should be viewed as the handloader and absolute, or true, gun magazines for ballistic informa- the logical outgrowth of the advanced pressure is the subject of this article. tion. The more experienced hand- technology which is permeating the Unfortunately, many handloaders do loaders work up new loads by slowly entire fabric of our lives. not have a proper understanding of increasing powder charges while the importance of rifle chamber presclosely inspecting fired cases for signs sures. This is not entirely their fault. of excessive pressures. Comparatively Many loading handbooks give the speaking, modern handloaders and reader only a smattering of pressure shooters have a greater and more re- information. Let us look for a moment liable body of ballistic information at how pressure is produced and reavailable than did their counterparts view its importance in terms of safety of 40 years ago. Yet a good look at and accuracy. current ballistic information, upon Production of Pressure which so many must rely, indicates a need for more and better data if per- When the firing pin strikes the formance and safety factors are to be primer, the primer compound is detoimproved. nated, producing a spurt of flame that Such a conclusion should not be enters the cartridge case through the looked upon as a severe criticism of flash hole in the primer pocket. This ignites the charge of smokeless powder L. E. BI l is o of Cmicl A grant in aid has been established at the L. E. Brownell is Professor of Chemical and University of Michigan by E. I. du Pont de Nuclear Engineering and Supervisor, Du Pont Nemours & Co. to help continue these studies. Ballistic Grant, University of Michigan, Ann Fig. 4. Ballistic test apparatus used in Du The research described in this article was W.Arb Yor k is an engineering consultantch. Pont ballistic grant. Note oscilloscope sweep supported in part by the Michigan Memorihal M. W. York is an engineering consultant, Phoenix Project and the Department of ChemiYpsilanti, Mich., and research assistant, Du of pressure vs. time and strain gauge con- cal Engineering at the University of Michigan. Pont Ballistics Grant, University of Michigan. nections on barrel. M.W. York is reloading. Ann Arbor, Michigan. Reprinted from Handloaders Digest, 3rd Ed., 1966 by special permission from Gun Digest, Inc. I-3 3RD EDITION 53

I-4 0 o:o "/ - "4 * b m Time, milliseconds (a) H Time, milliseconds (so) O- boattail bullet, 42 grains b. Piezo data: 110-gr. bullet, 51 grains IMR 4895 IMR 4895 Powder, Pmax = 60,500 psi. MV = 0 Powder, Pmax 63,000 psi, MV = 3370 fs. Copper A 2680 fs. Copper crusher data: MV = 2620 fs, ~ crusher data: MV = 3320 ts. Crusher value = crusher value = 49,700 psi. 50,700 psi. O it. e to:0.545 pg 1 -~~0I' C. 0.10.0.. 30. psi vu 5~H Time, * Time, milliseconds (mn)'* c. Piezo data: 180-gr. metal cased bullet, 35 grains ~ d. Piezo data: 110-gr. bullet, 41.5 groins IMR 4198 ~ IMR 4198 powder, Pmax 70,500 psi, MV powder, Pmax = 67,000 psi, MV =-.3310 fs. 2620 fs. Copper crusher data: MV = 2540 fs, Copper crusher data: MV = 3300 fs, crusher Crusher value = 60,300 psi. value = 57,100 psi. Fig. 2. Piezometric measurements of absolute pressure vs. time for 308 cartridge and 22" rifle barrel. (Reproduced by special permission of Du Pont Co.) 54 HANDLOADER'5 DIGEST

I-5 ~ 1 i i I il.. I. i....... a. Five replicate firings of 43 grains Hi-Vel 2 pow- b. Three replicate firings of 53 grains IMR 4064 der, 180-gr. Sierra bullet with new Remington powder, 150-gr. Remington SPCL bullet with new 30-06 cases and Remington primers. Remington cases and Remington primers. Fig. 3. Negatives of displays on Model 564 oscilloscope showing excellent duplication of replicate firings of maximum charges. in the cartridge case. The gases pro- capacity plus bore capacity) increases. of Hi-Vel 2 powder used with a 173-gr. duced by the burning powder are con- This tends to decrease pressure. As a GI bullet fired in a 1903-A3 Springtained in the case except for a small result of these opposing effects, cham- field 30-06 rifle. Chamber pressure is leakage after the case neck expands ber pressure usually reaches a maxi- about 5000 pounds per square inch to free the bullet and before the bullet mum after a few inches of bullet travel (psi) when the bullet starts to move begins to move down the barrel. This and then begins to drop as the bullet and reaches a maximum after about confinement of the initial gas pro- advances. Fig. 1 illustrates the manner 3 inches of bullet travel. Thereafter, duced, plus the generation of heat from in which chamber pressure varies as pressure decreases even though only the burning powder, tends to produce the bullet moves within the bore. Fig. a portion of the powder has burned. a rapid rise in pressure before the 1 shows curves obtained during tests This occurs because of the rapid inbullet moves very far. However, the at the University of Michigan for data crease in space behind the bullet bullet gains velocity as it travels down on pressure and bullet velocity versus which results in a reduction in chamthe barrel and the total volume avail- distance of bullet travel.' The curves ber pressure even though some powder able to the confined powder gases (case are typical for a load of 45.8 grains still is burning. In most cases of cor0E 1 TETRONDIX 564 OSCILLOSCOPE STRAIN GAGE STRAIN POWER SUPPLY GAGE CIRCUITIRY PROIECTIVE 3/8" STEEL BOX SHIED PLATE STRAIN RIFLE RIFLE GAGES MOUNT SAND FILLT BARRELS RIFLE AND MUFFLER SUPPORT | | BARREL SUPPORT Fig. 5. Ballistic test apparatus as modified, August, 1964. 3RD EDITION 55

I-6 rect loading, all or nearly all of the ber pressure information is unavailpowder is burned before the bullet able to the handloader. In such cases reaches the muzzle. In the case of a Breech Pressure Device experienced handloaders feel their fast-burning powder, all the powder In the Works way through this darkness (created is burned in the bore and the pressure by the lack of pressure knowledge) decreases even more rapidly, because In addition to ballistics research by studying the appearance of primers after "burn-up" no additional powder supported by E. I. du Pont de and cases after firing. Another sign, gases are produced. Nemours and Co., Inc., Mr. York difficult removal of cases, may indicate The Importance of Pressure and another engineer, Mr. Walter excessive pressure and a dangerous The Importance of Pressure Hackler, are currently working on load. Although these signs are better When accidents occur at the breech a portable rifle breech pressure than no information, they are no subend of a gun, the cause can often be measurement instrument that can stitute for measured pressures. They traced to excessive chamber pressure be used without requiring any modi- give only indications of excessive presin the cartridge and gun involved. fication of the rifle. This measure- sure and cannot be relied upon in all Many modern high-powered bolt-action ment system is being specifically cases to protect the shooter against rifles operate with chamber pressures designed to bring the benefits of a dangerous loads.2 (crusher values) of approximately modern ballistic laboratory to the Muzzle velocities are, of course, in50,000 psi. This represents a tremen- serious shooter and handloader at fluenced by chamber pressures. Howdous force which, if released in the a reasonable cost - perhaps as low ever, the influence is not as great as breech outside the cartridge chamber, as a few hundred dollars. Inquiries one might expect. For instance, we can disintegrate the receiver, magazine should be addressed to Mr. M.W. might ask how we can obtain accuracy and surrounding stock. This not only York, 547 Ivanhoe, Ypsilanti, Mich. with loads in which the maximum wrecks the gun but can seriously pressure can vary 10 per cent or more injure or kill the shooter. _ I from shot to shot using the same gun Accidents can also result from ex- and the same loads? Fortunately, the cessive head space of rimless cases the use of the wrong type of powder muzzle velocity of the bullet depends which leaves the cartridge case with- can induce failures. Regardless of the primarily upon the energy transferred out sufficient support for a chamber conditions that result in failures, it is from the powder gases to the bullet pressure that might be safe in a me- the chamber pressure that produces rather than upon the shape of the chanically correct gun. Oil or grease the force of destruction. Thus, whether pressure-time curve. If the same powin the chamber can reduce the fric- we are designing the gun, testing it, der charge is used and all the powder tion between the case and chamber or shooting it we must always be con- is burned in the barrel, the same walls so that the pressure puts a cerned with maximum chamber pres- amount of chemical energy will be regreater part of its force on the bolt. sure. Safety depends upon keeping leased by the powder. The portion This can cause a failure in some of this maximum chamber pressure with- of this chemical energy that is transthe weaker types of actions. Plugged in safe limits. ferred as kinetic energy to the bullet bores, excessive powder charges, or In many instances adequate cham- remains fairly constant for a given 70000 60 000 10000 o 0 000 _ OX X s0000 60000 70Xoooo o000O ABSOLUTE PRESSURE, psi Fig. 6. Calibration curve for typical ballistic crusher values vs. absolute pressures. 56 HANDLOADER'S DIGEST

I-7 8 r I49-. J. i t I I I IllfI $4 _ _ i' 44 gr. (avg~,~,tl t rZero trace i 100. s -Time, A@ s Fig. 7. Maximum pressure firings and procedure for calculation of absolute pressure. length of barrel and chamber size. hundreds of different loads we must complex equations are not suitable for'rherefore, the muzzle velocity for a use a separate correction factor for use by the handloader.5) given rifle bullet and powder type will every variable with every load. We Although tables of maximum and depend primarily upon the weight of end up with the impossible situation moderate loads without pressures may powder used. of thousands,of correction factors. serve the needs of most handloaders, The points mentioned above have A major reason for the fluctuations these tables are a dead-end street so been verified by test loads. Twelve in maximum pressure is the variation far as improvement in ballistic knowlloads using Hi-Vel 2 and additional in the distance the bullet travels be- edge is concerned; the data cannot be tests with other powders in the 30-06 fore the pressure peaks. For example, used in any realistic evaluation of cartridge were reported in the Ameri- if the powder burns rapidly because factors influencing maximum prescan Rifleman.3.4 In these tests bullets of a "hot" primer or large primer hole, sures. Only by knowing the true values from various suppliers weighing 110, the pressure will peak before the bul- of the maximum pressures can we be 125, 150, 180 and 220 grains were used. let travels as far as it normally would. sure of our factor of safety. All were tested in the same barrel This produces a pressure higher than using cases and primers of one manu- normal. With a "cold" primer or small Limitations of Ballistic Methods facturer. The mean 10-round disper- primer hole, the powder will burn more Based on Crusher Values sion was 66 feet per second or about slowly, the bullet will travel farther Ballistic information currently re2 per cent in velocity, but 5000 psi or before the pressure peaks, producing ported and available to the handloader about 11 per cent in pressure. a lower maximum pressure. Also, if is, in nearly all instances, based on Handloading Tables vs. the bore is worn or the bullet offers data obtained by the use of the rifle Ballistic Correlations only slight resistance to engraving by "crusher" gauge. These data are obBallistic Corelations the rifling, the resistance to bullet tained by firing cartridges in special Tables in handloading handbooks, travel will be less and the travel will test rifles designed to accommodate with or without data on maximum be greater before the pressure peaks. this gauge (See Fig. 8.) This type of pressures, have been used for half a The combined effects of the many measurement suffers from disadvancentury as a means of determining variables influencing the maximum tages and inherent inaccuracies* suitable powder charges. This proce- pressure can be taken into account by The crusher gauge has a long hisdure is simple to use but prevents determining the influence of each tory It was developed and first used correlation of the many factors in- variable on the distance of bullet travel in black powder days to p rovide a volved. Differences in individual prim- to the pressure peak. From this the means of indicating the magnitude of ers, in primer manufacturer, in diam- effect on the maximum pressure can the maximum chamber pressure in eter of the primer flash hole, in cham- be predicted. Thus, only by analysis firearms. For handloading purposes, ber volume, in bullet characteristics, and correlation of the data and by the pressures indicated by the crusher in bullet seating depth, etc., can in- use of ballistic equations rather than fluence maximum pressure. If we ex- loading tables can we hope to make rate to provide information on the press the maximum pressure for vari- significant progress in the predictions ous cartridges by a single equation, of maximum pressures. The Ballistic we can apply correction factors for Research Laboratories at Aberdeen these variables. But as long as we con- have developed useful ballistic equa- *See comment by Dr. W. F. Jackson, Astinue to use handloading tables with tions for small arms, but these rather saboriate Director, Explosives, accompanying thDivision, Du Ponticle. 3RD EDITION 57

I-8 combinations of loading components and control of temperature during fir- pressures rise to higher values. Therethat give safe loads. This is essentially ing is important in tests made with fore, we find that almost universally the basis for the loading information copper crushers. As the copper cylin- the reported crusher-gauge pressures in the NRA Illustrated Reloading der is compressed the copper becomes for high pressure loads are lower than Handbook.4 However, data based on cold worked and the cross sectional actual true pressures. This is indeed the crusher gauge technique are not area is increased. Thus, the measured unfortunate since the shooter should considered sufficiently accurate for deformation by compression is not be more interested in obtaining true use in ballistic correlations. directly proportional to the force ap- chamber pressures for high pressure The crusher gauge and its use are plied by the chamber pressure. Fric- loads than he should for the lower well-described in Norma's Gunbug's tional resistance of the steel piston pressure loads. Guide.6 The limitations of crusher used to transfer the force of the gas measurements are mentioned briefly. to the crusher absorbs energy. This is ther Methods of Meaur The copper crusher is a rather crude an added source of error. All of these Chamber Pressure device which is used to determine factors decrease the accuracy of the Two other techniques, the strain plastic deformation above the elastic crusher gauge. gauge and the piezometric pressure limit of the copper used, and therefore The crusher gauge technique is gauge, have been used to measure is a measure of the work expended based on the plastic compression of a chamber pressures with greater accuduring that deformation. Copper cylinder of soft metal by the pressure racy than with the crusher gauge. We crushers are not absolutely uniform in in the chamber of the rifle. The change have been informed that many powder their physical properties, and some er- in the thickness of the crusher is then manufacturers now are using "piezo" ror is introduced by variations in yield used as an indication of the pressure gauges to obtain better pressure data. point and cold working characteristics to which the gauge was exposed. The The use of strain gauges appears to be beyond the yield point. Physical prop- amount of deformation of the crusher less extensive. Dr. E.L. Eichhorn, erties such as yield point and modulus for each pound of pressure is greater ballistician,7 states that strain gauges of elasticity also are temperature-de- when the chamber pressure is rela- are used by RWS Genschow, Nurnpendent. For this reason measurement tively low than it is as the chamber berg, Germany; Svenska P.F., StockNo. (Outridge 1 ter betge ]allet Cruser Souroe oa Pr]esne sow". a m pIsle Prespe Sani. ao (gr 1) IaMdlg Crusher Data by Strain Stran Vo1l. by Pleso Pleso 0.N. pai Oam, pal Om Data tt/se 0cg, psi Data 1 308 1895 2. o. w 19,700. 0 Dmot*.. _.. 2650 60,580 o.1. DsPt*(trs. Z ) bot tail 2 30oI 119g 35.0 l8aw 60,8oo0.x. DPa-t* 2580 70,000 Z.1. DWxat*(l. Z ) boat ta 3 30o 489 51.0 Olgr 50,700 3.I. DuPont -. 3345 63,100o 3.1. DWuot*(1g. ) ipitaer 1 30 41198 141.5 a r 57,100.I. DuIOIt*.. —-- 3305 66,860 3.1. DuPot*(F. i ) epituer 5 30-06 3031..0 22r 52,700 o.I. Dulmtn 62 000 u. at N. 2352 63,600 Du Pont Lot 1221 8PCL 6 3o-06 3031 37.0 O22 110,700 3t.. Dot* 110,000 U. of M. 0278 8,0ooo Du Pont 7 30-06 3031 29.o 22gw 30,400 X.1. IDuPnt 33,0o0 U. oa M. 1681 28,000 DIu Pont Lt 1221 aCL 8 30o-06 3031 18.o 220r 6l,900 g.I. xDusont* 90,800 90,800 B.1. Dwont Lot 21 CL 93,100 B.. DuPont 95,300 3.X. Dusont 9 30-06 3031 49.0 220gr 6T7, 00.1. Dudot*y 97,7o0 3.X. puPot Lot 12211 PCL 100,000 X.1. DuPont 102,300 3.X. DuPot 10 30-06 3031 11.0 220 (52,700) tn as ar 6,ooo000. oat..... (63,600) tat 2 )0 LCL t 1224 11 Soo6 3031 37.0 O22g (0o, 700) taken as for 10,000. ator.................... Lot 1230 8PCL Lot 12211 12 3o06 3031 29.0 O20gr (30, 7o00) kn as for 29,000 U. at. —... (8, 000)............ ot 123o a Lot 12i21 13 30-o6 3031 3T.0 a22 (0,700) tkn a for,300. or.... (, 000)............ Lot 191 CL Lot 122 1 30-06 3031 29.0 22r (30,700) kna fo 30,000 U.. ao ------........... Lot 191 CL ot 15 30-06 t1 23.0 o2 20,000 P.mrpo 19,600 U. or. 1 1................ #2 (oor.prlar) 16 3o-o6 111.e 38.0 22r 3,000 ~P.mrpee 39,000 o U. at. 21t0.................. fe 17 30-06 1iv1 13.6 a 51,000 P.Sarpe 6, 50 U.. 131.................. i8 3o-6 iVee 37.0 8 31,150 I.R.A.+ 1,,600 1. at. 97.................. 19 30-06 311Vl 11.0 22r 18,100.Rs.A.+ 51,700 vU. oc a. 1357............ fe Wm ermpofteeA by speoeal pertieion. (Data fiet bras ht to attetion by Dr. Or L. itehhma, Blllttle ) (9) HWryepnew oeAmlation. *emepzaaa by elpooal penleeto on u-tIamey a!Iink an VaD.11 Co., Ne You, l.(11) +b folavin edit in bs beea zpmetd for spelal pzeieom to reprint data on 3o-o6 le T eato wal i1 wprlnet tre U AIICAJ IRIqBUM, ottioSal Jarnl a tba oaticeel ItRile A oeaiton oa Arearsa, 1600 Sa Zeisyt Avewmm, I.V., aekbnl t 6, D.C. N UMICAN UID gee..tblyO to to w tbs oo elfb ora of ad mill I A iPimbeN. irebzip i1 syalable to ittisemh at oa rerput." Table I - Calibration data for crusher reodings s. obsolute chamber preuure. 58 HANDLOADER'8 DIGEST

I-9 Oscilloscope Strain Gages Removed Section of Barrel and Receiver Showing Location of Strain Gages Oscilloscope Input Strain Gage Le Strain Gage Rifle Mount Circuit Note: Power Supplies are not shown Diagram of pressure measuring system. holm, Sweden; Norma, Amotfors, stress in the barrel does not exceed excess of 60,000 psi. Belted cartridge Sweden; Hembrug, Ijmuiden, Holland; the elastic limit. If the elastic limit is cases for magnum loads are consistand CCI, Lewiston, Idaho, and we exceeded, we will have a plastically ently loaded to produce actual absoknow they are used by Detroit Test- and permanently deformed (bulged) lute pressures in excess of 70,000 psi. ing Laboratory, Detroit, Mich. barrel. Such conditions of plastic defor- However, it is most important that the Both the strain gauge and the piezo mation exist only in destructive test- absolute chamber pressure and the gauge can be used in conjunction with ing and are not applicable to the values from crusher tests not be conan oscilloscope to give a pressure vs. studies in the present tests. fused. The possibility that crusher time curve while the bullet is in the At chamber pressures between 20,000 values might be confused with real barrel. Using correct instrumentation, and 40,000 psi fair agreement is pos- pressures may have been one reason both systems can be made to produce sible using the crusher technique and for continuation of the use of crusher a response that is linear with respect either the piezo or strain gauge. At values by the powder companies. In to pressure. In the case of the strain pressures of 50,000 psi and greater, research at the University of Michigauge the following relationship holds: the crusher does not give true abso- gan the intent is to work with funp r d2 (2 —a) lute pressure. The magnitude of this damentals and facts. References to e = -d-I = kp difference is not known by most hand- pressure are to the true absolute E do2-di 2 loaders and gunsmiths. A handloader pressure rather than the crusher The derivation of the equation above may be firing a cartridge with a charge values when the term pressure is used. is given in Technical Report #1 of of 42 grains of IMR 4895 powder and Because of the inherent inaccuracies the DuPont Ballistic Grant, Absolute a 180-gr. bullet in a 308 Win. car- associated with the crusher gauge Chamber Pressure in Center-Fire tridge. This load gives a pressure of technique of pressure measurement, a Rifles.s It is based on Eq. 14.18 of 49,700 psi as determined by crusher more accurate means of obtaining true Reference 9. For a given gun the measurement. The true pressure for chamber pressure measurements is value of Poisson's ratio A, modulus the identical load measured by piezo being utilized for all ballistic research of elasticity E, barrel outside diameter gauges is 60,580 psi.7,1o Some reload- at the University of Michigan. The do, and chamber diameter d, are con- ing handbooks list even higher charges University method incorporates the stants. Therefore, the unit strain e than 42 grains for this powder and use of strain gauges attached around will be directly proportional to the cartridge. Fig. 2 shows piezo gauge the circumference of the chamber porchamber pressure, p. If the strain data reproduced by special permission tion of the rifle barrel to obtain the gauge signal is directly proportional to of Du Pont." chamber pressure in the rifle. The unit strain e, the voltage change in the strain gauge and associated electric circuit will be linear with respect to circuit provides a voltage signal that pressure. We should point out that the Measurements has a direct relationship to the pres— mouius Jt eIastlcity. Rj. (sometfmes The conepet thst absoje pressures sure ivtisz the rie chamher Furthercaled Voung$s modulus), is constant over 55,000 psi are unsafe should be more, the change in voltage per pound for a given material at a given tem- dismissed. Commercial rimless car- of chamber pressure is essentially conperature. Hooke's law states that stress tridges for modern rifles are consist- stant over a wide range of chamber is dirrctJy p i.pr o.lD., si, s,;rai' e? 2y]a',ded w/tk c~rge' rr~S fsr-d'ce pressures. There(ore, accurate chamand definitely is applicable if the actual absolute chamber pressures in ber pressure measurements are pos3RD EDITION 59

I-10 sible at all pressure levels. This strain gauge and current are used with an oscilloscope and camera to record the Defen of the Crusher Gauge voltage change from the strain gauge "I must put in a word in defense "Conversion to a more scientific and hence the rifle chamber pres- of the crusher method - not be- method of pressure measurement sure. The oscilloscope records voltage cause of its accuracy, which it does has met resistance in both industry as a function of time so that the use not have, but because it has been and the military because of the of the oscilloscope provides the added a simple indicator used in a rea- tremendous backlog of crusher data advantage of being able to determine sonably standardized fashion for available and the relative complexthe chamber pressure at any given many years by both commercial and ity of piezoelectric, magnetostrictime after the cartridge has been fired. military ammunition testers and tion or strain gauge equipment Fig. 3 illustrates the type of picture developers. We recognized at least needed for the methods that operobtained with this strain gauge tech- 30 years ago that the crusher meth- ate within the elastic limits of the nique with 2 different powders and od would have been better served pressure sensing elements involved. replicate firing. if "psi" had never been appended "I do not believe the powder comAnother distinct advantage of the to the numbers on a tarage table. panies have really been reluctant strain gauge is that it can be readily In the first place, the tables are to bring this situation to the attenused in any sporting or military rifle prepared by calibrating compres- tion of the handloader. Rather, for without having to modify the rifle in sion as a function of load on a dead many years, both the powder and any manner. This is a definite advan- weight press. Even though the cop- ammunition companies refrained tage compared to the crusher gauge per has been carefully treated dur- from discussing with the handtechnique, which requires a special ing fabrication, and the deforma- loader pressures measured by any test rifle or at least the permanent tion under static load determined method. Only recently, with so modification of the rifle that is to be with considerable precision, it is many scientists and engineers intested. The strain gauge measurement well-known that the same deforma- volved in shooting as a hobby, have technique is also relatively inexpen- tion is not encountered under dy- the powder companies found it desive in terms of the equipment in- namic loading. sirable to share their information volved. Cost of the strain gauges and "The results of crusher tests in on pressures with the handloader. associated circuit is less than $30. guns might better have been re-'Pressure' I should put in quotes, However, in the tests being conducted ported as millimeters or mils from because the data so far presented at the University of Michigan an oscil- the beginning. The fact that in- to the handloaders are crusher data. loscope is used to record the voltage crements of compression for a given "Your paper should go a long way output from the strain gauge circuit increase in load are not constant toward educating the shooting fraand this is a relatively expensive piece as the load is raised is not surpris- ternity in this complex area. I beof equipment. Nevertheless, for the ing in view of both work hardening lieve, however, that there is still ordinary rifleman, use of a recording and increase in diameter as com- more to be done before any method system and a strain gauge circuit can pression proceeds. The tarage table can be called the last word. Calibe constructed for less than the cost does take care of this. The real bration of the strain gauge method of a moderately priced chronograph discrepancy lies in the difference by more direct means than those currently being used by many shooters. between defection under dynamic used so far would be desirable. Fig. 4 shows a photograph of the vs. static loading. Location of the measuring point test apparatus with Mike York, the "In spite of this situation, the along the chamber is another factor first Du Pont fellow, loading a test community of gun, ammunition, and that deserves attention. rifle. A diagram of the test apparatus, powder makers, aware of the fact "All of this I am writing only to as modified in August, 1964, is shown that they should bite their tongue suggest that your readers may get in Fig. 5. At the right side of Fig. 5, every time they attach "psi" to a too dismal a view of the crusher two sand barrels are indicated. They crusher measurement, have found method from your presentation." are used as a bullet trap and a shield the crusher method a handy day- W. F. Jackson for personnel working on the project. to-day procedure for powder and The plywood box surrounding the rifle ammunition test, fairly reproducand muffler effects some sound control ible at different stations, and by The above letter was written to Mr. Brownell by Mr. W. F. Jackson, Assistant and also acts as a personnel shield process of trial and error used it Director, Research and Development Diviagainst minor fragments of bullets, as a guide in gun and ammunition sion, E.I. du Pont de Nemours & Co., after reading this article. It is printed here to etc. The %3/ steel plate directly be- design. give a broader view on the subject. - Ed. hind the bolt protects the immediate working personnel against a ruptured displays the pressure in the chamber measurements. The calibration is gun or ricochetting bullets. The rifle as a function of time. based on the ratio of piezo gauge presmount is constructed of steel. It re- Table 1 reports some of the absolute sures and crusher values determined ceives the full recoil via the recoil pressure measurements that have been in the Carney's Point Development lug and serves as a free floating stock obtained at the University of Michi- Laboratory of the Explosives Departin which no pressure is exerted on the gan and data from various sources ment of E.I. Du Pont de Nemours barrel. For simplicity, the gun is used to prepare a general relationship and Co., Inc.,10 and firings of the triggered mechanically by a pull cord between absolute pressure and crusher same loads at the University of Michithrough the shield.A storage oscillo- values. The data in Table 1 are plotted gan. scope is used to record the strain in Fig. 6. It should be noted that dif- Proof Loads gauge signals. This oscilloscope has the ferent designs of crusher gauge apadvantage of storing any trace so that paratus give crusher values that vary To obtain data in the region of data can be recorded or discarded somewhat for the same loads. Also, 90,000 to 100,000 psi for a calibration without wasting camera film. instruments for measuring absolute curve of crusher values vs. absolute A simple strain gauge circuitry is pressure, including piezo and strain pressure, two experimental loads were used and involves a resistance in series gauge, require calibration. Differences especially prepared and tested with with the strain gauge. After detona- in calibration procedures will result in copper crushers at Carney's Point tion of the primer, the pressure in the inconsistancies in evaluation of the Laboratory. The maximum pressure chamber begins to rise and the strain same pressure. Thus, the curves shown was selected to simulate the proof gauge voltage changes. A small voltage in Fig. 6 have limited application and loads used to test U.S. Government change triggers the scope and its trace may not correlate data from some rifles. In these tests a proof load giv60 HANDLOADER'S DIGEST

I-"1 ing a crusher value of 67,000 psi has From Norma's been used for years. Because of the Gunbug's Guide very high pressure, pre-deformed crushers are used in the government test. However, since we did not wish to introduce the additional factor of cold working of the crushers, simple copper crushers were used in all the Du Pont measurements. Du Pont reported a crusher value of 61,900 psi using 48 grains of IMR 3031, Lot 1224, a 220-gr. Remington Core-Lokt bullet and Remington cases and primers. Another load, identical except that 49 grains of powder was used, gave a crusher value of 67,400 VN psi. (Warning: These are proof loads and " should never be fired by an individual CASE WALL holding the gun. A proof-testing rack with suitable protection for the op- Diagram of crusher gauge. A hole 0.20" in diameter is drilled into the chamber erators should always be used with about one inch from the head of the case at a right angle to the bore's axis. The tertos sfbe used with )hole is carefully lapped and a tight fitting steel piston is inserted above a gas tests of proof loads.) check which protects the piston from hot powder gases and prevents gas leakage. Du Pont did not measure pressures A small cylindrical piece of copper of known dimensions and physical characteristics with pieizo apparatus because the up- is placed on top of the piston and held by a heavy screw from the pressure gun per limit for this equipment is about frame. When the shot is fired, gas pressure punches out a disc of brass in the wall 70,000 psi absolute pressure. There is of the cartridge case and the pressure moves the piston upward, compressing the no such limit for strain gauges. copper crusher pellet beyond its yield point. By comparing the copper pellet's Strain gauge results for these loads length after firing with its original length, and use of a conversion table, pressures gave, as an average of three shots with in (so-called) pounds per square inch are obtained. (Actually, as Dr. Brownell gaveh, as,0an average o three shots wih notes elsewhere in this article, true psi is not had via the crusher gauge system each, 93,066 psi with the 48-gr. load but the values still are used - and useful - as comparison means.) and 100,000 psi with the 49-gr. load. These results, obtained at the Univer- absolute pressure of 90,800, 93,100 presented later. The results of these sity of Michigan, are shown in Fig. 7. 95,300 psi respectively (figures rounded studies will be reported at suitable inThe data are listed in Table 1 and off) by multiplying the scaling factor tervals and copies of these reports are plotted in Fig. 6. The four lowest of 4540 psi per unit by the value of the with original experimental data will traces of Fig. 7 show four calibration vertical units at peak pressure. Simi- be made available to the Du Pont firings using 44 grains of IMR 3031 larly, for the load of 49 grains we ob- Company and the general public withLot 1224, and the 220-gr. SPCL bullet tain 97,700, 100,000 and 102,300 psi out cost via the editorial offices of the load. This load was found by Du Pont absolute pressure respectively. It is various journals. The reports may be to produce a crusher value of 52,700 important to note that at the highest obtained by individuals for a small fee psi and a piezo-gauge value of 63,600 pressure of 102,300 psi absolute, the to cover the cost of reproduction psi absolute pressure. The lowest of crusher value is only 67,400 psi - (estimated for this report as $1.65 the four curves for this loading was which gives a difference of 34,900 psi. per copy).* ~ fired first in a comparatively cold rifle This corresponds to a difference of and produced an oscilloscope peak of over 50 per cent with respect to the only 13 vertical units. Then three crusher value. *For Absolute Pressure in Center-Fire Rifles. write to Ulrich's Book Store, 549 E. University, loads of 48 grains of IMR 3031, Lot We have observed that cartridges Ann Arbor, Mich., or to Ann Arbor Arms and 1224, were fired to produce three loaded to 0.1-gr. accuracy on powder Sporting Goods, 1340 N. Main St., Ann Arbor, nearly replicate curves with peaks of scales give good reproducibility using References 20, 201/2 and 21 oscilloscope units full charges as shown in Fig. 3. Al- 1. York, M. W., "The Non-Destructive above the zero-voltage reference line. though the charges of 48 and 49 grains Measurement of Rifle Chamber Pressure The headspacing was checked during were weighed to 0.1 grains for the fir- Th M.gh the Use of Strain Gagesnd thesis for C. M. 690, Dept. of Chem. and Met. Eng., firings and was found to be beyond ings in Fig. 7, the 44-gr. loads were University of Michigan, 1963. the No-Go but less than the Field loaded with a powder measure. We 2. Naramore, E., Principles and Practice of Loading Ammunition, Georgetown, South Carogauge limit and the tests were con- believe this accounts in part for the lina, 1954. tinued. Then three maximum loads of spread in the 44-gr. firings. 3. Waite, M.D., "Loads for the 30-06," The American Rifleman, Washington, D.C., October, 49 grains of IMR 3031, Lot 1224, were 1956. fired to give three similar curves peak- Other Studies 4. NRA Illustrated Reloading Handbook, Washington, D.C., 1960. ing at 211/2, 22 and 221/2 oscilloscope 5. Vinti, J.P., and Chernick, J., "Interior units respectively. Next, three addi- One of the chief objectives of the Ballistics for Powder of Constant Burning Surface," Report No. 625, Ballistic Research Labotional calibration loads were fired and research studies supported by the ratories, Aberdeen Proving Ground, Md., Feb. these firings were 1, 11/4 and 13/4 units Du Pont Grant at the University of 15, 1947. 6. Gunbug's Guide, A.B. Norma Projektilhigher than the original firings. A Michigan is to develop and explore fabrik, Amotfors, Sweden, Norma Precision, value of 14 units was taken as the new means of investigating the phe- South Lansing, N.Y., 1962. 7. Eichhorn, E.I., personal communication. average for the calibration firings. nomena of internal ballistics in small Manager, Professional Services, Burroughs arms, particularly internal ballistics Corp., Equipment'and Systems Marketing Div., ~Examnple Cal~cula~tio n of Pras~essure,% P. 1..Pasadena, Calif.. 1964. Example Calculation of Pressure for the centerfire, high powered rifle. 8. Technical Report #1, Du Pont Ballistic Perhaps examples of the procedure The new techniques, among other Grant, Absolute ('hamber Pressure in Centerfor calculation of pressures would be tests, involve the use of gamma radia- MichlesMa nell e a Ann Arbor appropriate here. If we divide 63,600 tion and strain gauges for simultaneous 9. rownell. I,.E. and Young, E.H., Process psi by 14 units we obtain 4540 psi determination of chamber pressure, of ip esin - Vessel sin. New York, absolute pressure per vertical unit on bullet location in the barrel and pres- 10. Dunn, G.F., personal communication. the oscilloscope sweep. For the 20, Du Pont Development l.ahoratorv. (Carnev s 201/2 and 21 vertical units for the New methods of correlation based on 11. Jackson, W.F., personal communication. 48-gr. IMR 3031 loading we obtain the use of dimensionless groups will be &Co, 1c. Wi lmngto l e Nemours 3RD EDITION 61

APPENDIX II CHAMBER PRESSURES IN SMALL ARMS by Berwyn J. Andrus Thesis Submitted to the Faculty of the University of Utah in Partial Fulfillment of the Requirements for the Degree BACHELOR OF SCIENCE University of Utah August, 1954 Reprinted by the University of Michigan with special permission from the University of Utah, Salt Lake City, Utah, August, 1967.

TABLE OF CONTENTS TABLE OF FIGURES............. v AI'STRACT... ~ ~. ~ ~ ~ ~ ~ ~ ~ ~ ~... ~.~~ vii INIAOl)UCTION..................4. I I- V INTROi)UCTION.. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ II-1 PURPOSE....- * * * * * * * * II-1 SCOPE I......... a.,... II-2 REVIEIW OF LITERATURE.................. II-3 PRESENT MtETIHODS OF [RESSURE MEASUREMENT.....* II-3 Copper Crusher Gage............ II-3 Piezoelectric Gage.............. II-4 I'ASIC ELEMEZNTS OF SR-4 STRAIN GAGES...... I I-5 TIIEORY OF TIIICK-WALLED PRESSURE CYLINDERS...... II-7 EQUIPItNT AND A'PPARATUS, aS.............. II-9 TYPE OF GAGE.............. II-9 RECORDING EQUIPM'ENT..............., II-10 The Bridge Amplifier............. II-10 The Oscillograph.......... II-11 TIh Trigger Circuit..... II-13 The Recording Camera. II-14 The Constant Voltage Transformer II-14 TIIE GUN..,.,..,,,., II-14 TESTING PROCEDURE., I.... II. -15 PR'PARATION OF TIIE RIFLE,........ II-15 SETTING Ul' THIE EQUII'MENT....,...,. II-17 FMINlII'Gr.,................. II-20 RESULTS........ II-25 CALIBRATION..... II-25 CALCULATION OF PiESSlUES............... II-26 S[~iJlANy AND CO NCLU SIONS. IONS.......... 11-29 LIST OF REFERENCES.,,...... i II-32 iii

TABLE OF FIGURES Fifg.. I)iagram of Piezoelectric Gage'. Copper Crusher Cage is similar with copper cylinder in place of crystals II-4 Fif. 2. SR-4 Strain Gagce C-5-1. II-9 Fig. 3. Simple Whieatstone Bridge.......... II-11 Fig. 4. Tahe Trigger Circuit................ II-13 Fio. 5. Cross-section of chamber and location of gages..... II-16 Fig. 6. Instrumental Set Up...............,,.. II-18 Fiog. 7. Rifle Set Up.................., II-19 Fig. 8. Equipment with dials set for firing...... II-22 Fig. 9. Equipment with dials set for calibration...... II-23 Fig. 10. Calibration Signal......... II-25 Fig. 11. Pressure Curves with Calibration Signal...... II-27 Fig. 12. Pressure Curves with Calibration Signal.... II-28

ABSTRACT The purpose of this thesis is to attempt to determine the feasi. bility of using SR-4 strain gages to measure chamber pressures in a rifle. The problem is approached as a structural strain gaging problem involving a thick-walled cylinder and the instrumental set-up is described in detail. The pressures determined by this method were within the accepted values for the type of ammunition used. The overall accuracy of the method was not determined, but the accuracy of the strain gaging equipment was estimated. The sources of error discussed include the inconsistencies between the ideal thick-walled cylinder and the chamber of the rifle. A static load test of the chamber pressure is recommended as a possible means of evaluating the errors in this approach. From the findings of these experiments it is believed that further research will enable the expansion of the use of the SR-4 strain gage to regular use in the ballistics industry. vii

INTRODUCTION There are three reasons for making chamber pressure measurements: It is necessary in cartridge manufacture to make routine control checks to insure uniformity of production; in ballistics research chamber pressures must be determined accurately in order to test theories of internal ballistics; and finally, it is often desirable to determine the safety of ammunition in a sporting rifle barrel without tests that destroy the barrel for sporting use. The first two reasons are solved at present bIy instruments that require a specially chambered rifle with a hole in tie side so that a piston can be exposed to the chamber gases. This requires expensive machining and is good only for a limited number of shots. There are serious doubts as to the accuracy of this method because the chamber is altered which theoretically will alter the pressure developed. Furthermore this approach is of no value for testing sporting rifles because the rifle barrels are destroyed by machining the hole into the chamber. PUll POSE The primary purpose of this thesis is to determine whether SR-4 strain gages fastened to the outside of the chamber walls can be used to determine the internal chamber pressures. Basic theory, techniques, and electronic apparatus are described. The problem is approached from the standpoint that the chamber of a rifle is a thick-walled pressure cylinder and that the circumferential strain on the outside surface of the chamber can be related to the IIl

II-2 internal pressure by utilizing thick-walled cylinder formulas. SCOPE The scope of these experiments is limited to the determination of the feasibility of this method and obtaining information as to whether or not the strain gaging equipment available is sensitive enough to give good results. The instrumental set-up is explored in an attempt to correct trouble spots or note for future work to correct. The experiments are limited to one rifle and to one type of cartridge. No attempt is made to determine the overall accuracy of the method. Errors in the common thick-walled pressure cylinder theory are not considered. The complex theories of interior ballistics are not considered except for a brief description of other means of pressure measurement in current use. Because this was approached primarily as a strain gaging problem the history and operation of the SR-4 strain gage and recording equip. ment are presented in detail,

II-3 REVIEW OF LITERATURE There are many textbooks and other publications which cover the fields of interior ballistics, strain gaging, and the theory of thickwalled pressure cylinders as individual subjects. The combination of these three is unique at present and no published literature was found on this approach to chamber pressure measurement. Mention was made in a publication of Baldwin Locomotive Works (3), manufacturer of the SR-4 strain gage, that the gage has been used in ballistic work with shotguns. I'ESENT MIETHIODS OF' rREESIURE 1EASUREMENr There are two methods in current use for determining chamber pressure. Both have the disadvantage of being destructive to the rifle used, as their principle of operation requires a drilled and lapped hole into the chamber to admit a piston that is acted upon by the powder gases. op Crusher Gag e The copper crusher gage is used when only the maximum pressure is desired and details of the time-pressure curve are not important. It was invented by Noble in 1860 and is used successfully for routine checks in the production of cartridges (5). Essentially the gage consists of a copper cylinder that is restrained at one end and compressed by a piston that is exposed to the powder gases on the other end. A table showing the reduction in length as compared to the pressure is prepared by calibrating the cylinder

II-4 &v',#y Pas9 on... /smov/aed Bridfe BoAcht Ie lt cl B lseel *u r Ckdr BahAe/,t Ca sr. _-vr ft Crysils $,lvt Shell ~eerecte. ---- S-5reel Loodal Plote So/tr rel Connectors S1 Li Ple Fig. 1. Diagram of Piezoelectric gage. Copper Crusher gage is similar with copper cylinder in place of crystals. (7) with dead weights. For pressures below five tons per square inch a lead cylinder is usually employed.' i eze lectri Gage This gage makes use of the fact that certain types of crystals have the property of developing an electrical charge when they are placed under load. The amount of charge is proportional to the amount of load placed on the crystals. The crystals are cut perpendicular to their electrical axes and secured with conducting cement to silver foil connecting the negative and positive sides of each crystal to the next, as shown in Fig. 1.

II-5 The complete apparatus for pressure measurement work consists of the following: a means for transmitting the chamber pressure to the piezo gage, the piezo gage proper, a resistance-coupled amplifier to convert the electrical impulse developed by the gage to a measurable current, and an oscillograph-camera combination to record the impulse(7). This method can be used for time-pressure studies and is used more in ballistics research than routine checks. BASIC ELElMENTS OF SR-4 STRAIN GAGES By definition strain is the change in length of an article subjected to a load, divided by the original length from which the change was measured. The comnon unit of expression is inches per inch. This is what any strain gage measures. The SR-4 strain gage is not a mechanical measurement of strain but is an indirect measure utilizing a principle of electrical resistance that was discovered by Lord Kelvin. lie found that the electrical resistance of wires changed when they were stretched and that the change in resistance was proportional to the amount the wire was stretched. lie also noted that different metals had different changes in resistance for the same change in length (6). The SR1-4 gage of present form is the result of work by 1'., Simmons at the California Institute of Technology anrid A. C. Hu(Je of the Massachusetts Institute of Technology. They concluded that if wires could be bonded to tile material to be tested firmly enought thiat no slippage would occur they could measure the strain by measulrin%, the change in resistance of the wires. The problem of bonding the wires

ii-6 to the material was solved by using wires 0.001 inch in diameter, because the ratio of circumferencial area to cross-sectional area increases as the diameter is reduced (3). In order to use a long length of wire in a small space the wire is formed into a grid by looping it back and forth. The grid is then bonded in a matrix that can be applied to the material for test. Each lot of gages has certain characteristics. These are gage factor and gage resistance. The gage factor is also known as t1itc sensitivity factor and is the ratio of the change in resistance of the wire to the strain it undergoes. The change in resistance is expressed the same, as strain-that is-the total change in resistance in olhms divided by the original resistance in ohms. Thus the g!age factor or sensitivity is expressed in ohms per ohm over inches per inch. A higher gage factor indicates a greater sensitivity. The resistance of the gage is expressed in ohms and both characteristics are printed on each package of gages. Two types of gages are sold in three types of matrices. One type of (jage is made from a copper-nickel alloy and the other from a heavi i) cold-worked elinvar. The latter has a higher gage factor but is also highly sensitive to temperature changes. The matrices availabl) are paper and Bakelite, the latter for use at higher emperatures.'I 1i newest type is an extra thin paper base used where speed of application is desirable or where clanri;n(! for long periods of time is impossihle. Many different shapes of grids are available in all types and over i()t0 different gages are offered (3).

TIlOIIRY OF THICK-WALLED IRESSMtE CYLINDERS Cylinders subjected to pressures are divided into two general classes called "thick-walled" cylinders and "thin-walled" cylinders. The difference being that stress distribution through the thickness of the wall may be assumed constant without appreciable error when the cylinder wall is thin compared with the internal radius, hut when the wall is thick errors in this assumption become too large to be acceptable. Just where the dividing line is depends on the accuracy required. In the case of this rifle chamber the ratio of thickness to radius is 1.65 and the error would be in the order of at least 120 per cent(8). This is too large for this type of work so the thick-walled theory must be applied in this case. The solution of both cases makes use of several assumptions: A perfectly elastic, homogeneous material; and a prismatic, circular cylinder infinitely long. The solution of the thin-walled case is arrived at by simple statics because the stress across the section can be assumed constant. The thick-walled case was solved orginally by Lame' and makes use of the fact that for any thin section of the thick-wall, the thin-vll11 formula will be true (9). By use of the properties of the material of the cylinder, Lame' derived a workable set of formulas for thickwalled cylinders. The formula he arrived at that relates circumferential strain to internal pressure is shown on the following page together with its application to this problem.

II-8 o2 R- 2 PlR2 - P2R2 2 + ( P1 P2 ) p S - t 2 2 R2 RI Where: 1P - Internal pressure P2 = External pressure Rn = Internal radius R2 = External radius p = Radius of point being considered St = Stress at point being considered When P2 = 0 and P = R2' the equation becomes: 2 2 P 1R R 2 2 2 1 A = Strain, inches per inch E = Modulus of Elasticity = A Solving for Internal Pressure the equation becomes: 2 2 ( A E ) (R2 - R1 ) P = Substituting the values of E, R2, R i, it becomes: P1 = A microinches per inch X 09.395

II-9 EQUIPlENT AND APPARATUS The equipment and apparatus used in this work are the property of the University of Utah with exception of the rifle barrel which was furnished by Mr. P. O. Ackley* for use in research. All of the preparatory work was done in the Structures Laboratory of the Civil En(gineering Lcpartment and the test firing was done in the basement of the Civil Engineering Building. The individual pieces of equipment are described below. TYIE OF GAGE The duration of strain is so small in this work that temperature change affecting the gage during the recording period can be assumed to FELT produce negligible effect, which allows the use of the higher sensitivity of the "C" type gage. A convenient gage is tile C-5-1, Fig. 2, Mj Ibecause its 1/2 inch gage length makes it possible to apply two gages on the circumfer. ence and is of the special group of extra thin Fig. 2. SR-,1 Strain paper back gages for quick drying. This was Gage, C-5-4. desirable because of the difficulty encountered in clamping Sgages on the curved surface while they were drying. The C_1l required )beinqg held in place with tile fingers a few * llr, P, O. Ackley, uarrel Maker, 2235 Arbor Lone, Salt Lake City., tah

II-10 minutes and was ready, for use almost immediately. The gages used had a gage factor of 3.26* 2.0 per cent and a resistancce of 350 ohms i 5 ohms. The gages were mounted with their centers diametrically opposite and their length-wise axes on the same line. They were wired in series so that they acted as one gage with a resistancc of 700 ohms and a gage factor of 3.26. RECORDING EQUIlIMNT The recording equipment includes those instruments that were used to transcribe the changes in resistance of the wires in the gage into a for.; that is readable in terms of strain which could be ccrverted to internal pressure. The equipment consisted of a Bridge Amplifier, Oscillograph, Trigger Circuit, Oscillograph Recording Camera, and Constant Voltage Transformer. The nridge Amp-lifier This instrument is manufactured by Ellis Associates and is known as the BA-1. It contains the elements of a Wheatstone Bridge with variable resistors and a four step amplifier. The Wheatstone Bridge is shown in simple form in Fig. 3. The gage is shown in one leg of tile circuit and with no strain on tlhe gage, or any other point desired to use as reference, the blridge is balanced so that the voltage Eo is zero. Wllen thlc gage is subjected to strain the resistance is chan(lc(l and there is tihen an electrical potential at Eo. This small chantge in potential is impressed into the a;mplifiers which enlarge it depending upon the staoe of amplification lsedi. The output of the BA-1 is connected to the "Y"' input of the oscillotiranrl.

II-l1 E0 POWER SUPPLY Fig. 3. Simple Wheatstone Bridge. For purposes of calibration there are resistors of known vaiuc in the BA-1 that can be placed in the position of the gages by turning a switch. Thus calibration can be done at any convenient time during the test. Tlce fO)sjilloraph The oscillograph used in this experiment is a I)uMont 30411,. it is used for tie purpose of putting into wave or (!rapll form tile vlt,!;.' tlhat is imniresscd into it. In tlhis case the ilprCesset o oltit;C i s!s rOil thtc [A-1 amplifier. The oscilloJraplh Ihas two sets of parallel plates set p,1elvpeliicd l i:r

II-12 to each other to form a box through wliichl electrons must pass. Tihe impressed voltage from the amplifier is fed to one set of plates and tlhe voltage on the other set of plates is controlled by the setting on tile oscillograph. Depending on tile voltage impressed on the plates, thle electrons moving throu1ghl are deflected a certain amount. Tile electrons are "ained" at the sensitized screen on the front of the cathod ray tube, and with only the set of plates controlled by the oscillograph carrying voltage, the electrons make a luminous spot on the screen that moves straight across the screen. The speed of travel of tlhespot can be controlled from two crossings of tile screen per second to 30,000, which appears to thle eye to be a solid line. When the voltage change that originates with the change in resistance of the strain gage is impressed on the other set of plates, the traveling spot is either deflected up or down depending on whether the strain is compression or tension. The amount of deflection of the spot is dependant on the amount of strain that the gage and material it is attached to undergo. During tile time the pot is being deflected up or down by the voltage from the g9a!le it is moving across the screen with a constant speed so it produces a wave form that can be used to time the chlange in strain on the gage. Because in work of this nature the spot would be traveling so fast that even a camera would not record it, the screen has a certain amount of retentivity so that the trace of the spot remains after the beam of electrons has passed.

II-13 T Tr i.er Circuit Since the traveling spot on the oscillograph keeps moving across and repeating instantaneously there is the need for allowing only one pass of the spot each time the gun is fired to keep the scope from being cluttered up with several traces of the spot because the camera had no shutter speed that could be timed to take only the trace at the time the rifle was fired. SWlTCH EXT. SYNC........ 0 ~ ~ ~ --,.... 0,01 MFD 10 MEGOHM I- VOLTS 10 000 OHM 2 I20,0 GROUND Fig. 4. The Trigger Circuit The oscillograph can be set on "driven" which means that the spot must be driven across by an external voltage source. The trigger circuit was used for the purpose of developing sufficient voltage to drive tihe spot across the oscillograph, The control of the circuit was Ihandled by a micro-switch mounted on the rifle so that it was tripped by the firing pin as it moved forward. By controlling the "Sync Amplitude" it was possible to get a single trace on the scope, The circuit used is shown in Fig. 4,

II-13 T, Trifcer Circuit Since the traveling spot on the oscillograph keeps moving across and repeating instantaneously there is the need for allowing only one pass of the spot each time the gun is fired to keep the scope from becing cluttered up with several traces of the spot because the camera had no shutter speed that could be timed to take only the trace at the time the rifle was fired. SWITCH EXT. SYNC. 0.01 MFD 10 MEGOHM I2 VOLS 10,000 OHM GROUND Fig. 4. The Trigger Circuit Tile oscillograph can be set on "driven" which means that the spot must be driven across by an external voltage source. The trigger circuit was used for the purpose of developing sufficient voltage to drive tie spot across the oscillograph. The control of the circuit was handled by a micro-switch mounted on the rifle so that it was tripped by the firing pin as it moved forward. By controlling the "Sync Amplitude" it was possible to get a single trace on the scope. The circuit used is shown in Fig. 4.

II-14 Tlhe _[ccordini Camera The recording camera is a 35 mm camera that operates tlhroughi a mirror and is mounted directly on the front of the oscillograph. li th1 the trigger circuit in operation the shutter is opened before tile rifle is fired and closed after firing because only one trace shows on the scope. The camera is Model Number 35, manufactured by Allen B. DuMlont Laboratories, Inc. The film used was Super XX because it was desirable to have a fast film. Even with the small amount of retentivity and the diaphra(,ui wide open it was necessary to use fil intensity to get good pictures. Thel Constant Voltage Transformer A constant voltage transformer was used to insure a constant voltage source to the oscillograph. This eliminated the interference on the oscillograph due to slight variations in standard line voltage. TIlE GUN The gun was a standard 30-06 barrel mounted in a special jig that is used for test firing. The standard Army M-12 ball ammunition marked SL 45 was used in all test firing.

II-15 TESTING PROCEDURE Since no published material was found to set forth definite procedure the work of this thesis was governed by standard procedures only so far as placement and operation of the SR-4 gages. The technique that produced results is the one set forth below. Certainly many refinements could be made, but this procedure did yield good results without using any equipment that would not ordinarily be available to any laboratory equipped with strain gaging apparatus. PREPARATION OF TIlE RIFLE The modulus of elasticity was determined on the barrel blank after the barrel was bored but before the outside was turned to final dimensions. A twelve inch section was turned smooth and the strain measured by means of two Hluggenberger Tensometers. This was performed by Professor R. L. Sanks and the modulus of elasticity was determined to be 29,600,000 psi. The barrel blank was then chambered, turned to finish size and fitted to the test jig that was used for the firing. The measurements of the barrel were taken by means of micrometers. The inside diameter of the chamber was determined by using a fitted dowel and checking against fired shells. The chamber is shown in cross-section in Fig. 5. A base to mount the test firing jig on was made out of a four by four piece of wood. The jig was held in place by recessing into the wood and holding it in place with 1/4 inch U-bolts. Holes were provided in each end of the four by four for 1/4 inch rods to be driven into the

II-16 -G AGE to IF ~ R RR ~-' OF'- ----......G.......I -I-N —-- R 0.233" 0.68" 0.68" R 0. 225" I0.60' - 1.36 —- R 0.220" Fig. 5. Cross-section of chamber and location of gages. ground to hold the block in place during firing. The gages were applied to the rifle last in order to minimize any chances of damaging them. Two C-5-1 gages were mounted with their centers diametrically opposite and their lengthwise axes on the same plane. loth center lines were scratched on the outside of the chamber and then the surface was cleaned throughly with acetone. Standard SR-4 cement was applied to the surface and the gages were applied. They required holding in place with the fingers for about one minute. They were allowed to dry overnight before using. After applying the gages and before and after firing the resistance of the gages and the resistance to ground were checked. The resistance

II-17 of the.gages was the same before and after, and a resistance of over 1,000 meg-ohms was present at all times to ground. A ground wire of shield cable was attached to the barrel next to the gages by means of a hose clamp. SETTING U(P THE EQUIPAENT Some preliminary firing on a different rifle was done at the Police Rifle Range but weather conditions, power difficulties and the effect of the shock from the firing upon the oscillograph made it obvious that it was not practical to do the firing in the open. The tunnel in the Civil Eni neering Building basement is an excellent location. The rifle is set up in the tunnel and all equipment is kept on the main floor where there are no vibrations to interfere with the equipment. The rifle was mounted with the four by four against a concrete wall and the 1/4 inch rods driven into the ground. A four by four was grooved and placed under the barrel with a sandbag on top of the barrel to hold it firmly in place during firing. The only way to load the gun involved unscrewing the barrel so the lead wires to the gages and the ground connection were brought to a plastic board with cap screw terminals. The bottom of the terminals were soldered to the wires leading up the outside of the building to the Soils Laboratory where the recording equipment was located. All ground wires were connected in common to a water pipe by a hose clamp and lead sheet to insure complete bleed off of any ground current.

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II-20 Four b9s of WeAt sand w;ere placed three to four feet in front of the rifie to Stop tha bullet. All of) Pe -ezo rv U. equip renet wa s et u tp the,Sol a Loa ratory or one tabiT except the conlstnt vo1'agO Vra sTohrierp which w-s seu up in "aothe-r L:-oal because the operaZt:on of a~y' typie 4of coil near an oscl1og~raph p~'oduces anterference o the ~copa, The oscillograph, BA-1. Trion'e C ir~te, and Camara were set up as shown an Fig. 6. FIRIING Firing was accioplished by're* an~ of a wire running rthwroul gouide loops to a ieve.r attsded to the firing ping At first an at.S et was mnade t us the tige c uit to obtain a i but s e ae t since the f~it-l, pin would have to t ip lhe micro-sWt ch at the c9recrt time in order ro fi re the:ifle while the.V)t ias traveling across nothing was shoin n o the screen but a straight line. That meant',at either the t-rigjetr circuit was out of tir~ne or else no results were coming from the Gages. To determine which it was the trigsjer circuit was removed anrd the osciograph l eft on'recur"'hicch allowed the spot to contlinualy travel across and repeat instantaneously. The next shot fired gave a gocd trace on the scope and by a rough calculztion seemed, to be of about the correct magnitude. This first riring was donae with thie oscillograph set for Sweep Speed of 20 and Sweep Range of 10-50. tZis as n emh too slw to get the c hrve spreead out across the scone. By trial and error a setting of 1X0 cn the Sweep Speed and 5.250 on

II-21 the Sweep Range was found to be the most desirable. This speeded up the trace so much that to get results on the film it was necessary to use a diaphragm opening of fl.5 and set the intensity of the trace up to "full." The problem of adjusting the micro-switch so that it would trigger the scope off at the right time was solved by trial and error. It was fortunate that the first adjustment was close enough that by adjusting the controls on the oscillograph the trace was moved to the center of the scope. The calibration signal with the 1 M resistor from the BA-1 was tih nK s;t convenient one to use in this case. The decision is made by roughly approximating the strain anticipated and then using the resistor that produces a calibration signal near the same height as the expected pattern. After the correct resistor is used the "Y" amplitude control on the oscillograph can be used to make fine adjustments so that each division on the scope will be an even value of strain. In this case the adjustment was made so that each division on the scope grid represented 20 micro-inches of strain. The calibration signal can be shown at any time during the test by throwing the chopper switch from "Dynamic" to "Calibration". It was taken several times during the firing to make sure that the calibration had not changed. The equipment with the setting of the dials for both firing and calibration are shown in Fig. 8 and Fig. 9.

II-22 tiii: i' ~: 9r. ~ ~ ~. L~~~~~~~~~~~~~~~~~~~~~~~4.12 4 tc *t to~_ I l l| l _"" 1 g "Ot4 C)s.: s. 1 1 I4' —- ~ —f t_ t rl f t — - _'j-;' 1.-..2D.rBd c:l X i.t o:! iiE i;iE Ei!EEE sjiEot~E!i:lE EE: Sw r::::::::::*

II-23 i......... o ~ 0:!iii!?,:,!:??i,????/?'.~'*:::,' *,,,:z.:_ - ir: ~ ~ ~ ~ ~ ~ ~ / U i! | | I | b *S.~~~~~~~~~~~~~~~~~~~~~~~*s *S | | _ ~~~~~~~~s~ 4 | w~~~~~~~~~~~~~~~~~~~~~~~~~~ X~~~~~~~~ EI drq g S I N S~~~~~~~~~~~~~~~~~~~~~~~~~~~~-,i i ca # I~~~~~~~~~~~~~~~~~~~~~~~~~~se

RESULTS The results of these experiments indicate that this is a feasable approach to the problem of chamber pressure measurement when considered within the limits of these experiments. The equipment has demonstrated that it is of sufficient sensitivity to handle the strains involved without resorting to critically high amplification. The performance of the equipment used was very satisfactory. CALIBRATION The calibration signal was siwrn three times during the final firing, and was the same each time which results in only one caliSee Fig. 10, Page 25. bration value for all pressure measurements. At the enlargement used for the pictures the calibration signal is 0.70 inches high. The amount of strain that it repsents is calculated as follows: Resistance of Gage Strain, inches per inch= Gage Factor X Calibration Resistor Used Strain, inches per inch= 700 ohms 3.26 ohms/ohm over inches/inch X 1 megohlim Strain, inches per inch- 0.000214f - 214.0 micro inches per inch. 1I-24

II -2 5 _ E ~~~~~~~~~~~~~~~~~~~~~~~~~~~ _lE_~~~~~~~~~~~~~~~~~~~~~~ i~i"~~ ii Fi.1.Cl irto Sina

II-26 The st riil on the outside of tile chamber during firing is equal to the licilht of the pressure curve divided by the height of the calibration curve t ies 214.8 micro inches per inch. CALCULATION OF PRESSURES The internal pressure is determined by substituting the strain as calculated from the above calibration into the formula derived from the basic thick-walled pressure cylinder theory on page 8, This calculation as performed for shot number one of Fig. 11 is as follows: Internal pressure = Strain, micro inches per inch X 09.395 Internal pressure = Height of pressure curve X 214.8 X 89.395 hleight of Calibration Internal pressure = 27,430 X Height of pressure curve. Internal pressure = 27,430 X 1.58 = 43,300 psi. The photographs of all shots are shown in Fig 11 and Fig. 12. with the calculated pressure. The range of pressure given for the ammunition at the time of purchase was 40,000 to 45,000 psi. All calculated pressures were in this range.

Fig. 11. Pressure Curves with Calibration Signal.

II-28 P1 41,200 ps i1 - 42,200 psi $Si na Ir He i?,~ t~. 7 i.nc...s. Fig. 12. Pressure Curves with Calibration Signal.

SUMIAR Y AND CONCLUSIONS These experiments have shown that the instrumentation is of sufficicnt sensitivity to record the magnitude of the strain on the out. side of the rifle chamber. The accuracy of the strain measurements could be within the accuracy listed for the Gage Factor on the gage — two per cent — but errors in amplification, calibration and reading probably make the error in the strain measurements in the order of five per cent. The actual strain gaging technique used has been proved under so many different conditions that there is very little reason to suspect any large errors from this source. A possible source of error is the measurements of the Chamber, although they still should be within five percent. The measurements of the interior of the chamber could be made more accurately than those of these experiments by using a chamber castirlr Fetal. The logical source of serious error is in the difference betwcen the ideal conditions of the thick-walled pressure cylinder theory and the actual case of the gun chamber. The chamber is not of infinite length nor is it of uniform cross-section. Also, there are abrupt changes in cross-section near the strain gages- threads at one end and the shoulder at the other. In short, there is no formula at present that will give the relation of circumferencial strain to internal pressure precisely. There is also the question of whether or not the cartridge case should be included in the thickness of the wall of the cylinder. The fact that the calculated pressures were in the range of anticII-29

11-30 tlated pressures indicates that the errors are not of great magnitude. The simplest procedure for making a check of the results of these experiments would be to make a static pressure load test on the chamber I,)y sin(J some sort of fluid under pressure and recording the strain In(llcated. Comparison of these two along with research to determine the effect of the end areas and the short length of the cylinder should enable the determination of the possibilities of using this method in I)oth industry and research. The writer feels that the shrface has just been touched in the possibilities of using the SR-4 strain gage in ballistics industry and research. At present further research has been undertaken at the University of Utah to enlarge upon these experiments, to determine the backthrust on the bolt and the possibilities of using a static load test on the chamber as a means of verifying the results obtained during firing. Time calibration of the pressure curves and location of the bullet within tile barrel with respect to the pressure curve are all possible with the use of SR-4 strain gages. The possible applications of the SR-4 gage in the field of ballistics are almost unlimited and the writer feels that eventually this approach will be used in extensive ballistics research. There is a possibility that this method may replace the copper crusher gage for production checks. Then, too, there are advantages offered the sportsman who uses non-standard or "wildcat" calibers. Several of our present outstanding calibers are the result of "backyard" experimenters. Few or none of

II-31 these experimenters have the means to afford standard pressure tests, and, therefore the safety of their experiments is doubtful. The SR-4 larJc may make the testing of any "wildcat" quite practical and inex-. ipensive.

LIST OF REFERENCES 1. Baldwin Locomotive Works, Philadelphia 42, Pennsylvania. 1low to Apply SR-4 Strain Gages'. Bulletin 279-B. 1949. 2. Baldwin Locomotive Works, Philadelphia 42, Pennsylvania. Strain Gage Primer. Bulletin 283. 1949. 3. nBaldwin Locomotive Works, Philadelphia 42, Pennsylvania. SR-4 Strain Gage. Bulletin 179. 1949. 4. Balfour, C. Mi. Pressure Measurement in Ballistics Research. Engineering. 134:No.3476:231-2. August 26, 1932. 5. Corner, J. Theory of the Interior Ballistics of Guns. New York, John Wiley and Sons. 1950. 6, Dobie, W. V., and Isaac P. C. G. Electric Resistance Strain Gages. London, English Universities Press Limited. 1948. 7. Ervin, C. T. Piezoelectric Gage for Shotgun Pressures. Franklin Institute Journal 213: No.5:503-14. May 1932. 8, Laurson, P. G., and Cox, W. J. Mechanics of Materials. New York, John Wiley and Sons. 1947. 9. Murphy, Glenn. Advanced Mechanics of Materials. New York, McGraw-Hill Book Co. 1946. II-32

APPENDIX III Elements of Internal Ballistics I. Russo-German Methods by Professor L. E. Brownell; PhD Departments of Chemical and Nuclear Engineering and Faculty Supervisor, Du Pont Grant for Studies in Internal Ballistics Copyright 1966 The University of Michigan Ann Arbor, Michigan

TABLE OF CONTENTS Page Preface III-1 1. Introduction III-3 2. The MKS Mass System and Its Units 111-5 3. Muzzle Energy and Some Basic Concepts III-11 4. Smokeless Powder Propellants III-17 5. The Burning of Smokeless Powder at Constant Volume III-21 5.1 The Equation of State and Abel's Equation III-21 5.2 Powder Constants I I I- 31 5.3 The Burning Process III-34 5.3.1 Linear Burning Velocity II1-34 5.3.2 Dependence of Burning Rate on Size and Shape of the Powder Particles I I I-41 5.3.3 The Basic Equation of Pyrostatics III-54 5.3.4 The Linear Burning Law of Muraour and Aunis III-59 5.3.5 Experimental Investigation of Progressivity III-61 iii

1. RUSSO-GERMAN METHODS PROCEDURES FOR THE SOLUTION OF THE MAIN PROBLEM OF INTERNAL BALLISTICS by L. E. Brownell The "main" problem of internal ballistics is the solution of the relationships between chamber pressure (p), bullet velocity (v), bullet barrel-travel length (f) and time (t). European ballisticians such as Charbonnieri of France have usually related these variables to "y," the fraction of powder burnt at any time, "t." American ballisticians for many years related "p," "v," and "t" to bullet travel length "'" using the empirical methods developed by the French artillery officer Leduc.2 Both of these procedures have certain disadvantages. The equations of Charbonnier are based on the energy balance. Various modifications of his equations have had worldwide use in solving problems in internal ballistics. A major difficulty in using Charbonnier's equations, however, is that the fraction "y" of powder burnt is not zero at bullet start where v = 0 but "p" is "po," a definite pressure required to "start" the bullet on its path. The methods of Leduc are not based on the equation of the conservation of energy (First Law of Thermodynamics) and so they lack generality. New equations and constants must be determined for each type of gun. In 1903, the Russian ballistician Drosdow first proposed that "x," the fraction of powder thickness burnt after bullet start, be used as the independent variable rather than the total weight fraction "y." He let "e1" be half the initial wall thickness of tubular powder and "e" be the thickness at any time after burning starts and, let "z" be the fraction, e/el, and "zo" the fraction at bullet start. Then he defined x = z - zo. The value of "x" is zero at bullet start, and will be 1 - zo when the powder is all burnt. Thus, the value of "x" increases from 0 to a maximum as does the bullet velocity. This simplification reduces the definition of velocity I Charbonnier, P., "Ballistique interieure," Droin, Paris, 1908. 2 Challeat, J., "Theorie des affute a deformation," Rev. d'art, LXV, p.184, 1905. 3 Drosdow, N. F., (in "Innere Ballistik" by Serebrjakow,) (Moscow, 1949,) Moscow, 1903.

to a constant times x. This in turn simplifies the integration of velocity to give bullet-travel length (i). Serebrjakow4 developed the method of Drosdow further and explored the expression of burning rate of the powder as directly proportional to chamber pressure. In 1961, Wolff, a professor in East Germany, reviewed the Russian methods in Innere Ballistik 5 prepared for use in Officer Schools of East Germany and Russia and his book was translated by L. E. Brownell6 in 1966. 4 Serebrjakow, M. E., Innere Ballistik, Oborongis (Russ.), Moscow, 1949. 5 Wolff, W., Innere Ballistik, Deutscher Militarverlag, Dresden, 1961. 6 Wolf, W., Innere Ballistik, translated by Brownell, L. E., (Ulrich's Book Store, East University Street) Ann Arbor, Michigan, 1966. vi

PREFACE The international MKS (Mass) System of units was officially accepted in the (East) German Democratic Republic on August 14, 1958. This change is of special importance to the study of internal ballistics, because the general use of new units necessitates a change from the "engineering" system previously used in ballistics. Some of the benefits associated with this new system can be seen in the following example. The impulse, a basic unit in the rocket field, in the engineering system has the dimensions: "kg s/kg;" it is written in this manner to avoid writing "s" only. In the new system the impulse has the dimensions: "m/s," which is sufficient without additional description. The advantage of the new system become more obvious throughout this text whenever calculations with combined units are given, and becomes especially significant in the solution of the "main" problem of internal ballistics. Chapter 2, "The MKS Mass System and Its Units," is included to assist the reader in becoming familiar with the new system. All formulae and equations are written as unit-equations; and, conversions of non-consistent dimensions are clearly indicated. This book is not intended for the ballistic specialist, but rather for anyone connected somehow with weapons and ammunition who is interested in the processes inside a weapon, e.g., for officers and weapon-technicians. The field of internal ballistics may also be of importance to students in the field of thermodynamics since many problems related to thermodynamics still exist in internal ballistics. Only an elementary mathematical background is required to understand this book; seldom are basic concepts of differential and integral calculus necessary. The Appendix, however, contains two sections which require a somewhat greater knowledge of mathematics. The solution of the "main" problem of internal ballistics is certainly the heart of internal ballistics. Because of the limited scope of this book, only one ballistics method was selected from the many developed throughout the years, and this III-1

is a simplified and approximate one. The original plan was to demonstrate the ballistics solution of the French ballistician, Charbonnier. However, a detailed examination showed that Charbonnier's assumption that the covolume is equal to the reciprocal of the density, led to errors which increase during burning of the powder and which became excessively large at the end of the burning process. The same examination has shown that the method of the Russian ballistician, Drosdow, is more accurate. Therefore, the latter has been chosen; the more elaborate mathematical calculations involved in Drosdow's.method are more than justified by the greater accuracy obtained. The illustration of the Drosdow method used here is closely related to the method described in the more recent book by the Russian ballistician, Serebrjakow, Inner Ballistics, Moscow, 1949. Unfortunately, this book is not available at the present and no German (or English) translation exists. W. Wolff Dresden, Mairz, 1961

1. INTRODUCTION Ballistics is the study of projectile motion; internal ballistics is involved with the motion of the projectile in the barrel under the influence of the powder-gas pressure, whereas external ballistics treats the motion of the projectile under the influence of gravity and air resistance after leaving the gun barrel. This separation is made necessary by the differences in the forces acting on the projectile. For the intermediate case in which the projectile has just left the gun barrel, the projectile will be passed by part of the escaping gas and thereby influenced to a limited extent. The study of this transition state is not treated as a new branch, "transition" ballistics, because the effects on projectile velocity are not sufficient to warrant such a consideration. Precise knowledge of the forces acting on an object permits precision in the calculation of its motion. In the case of internal ballistics, the most important force to be considered is the pressure of the powder gas. Examination of the pressure process as well as the other processes in weapons is made difficult because of the high pressures (2000-3000 kp/cm2 or 30,000-45,000 psi) and high temperature (2500-3000~C, or 4500-5400~F) involved; also, the processes exist only a very short time. For example, the time which an artillery projectile spends in the gun barrel is between 0.001 second and 0.060 second. This time is determined by the burning process which in turn depends upon the gas pressure in the weapon. Several "laws" of combustion have been defined of which two will be examined more closely. Recent experiments conducted in Russia show that these "laws" are valid only under certain conditions (see Section 5.34 and 5.35), i.e., values which were considered constant are only constant under certain assumed conditions. The burning "laws" must be treated as approximations because the combustion process is far more complicated than they take into account. III-3

III-4 Internal ballistics can be divided into two main stages: the pyrostatic and the pyrodynamic periods. The pyrostatic period is involved with combustion at constant volume, such as in the manometric or pressure bomb, or in the chamber of the weapon before the projectile begins to move. The pyrodynamic period, however, deals with the events concurrent with the motion of the projectile. Experimental measurements with the powder in a pressure bomb give the data characteristics of the powder and provide basic information necessary for ballistic calculations. The examination of the pressure process and the motion of the projectile in the gun barrel lead to a system of equations, the solution of which is considered the "main" problem of internal ballistics. An exact solution requires extensive calculations. For convenience, other methods may be used which are based upon simplifying assumptions and require fewer calculations. However, such methods must be recognized as not being absolutely rigorous; some of the simplified assumptions can lead to large errors if the loading density is high. Internal ballistics for mortars and recoilless guns is appreciably different from that for cannons, howitzers and rifles and, thus, requires special treatment. An important problem of internal ballistics is the determination of the magnitude of the maximum chamber pressure and the muzzle velocity which depend in large part upon the weight of the charge of powder (load), projectile mass, combustion space, and the dimensions of the powder grains. In the sixteenth century, mathematicians Daniel Bernoulli and Leonhard Euler treated problems in internal ballistics. Euler established the theory of the elasticity of gasses. And, the invention of the ballistic pendulum, by Robins in 1740, was of prime importance as it permitted measurement of projectile velocity. Only recently has the magnitude of the forces acting on the projectile been appreciated. An extensive development of modern internal ballistics began at the end of the last century when smokeless powders, composed of nitroglycerine and nitrocellulose, were discovered. Since these powders can be formed in definite shapes and sizes, it became possible to control the rate of burning as well as the quantity of gas produced. The development of propellants for solid-fuel rockets is still in its early stages, but propellants for artillery pieces and small arms have reached a degree of perfection.

III-5 2. THE MKS MASS SYSTEM AND ITS UNITS In the past, the engineering (Force) system of units has been used in the field of ballistics. The international (Mass) system used here is that made public in Report No. 149 of the German Bureau of Standards of the (East) German Democratic Republic. At the tenth general conference of the Bureau of Standards in 1954, six basic units and their abbreviations were defined. Those of interest in internal ballistics are as follows: Dimension Unit Abbreviation Lenth the meter m Mass the kilogram kg Time the second s Temperature the degree Kelvin OK The adoption of this system provides a common set of units for both Physics and Engineering, and thus eliminates the simultaneous usage of the engineering system and the physics or absolute system. The use of two systems has caused extensive confusion in the past. The change described above is important in the field of ballistics and results in a number of advantages. Some comments are in order here to demonstrate the difference between the MKS Mass System and the MKS Force System. First the letters M-K-S are the abbreviations for "meter," "Kilogram," and "second." Note that in the MKS Mass System the kilogram is taken as a unit of mass whereas in the MKS Force System (called engineering system by Wolff) the kilogram is a unit of force. In the MKS Mass system the unit of force is the newton and is that force which will produce an acceleration of 1 m/sec2 with a one kilogram mass. A newton by definition is also equal to 105 dyns or about 0.225 lbs. of force. Work is equal to force time distance. Thus, in the MKS Mass System the newton meter also called the "joule" is the unit of work. One joule also is equal to 166 ergs and one joule per second is equal tothe "watt," the unit of power. This

III-6 simple system of units has been most widely used by electrical engineers but should be used by all engineers to eliminate existing confusion about the units of force and mass. The essential difference between the engineering and the physics systems is in the use of the unit of mass, the kilogram (kg) as a basic unit. Non-homogeneous derived units exist from simple product and quotient formation. If no other factor than unity (1) appears in the result, the derived units are called homogeneous or consistent with each other. In connection with the above considerations, the concepts of mass and force may be given a brief review. The "mass" of a projectile may be determined by placing the projectile on one side of a weighing scale and balance weights on the other. If the projectile is equal to the weights, the scale will remain in equilibrium, i.e., the scale shows equality of masses. This simple method determines the "mass," for example, of a 100 mm projectile to be 13.2 kg, and the "mass" of a 150 mm projectile to be 43.8 kg. The term "weight" would be wrong here because it signifies a force. Furthermore, the'mass" determination, unlike "weight," is independent of location; the same result would be obtained if the measurement were done at the North pole, at the equator or even on the moon. Thus, "mass" is an intrinsic property of a particular body. As indicated, the above discussion does not apply equally to the term "weight," and in this book, the term will be avoided.. A 100 mm projectile, placed on a table, exerts a force which becomes apparent by the bending of the table-top. This bending is rarely visible, but can be detected easily with sensitive instruments. Obviously, a 150 mm projectile exerts a larger force on the table. From Sir Isaac Newton we have the equation: F = jma where "F" is the force, "m" the mass and "a" the acceleration. On the basis of this relationship, a consistent unit for the force is defined. This unit, called a newton (N), is defined as follows: One newton is the force which gives a mass of 1 kg the acceleration of 1 m/s2. Thus, the unit of force becomes

III-7 1N = m kg s Units from other systems may be converted to the international system. Such derived units are not necessarily consistent. An example is the kilopond (kp) which is defined as equal to 9.80665 newtons. Thus, we have: 1 kp = 9.80665 N = 9.80665 kg-2 In earlier days the unit of force in the engineering system was one kg. In order to distinguish between the units of mass and force the term one kilopond (kp) was introduced. One pond (p) is 1/1000 kilopond. Hence: ip = 10 kp -3 2 = 9.80665 (10) kg s The determination of the unit of the force, kp (formerly kg), is arbitrary, whereas the determination of the former unit, the kg, was based on the normal gravitational acceleration at 450 geographical latitude. Both projectiles mentioned earlier would cause a different deflection of the table at the pole than at the equator, because gravitational acceleration is larger at the pole than at the equator. This reveals one weakness of the engineering system of units. Units are also required for pressure, work, energy, heat and power. The unit for pressure is newton per square meter, i.e., it is the uniformly distributed force of one N exerted on one m2. Thus: 2 -1 -2 1 N/m = 1 m kg s Often the engineering term,i2atmosphere (at), is used for pressure and is equal to 10,000 kp/m. Hence: 1 at = 101 2p/m = 98065.5 m 1 at = 10 kp/m = 98065.5rm kgs

III-8 For the ballistician not familiar with the k~ as a unit of force, it is convenient to remember that 1 kp/cm is equal to one atmosphere (at) or 14.7 psi. The expression of pressure in atmosphere of force per unit area also avoids confusing mass and force units. The product of force times distance is equal to "work." Work is equivalent to energy and heat. The Joule (J), the wattsecond (Ws) or the newtonmeter (Nm) each define the amount of work done by moving a fixed point the distance of one meter in the direction of applied force of one newton. Hence: 1 J 1 Ws = 1Nm 1 m kg s The unit of heat used is one calorie (cal). This is equal to: 2/ -2 I cal = 4.1868 J = 4.1868m kg s One kilocalorie (kcal) is equal to 1000 cal; hence: I kcal = 4168.8 m kg s "Power" is the amount of work done per unit time; its unit is the watt (W) and it is equal to: 1W = 1J/s = 1 m2 kg s The unit of one horse-power (PS) is also used and is equal to 75 kp m/s, or 735.5 W. In addition to the three basic units: length, time and mass, a unit for temperature, the degree Kelvin (~K), is required. The well-known centigrade degree is equal to the Kelvin degree. However, 0~C corresponds to 273 ~K, i.e., 15~C corresponds to 288~K. The term "degree" to indicate temperature differences may be used instead of "Kelvin degree." Thus, ~K is replaced by the notation: degree. The following table illustrates the most important derived units:

III-9 TABLE 2.1 Important Derived Units Derived Unit Units Involved Abbreviation Relation to Basic Unit Velocity meters/second m/s I/s m - s-! Acceleration meters/second sq. m/s2 1 m/s2 = 1 m s-2 Density kilogram/cubic m. kg/m3 1 kg/m3 = m-3 kg Force newton N 1N = lm kgs-2 kilopond kp 1 kp = 9.80665 N - 9.80665 m kg s-2 = lat cm2 = 10-4 at m2 pond p Ip 10-3 kp = 9.80665 (10)-3 m kg s-2 Pressure newton/sq. meter N/m2 1 n/m2 = 1 m-1 kg s-2 atmosphere at 1 at = 104 kp/m2 = 1 kp/cm2 = 98066.5 m-1 kg s-2 Work, energy Joule J 1J = 1 Ws = 1 Nm = 1 m2 kg s-2 and heat watt second Ws newton meter Nm calorie cal 1 cal = 4.1868 J = 4.1868 m2 kg s-2 Power watt W 1W = 1J/s = 1 m2kgs-3 horsepower PS 1 PS = 75 kp m s= 735.49875 W

III-10 Conversion factors are needed for non-homogeneous units The calculations shown in this book generally use the rounded values of conversion factors, i.e., 9.81 is used instead of 9.80665, and 735.5 is used instead of 735.49875; this is within slide rule accuracy. In all calculations, formulae and equations are written as dimensional equations, i.e., each physical value is given as a product of the actual numerical value and the corresponding unit. For the velocity "v" we have: v = it where "f" is the length of the completed path and "t" is the time in which the distance was covered. For example: - 16 km and t = 4 min; then: 16 km km v - 4 4 min min Note that 1 km = 1000 m and 1 min - 60 s; thus: 1000 m -1 v = 4 0 4(16.66) m/s 66.66 ms 60 s For a weapon producing a maximum gas pressure of 2200 atm, the pressure can be re-expressed as follows: 104 104 (9.81)N 2200 at = 2200 = 2200 2 2 m m 20 104 (9.81) m kg8 -) -2 = 2200 _~_ _(9.8 k = 2152 (108)ml kg S2 S m = 2200 (1kp = 2200 (14.7) lbs 220cm 32,340 psin cmn in

III-11 3. THE MUZZLE VELOCITY AND SOME ADDITIONAL BASIC CONCEPTS Firearms are heat engines which convert the chemical energy of the powder into the kinetic energy of the projectile. Although only a fraction of the available chemical energy is converted into kinetic energy, it is desirable to obtain the highest conversion percentage possible. The energy which the projectile possesses when leaving the muzzle, the "muzzle energy," is represented by: E =1 mv 2 (1) k 2 O where "m" is the projectile mass and "vo" the projectile velocity, or "muzzle velocity," at the moment when the projectile leaves the muzzle. At zero travel from the muzzle, this "muzzle velocity" is identical to the "initial velocity" in external ballistics. Equation (1) gives the kinetic energy of the projectile. A spinstabilized projectile rotates about its cylindrical axis and possesses rotational energy, but such energy can generally be neglected as it is often less than 0.5 per cent of the translational energy. The following examples will illustrate the calculation of the muzzle-energy and other closely associated parameters. The examples also illustrate the application of the mass system and the units mentioned in Chapter 2. Example 1. What is the value of the muzzle energy of a 150 m projectile with a mass of m = 44.2 kg and a muzzle velocity v = 680 m/s? Ek = 1(44.2 kg) 6802 m2/s E 152 -2 Ek 2 (44.2) (4.624) (105) m kg s 2 k 2 E 1.022 (10)7 J k

III-12 Example 2. What is the value of the muzzle energy for a carbine, if m - 9 grams (139 grains) and vo = 840 m/s (2760 ft/sec.)? 1 (0.009) (70.65) (10)4m 2kg s-2 k 2 E = 3175 J k Only about 1/3 of the energy contained in the powder is transferred as muzzle energy to the projectile. The remainder of the energy escapes as heat content and kinetic energy of the escaping powder gases. Based on Example 1 and a conversion efficiency of 1/3, determine the energy content (Ep) of the charge: E = 3(1.022) (10)7 J = 0.7323 (10)7 cal p E = 7323 kcal p The heat content of nitroglycerine powder is 1150 kcal/kg. In order to obtain the required muzzle energy, the following loading mass is necessary: 7323 kcal 11 50 kcal/kg The corresponding values for Example 2 are as follows: E = 3(3175 J) = 9525J = 2275 cal = 2.275 kcal p Using nitrocellulose powder with a heat content of 834 kcal/kg, the required loading mass is: 8342.275 kca = 0.00273 kg = 2.73 gm 834 kcal/kg (2.73 x 5.4 grain = 42.0 grains) gram The powder gases produced by burning the load exert a pressure on the base of the projectile. These gases accelerate the

projectile until it emerges from the muzzle. In order to obtain an approximate average value of the magnitude of the pressure, we can assume a mean or equivalent pressure of the powder gases that remains constant. Note that this is not correct, but is a convenience since the gas pressure increases to a maximum and decreases thereafter. This assumption permits calculation of an integrated mean gas pressure which will be of interest. A pressure of "p" acts on the base of the projectile. If "p" indicates the mean pressure of the powder gases, "q" the projectile cross section, and "to" the length of the rifle barrel, the work done by the powder gases is "pqfo".Neglecting friction losses (small for large guns), this work may be set equal to the muzzle energy: - 1 2 pq 2 Lmo (2) and the mean pressure becomes: mv y2 E 0 k _ o _ k (3) 2 q=O qfo Example 3. Using the same data as given in Ex. 1, what is the mean gas pressure if the rifle barrel has a length of 3.75 m? Ek 1.022 (10) J 5.4 (07 J4 (107kg = 15.4 (10) 5.4 (0) 0q~ 0.0177 m (3.75)m m ms Now: 1 at = 9.81 (104) k ms Thus: p - 1570 at (Note: The maximum gas pressure is significantly higher than the mean pressure p.)

III-14 Firearms use comparatively large amounts of energy. To show this, we shall calculate the kinetic energy of an automobile with a mass of 1800 kg, (3960 lbs.) moving with a velocity of 80 km/h. (The above velocity is v = 22.2 m/s, or about 50 m.p.h.) Hence: Ek = - 1800 kg (492.84) m /s2 = 4.44 (10) m kg s k 2 The muzzle energy of the 150 mm projectile of Ex. 1, was twentythree times as large. These large amounts of energy must be liberated in a very short period of time. The approximate time elapsed between the instant the projectile starts its motion and the instant it leaves the muzzle may be determined easily. The projectile motion starts with velocity = 0 and ends with velocity = vo. Thus, the arithmetic average projectile velocity in the barrel is: - O + Vo vo v = 2 =-2 (4) If the length of the rifle barrel is f, the approximate time for the transit becomes: t - m v m/s 2 (s) (5) In reality the true transit times are somewhat longer. Equation (6) is an approximate formula which gives more realistic values than Eq. (5): t 2.4 (6) vo For Exs. 1 and 3 in which I = 3.75 m; and v = 680 m/s, Eq. (6) gives: t = 2.4 (3.75m) = 0.0132 s 680 m/s

III-15 and for Ex. 2 with a length off = 0.48 m for the carbine barrel: 2.4 (0.48 m) t 840 m/s = 0.00137 s The examples above show that a significant difference in transit-time exists between a cannon and a small arm. Here, this difference amounts to a ratio of 10:1. The transit-time is of major importance in determining the lifetime of the weapon, i.e., the number of firings possible with one barrel before it is rendered useless. This is true because for the same gas temperature, the longer the transit-time, the shorter will be the barrel lifetime. The combustion of the powder in the barrel develops temperatures from 2000 to 35000C which exceed significantly the melting temperature of the steel (1300 - 1500~C). Thus, the longer the exposure of the interior of the barrel to the hot powder gases during a firing process, the shorter the lifetime of the barrel. Firearms have the characteristic of producing a large amount of work in a very short time. The work done by a force in a unit time of one second (s) is a unit of power such as the watt (W). Hence: J 2 g s-3 The power may be determined by dividing the work of kinetic energy by the time, or for the power, N: Ek N-7 Referring to Ex. 1, we have: 1N 022 (1o0) J = 0.774 (10)9 W = 774000 kW 0.0.32 s (kW - Kilowatt = 1000 watt) This is a very significant amount of power as shown by comparison with the output of a modern powerplant. Excluding some of the huge powerplants of the USSR (and the USA), the average power

III-16 of a typical powerplant lies between 100,000 and 500,000 kW. Converting to horsepower (PS) for the case of Ex. 1, where 73 5.5 W = 1 PS, gives: N = 0 744 (10) W 1.052 (10)6 PS 735.5 W (or about the same as the design power level of McNary Dam on the Columbia River, USA) i.e., approximately one million PS. The carbine, used in Ex. 2 also produces considerable power: N = 00137 = 2.32 (10) W = 2320 kW 0.00137 s or 2.32 (10)6 w 314 PS 735.5 W (or about ten times the horsepower of the larger 1966 automobiles of the USA)

III-17 4. SMOKELESS POWDER PROPELLANTS We have seen that firearms are heat engines of extraordinary power output, i.e., large amounts of energy are converted in a very short time. The combustion process is the responsible event, and, therefore, must occur very quickly. In order to obtain the high power required, a propellant called a "powder" is used. The term has nothing to do with the characteristic implied in "powderlike," but is an inheritance from the days of black "'powder." Before discussing modern propellants, we will consider briefly the precursor black powder which has been used for the past 500 years and continues in limited use today. The action of black powder can be demonstrated by the following simple experiment. A small amount (5g) of saltpeter (potassium nitrate, KNO3) is heated in a pyrex-glass beaker until it becomes liquid. A pea-sized lump of sulfur now introduced into the molten KNO3 will burn, developing an intense flame and giving off so large an amount of heat that the beaker will melt. Such rapid combustion is possible because KNO3 contains a large amount of oxygen which is given up almost instantly. A similar but slower process occurs if charcoal is used instead of sulfur. Black powder consists of a mixture of saltpeter, charcoal, and sulfur. The proportions of the mixture are selected for the intended purpose. One widely used formula for black powder consists of 75 per cent saltpeter (KNO3), 13 per cent coal (C), 12 per cent sulfur (S). The combustible materials are the coal and the sulfur, whereas, the saltpeter furnishes the oxygen. Through the addition of sulfur, the mixture becomes more easily ignitable. Thus, the burning of black powder is an ordinary combustion process with the exception that it occurs much more quickly than say that of coal in a furnace; an additional difference is that black-powder combustion takes place in a closed space at a constant or confined volume. Combustion occurs according to the following equation: 2KNO3 + 3 C+S -, K2S+3 CO2 +N2 The right side of the equation shows that the products of black-powder combustion are potassium sulfide (K2S), carbon dioxide (C02) and

III-18 nitrogen (N2). Carbon dioxide and nitrogen are gases; potassium sulfide is a solid compound part of which remains as a layer on the walls of the barrel, the rest escaping as smoke. Here, the solid combustion products amount to nearly 41 per cent and influence the overall combustion process in an undesirable manner. In the firing of modern powders, the propellant grains are converted completely into gas and are, in contrast to the black powder, "smokeless" (or better, "less smoke") powders. The basic ingredient of all modern smokeless powders is nitrocellulose (pyroxilin) which is a combination of cellulose and nitrate groups from nitric acid. Cotton is nearly pure cellulose and scrap-cotton can be used for the production of nitrocellulose. But, it is not necessary to rely on cotton alone, since pure cellulose may also be obtained from wood and straw. In the production of nitrocellulose, very small pieces of cellulose fibers are added to a mixture of nitric and sulfuric acids. The cellulose combines with the nitric acid and the end result is an explosive product, "guncotton." Explosives are not suitable as propellants because of their high conversion velocity; rapid conversion of explosives is called detonation. To avoid the almost instantaneous conversion of nitrocellulose into gas (which renders it unusable as a propellant) the fibers of nitrocellulose are gelatinized, i.e., the fibers of "gun cotton" are transformed into a uniform mass. This is accomplished with the help of a mixture of ether and alcohol in which nitrocellulose is soluble. The plastic mass thus obtained can be shaped and then dried to evaporate the solvent. However, one disadvantage of this method is that in the evaporation process the nitrocellulose powder becomes porous and hydroscopic. A second disadvantage is that of using a solvent which must be removed later. Therefore, years ago, attempts were made to find a solvent which could remain in the mass-such a solvent is nitroglycerine. Nitroglycerine is also an explosive, which, when mixed with diatomaceot earth, is known as "dynamite." The mixing process reduces the extren sensitivity of nitroglycerine so as to lessen its explosiveness; diatomaceous earth is very porous and consists of the shells of dead microscopic organisms, "diatoms." Nitroglycerine is obtained through a process, similar to that used for nitrocellulose, using the combined

III-19 effect of nitric acid and sulfuric acid on glycerine. In the reaction, the nitric acid combines with the glycerine. "Nitroglycerine powder" consists of a mixture of nitroglycerine and dissolved nitrocellulose; formulations for the powder vary in the percentage of each of these components as well as in the addition of a third component. This is "centralite," an organic nitrogen compound related to the urea, which improves the gelatinizing process, and increases the storage life and reduces the heat content of the powder. A controlled and lower-heat content is important since this allows a better adjustment of the powder to various purposes. Also, propellants with a high-heat content lead to a reduced barrel lifetime. As mentioned earlier, nitrocellulose and nitroglycerine powders burn completely to gases, and, in this respect, are superior to black powder. A further advantage lies in the possibility of forming the powder into definite shapes. Distinctions are made between the shapes of individual powder particles:' leaf-powder, strip or band powder, ring powder, tubular powder, and multichannel powder. Multichannel powder, often called just "channel" powder, consists of powder cylinders, each containing seven channels which run parallel to the cylinder axis. The length of. tubular powder grains in general varies with the length of the combustion space. The wall thickness is decisive for the length of the combustion period; hence, the ratio of the diameter of the empty space in the tubes to the total diameter should be about 1:2.5. Instead of strip or band powder common in Germany, some countries such as France and England use powder in the form of long, thin, full-cylinders. Since this powder has the form of a rope or string, it is also referred to as cordit (French corde=rope, string) or cordite (England). Since the powder grains burn in successive layers, the combustion velocity of the total charge can be regulated to a large degree through the proper selection of the shape. The mathematical treatment of the combustion processes is made possible simply through defining the shape of the burning surface. The internal ballistician may not only calculate the required powder charge for a weapon using a certain powder composition, but may also determine the necessary shape and size of the powder grains. Before the development of the "smokeless" powder, attempts were made to regulate the combustion process of black powder through selection of the proper powder shape. However, the individual grains pressed from black powder tended to fall apart during the combustion process.

111-20 After World War I, Germany developed new powder mixtures such as diglycol powder and gudol powder. The incentive for this was the catastrophic need for fat. Glycerine as part of a fat can be obtained essentially only from fats, and, during World War I, the total fat supplies in Germany were used for the production of powder. In the new powder mixtures, nitroglycerine was replaced by diglycol, i.e., diglycol and nitroguandin, respectively. The basic substances required to produce diglycol are coal and chalk. It has been shown that these new powder mixtures possess some favorable properties which are derived from their relatively low heat content. This results in better protection of the barrels giving them a longer lifetime, and in a reduction in muzzle flash. Diglycol powder also can be adapted to rockets. Inorganic propellants (their major component is ammonia) are also used since such propellants are significantly cheaper than nitrocellulose or nitroglycerine powder.

III-21 5. BURNING OF SMOKELESS POWDER AT CONSTANT VOLUME 5.1 THE EQUATION OF STATE AND ABEL'S EQUATION: The conversion of the propellant from the powder load to powder gases takes place in the gun during the definite, but very brief, interval of time called the combustion period. Factors influencing the length of the combustion period present the most important problems in internal ballistics. Analysis of the propellant combustion in the chamber and in the barrel of the gun is complicated because the acceleration of the projectile during the combustion period increases the space available for the powder gases, and this prevents the direct use of the simpler relationships of combustion at a constant volume. However, the problem can be overcome by using approximate procedures which permit the use of constant-volume data. In the determination of constant-volume data, powder is burned in a combustion space of an apparatus known as a pressure, or manometric,bomb. The pressure bomb in Fig. 1 consists of a hollow cylinder (1) made of high-strength alloy steel. It is closed at one end by a movable piston (3). The piston presses against a small copper cylinder (crusher gage) (4) which is held in place by a threaded closure (5). The powder is placed inside the bomb and ignited by an electrically heated glow-wire. The pressure created by the burning powder is transferred by the piston (3) to the copper cylinder (4) which it compresses. The dimensional change in the axial direction is used as a measure of the pressure exerted on the crusher body. This method measures only the maximum pressure. Pressure as a function of time can be determined by replacing the copper crusher with a piezo-electric quartz crystal or a strain gage used with an oscilloscope. Such methods will be discussed in more detail later.

III-22 Usually a series of test firings are conducted in the manometric bomb over a range of maximum pressures to determine the powder constants. In addition to pressure, other important variables include the identity of propellant, amount of load, bomb volume, and maximum gas temperature. A classic relationship known as the ideal gas law exists for ideal or perfect gases. Real gases, such as the powder gas at high pressure, do not precisely satisfy the ideal relationship. In general, the lower the gas density, the more closely the gas follows the ideal equation of state. As an example, the equation of state can be used to predict the behavior of air. The relationship between pressure, volume, weight of gas, and temperature is: pV = LRT (1) where "p" is the pressure, "V" the volume of the bomb, "L" the weight of the gases (taken as equal to the weight of the propellant load), "T" the absolute temperature, and "R" the gas constant which is a characteristic of the gas. Dividing Eq. (1) by L gives: pv = RT (3) where L = v (2) Here, V/L is the specific volume of the gas: 1 L v V The quantity "p" is the density of the gas. From Eq. (3) it follows that: pv= R (4) T

III-23 Thus, the gas constant (R) which is characteristic of a particular gas can be determined if the values of p, v, and T are known. Example 1. Let us determine the gas constant for air. For a pressure of p = 760 mmHg = 10332 kp/m2 x 9.807N/kp = 101,325 N/m and a temperature of T = 2730K = 0~C and an air density of p = - = 1290 kg/m3 v the gas constant for air is: R =_ 101325 N/m2 T 1290 kg/m3 (2730K) m T R = 286.9 286.9 s2K kg~K sK If the given weight (L) of a gas is heated at constant pressure (p) from T1 to T2, and the volume increases from V1 to V2, the following relationships hold: pV - LRT1 and PV2 = LRT2 Subtracting the first equation from the second gives: p (V2 - V1) = LR (T2 - T1) The product on the left side has the dimensions of N/m2 and m

III-24 Writing only the units: N 3 m = Nm = J m Thus, the left side represents work which is performed by the expansion of a gas having a weight of L kg if the gas is heated from T1 to T2. The gas constant can now be considered as equal to the work done by 1 kg gas when the temperature of the latter is increased 1~K under constant pressure. The gas constant of air is: R = 286.9 kgOK The dimensions J/kg ~K are consistent with the definition above. In the engineering system of units, gas constants with other dimensions and other numerical values are used. The dimension of work in this system is written as kpm. Thus: 1 kpm - 9.807 Nm = 9.807J Hence: JR = 286.9 _ 286.9 kpm R= 286.9 kg OK 9.807 kg ~K R = 29.27 kpm kg OK The above value of R is found in the literature of ballistics and thermodynamics. The equations of state, Eqs. (1) and (3), are accurate for ideal gases of low density. Therefore, since the processes in firearms involve high pressures and powder gases with large densities, the equations of state must be corrected. This correction is based on the assumption that the gas consists of molecules which are small with respect to their separation distance. Substituting

111-25 T = 0 into Eq. (3) will eliminate the term pv. But this implies that gas has no volume at some finite pressure, contradicting fact. If the gas has a large density, the volume of gas molecules becomes significant. Subtracting the volume of gas molecules from the total volume gives the value of the free volume available between molecules. Thus, the free volume (V) in Eq. (1) may be replaced by the term V-aL. Here, a is the volume of 1 kg of powder molecules. It is called the "covolume" and has the dimension, m3/kg. The product aL can be considered as the characteristic volume of the powder gas molecules. Eq. (1) when corrected for this volume becomes: p (V-aL) = LRT (5) If the powder is burned in a pressure bomb, a certain temperature, Tex (the explosion temperature), and a certain gas pressure are reached. Tex and R are characteristic constants for a particular powder. Combining both constants gives a new constant "f;' sometimes called the powder force and equal to RTex or: P f V-L (6) In order to use the values of f given in the existing literature, the dimensions of kpm/kg are used as shown in Table 5.1. Equation (6) is called Abel's equation after the ballistician, F. Abel who proved its validity for pressures up to several thousand atmospheres. If L is small compared to V, the product aL can be neglected and Eq. (6) becomes the equation of state for perfect gases.

III-26 TABLE 5.1 Values of "f" and "a" for Different Powders Powder Types f = tkg a = [ ] Single Base (Nitrocellulose) 84,000 —- 105,000 0.90 —- 1.1x10 Double Base (Nitroglycerine) 90,000 --- 120,000 0.75 --- 0.85x10 Diglycol 100,500 1.00x10 3 3 Black Powder 28,000 --- 30,000 0.5x10 Dividing the numerator and denominator of Eq. (6) by V gives: L/V P 1-aL/V The ratio L/V is called the loading density and is indicated by ~, or: A= L/V kg/m3 and Eq. (6) becomes: P = f (7) 1-aA The term "loading density" is applied not only in the study of powder constants determined using the pressure bomb, but also in the general study of firearms. This density is equal to the ratio of the powder load to the combustion space volume (Vo) in the chamber. The unit of dimension for Vo is the liter which can be set equal to one cubic decimeter dm3, or 1000 cm3 Equation (7) shows that the pressure is a function of the loading density (A) and that the pressure increases with increasing loading density.

III-27 Figure 2 shows a plot of the final pressure p (atmospheres), i.e., after the powder load is burnt, as a function of the loading densities (kg/dm3) of single-base (nitrocellulose) and double-base (nitrocellulose and nitroglycerine) powders. The upper curves of Fig. 2 indicate that the rate of pressure rise is not constant but increases with the loading density. If the covolume is neglected (a = 0), the lower linear curves indicated by dotted lines are obtained. The two sets of curves deviate strongly, particularly at high loading densities. Thus, covolume can be neglected only up to pressures of approximately 500 atm. Firearms (except mortars) have loading densities which are significantly larger than those used in the pressure bomb. Typical values for loading densities of various weapons are given in Table 5.2. TABLE 5.2 Typical Loading Densities of Various Weapons Weapon A, kg/dm3 Small Arms................. 0.80 -- 0.85 Long Range Weapons.............. 0.65 -- 0.78 Cannons..................... 0.55 -- 0.70 Howitzers, Full Load.............. 0.45 -- 0.60 Howitzers, Reduced Load............ 0.10 -- 0.35 Mortars........................ 0.03 -- 0.12 In spite of some of the high loading densities listed in Table 5.2, the high pressures shown in Fig. 2 are not actually reached in the weapons. This is because the projectile moves during the combustion period thereby increasing the volume of containment. Thus, ordinarily the maximum gas pressure does

III-28 not exceed 3000 atmospheres; the danger of producing excessive pressure if the projectile were hindered in its motion, however, should be obvious. Although the validity of Abel's equation has been proved for several thousand atmospheres, it has not been proved for extremely high pressures which would occur in pressure bombs with loading densities of 0.6 and 0.7 kg/dm3. Example 2. A charge of 25 g of single-base (nitrocellulose) powder is ignited in a manometric bomb having a volume of 200 cm3. The powder has the constants; f = 95,000 kpm/kg, and a = 1.1 x 10-3 m3/kg. What is the pressure in the bomb at the end of combustion? A_ 02 - 0.025 kg - 125kg/m 0.2 dm3 0.0002 kg/m 95,000 kpm/kg (125 kg/m ) p 3 3 3 1-l.lxl1O m /kg (125 kg/m ) 875x106 2 6 2 11.875x10 kp/m = 13.77x106 kp/m 0.8625 p = 1377 at 1380 atmospheres The powder force (f) and the covolume (a) can be found from the pressure measurements in the manometric bomb. Two equations are required for the simultaneous solution of the two unknowns. These equations can be obtained from pressure measurements using at least two different loading densities. The solution can be performed either analytically or graphically. Using the graphical-solution method, Eq. (7) may be rewritten as the equation for a straight line, or: p/A = ap + f (8) Values of p/A on the y-axis versus p on the x-axis may be plotted as shown in Fig. 3. A series of firings with different loading densities produces a range of pressures which lie on a straight line (g) corresponding to the points obtained from

III-29 Eq. (8) (see Fig. 3). The coefficient a is equal to the tangent of the angle d which is the slope of the straight line; hence: = - arc tan a The intercept of the straight line (g) with the ordinate axis gives (f). Figure 4 shows three characteristic straight lines, 1, 2, and 3, for three different types of powder: 1. Double-Base (Nitrocellulose plus Nitroglycerine) Powder f = 120,000 kpm/kg; a = 0.85 x 10 m /kg 2. Single-Base (Nitrocellulose) -3 3 Powder f = 95,000 kpm/kg; a = 1.1 x 10 m /kg 3. Black Powder f = 30,000 kpm/kg; a = 0.5 x 103 m 3/kg Since two points are required to define a line, pressure measurement at two different loading densities must be made to determine powder constants f and a. If the loading densities are h1 and A2 and the corresponding pressures are P1 and P2, then the points A and B are obtained with the following coordinates: A (p1/Al, pt) B (p2/A2, p2) These points connected by a straight line are shown in Fig. 5. The point of intersection of the straight line AB with the ordinate axis is the powder force f; the slope (angle d) gives the covolume according to the relationship: a = tan3 In order to obtain accurate values, very small loading densities should be avoided; also, A and B should be separated so that A2-A1 is approximately 0.1 kg/dm3 or more. Loading

III-30 densities of A1 = 0.15 and A2 = 0.25 kg/dm3 are recommended for single-base (nitrocellulose) powder, while values of A1 = 0.12 and A2 = 0.20 to 0.22 kg/dm3 suffice for the more potent doublebase (nitroglycerine) powders. Better accuracy can be obtained by making more than two test firings using the same kind of powder but different loading densities. If all of the points obtained do not lie on one straight line, an average straight line between all the points may be drawn such that the mean deviation of the individual points from the line is a minimum (method of least squares). The analytical procedure for determining the two unknowns a and f also requires a minimum of two equations which can be obtained by two firings using two loading densities, A1 and A2. This gives the following two equations (9a) and (9b): P1/A = f + ap (9a) and P2/A2 = f+ p2 (9b) Subtracting the first equation from the second gives: (P2/A2)-(P1/A1 )- a(p2-pl) It follows that: (P2/A 2 t(P/ ) a (p2/2(/) (10) P2 - P and (P1/A1) (P2/A2) (A2-A1) ( f = (11) P2 - P The analytical method may also be used with more than two test firings by utilizing the method oi least squares.

III-31 In order to obtain an accurate determination of f and a, the loading densities used must be sufficiently large —a lower limit of A = 0.1 kg/dm3 is recommended. Systematic experiments by A. I. Kochanow have shown that the straight line predicted by Eq. (8) is not obtained when very small loading densities are used (0.015-0.020 kg/dm3). Instead, the results (A, B, and C) lie on a hyperbolic curve which approaches asymptotically to the straight line g predicted by Abel's equation, (see Fig. 6). To explore this phenomenon, Serebrjakowl experimented with powder grains of different thicknesses using equal loading densities. His work showed that the maximum pressure was lower with the thicker powder than with the thinner powder. In Fig. 6, point 1 indicates the point of maximum pressure for the thicker powder, and point 2, that for the thinner powder. The reason for the deviation from Abel's equation was found to be related to heat transfer to the bomb wall. Thick powder has a longer burning time than thin powder and so it transfers a larger amount of heat to the bomb wall before reaching maximum pressure. Thus, the powder force (f) which equals RTex, and the measured pressures,are low. On the basis of such experiments it is possible to determine correction values for heat losses in the bomb. 5.2 THE POWDER CONSTANTS In addition to the powder force (f) and the covolume (a) discussed in Section 5.1, other characteristic constants are required to define a particular powder. The heat of combustion (Q) is one of special importance. This quantity is the amount of heat which is created during the burning of 1 kg of powder. The unit used for the heat of combustion. is usually the kcal (1000 calories), where 1 cal = 4.1868 m kg s-2 = 1 kcal/1000. Thus, the dimension for the heat from a kilo of powder is kcal/kg. The heat of combustion for double-base (nitrocellulose plus nitroglycerine) powder is about 1100-1200 kcal/kg, whereas diglycol powder has a heat of combustion of only about 740 kcal/kg. Therefore, to obtain the same power with diglycol as with double-base powder, a significantly larger amount of diglycol powder is needed.

111-32 The value of Q is determined experimentally by burning a known amount, L kg, of the powder in a calorimetric bomb. The calorimetric bomb is used in many fields besides ballistics to measure the heat of combustion of various compounds. Essentially it is a pressure vessel with a screw top through which an insulated wire is introduced for powder ignition. It has a thinner wall than the manometric bomb and cannot be used at very high pressures. The combined assembly of bomb plus a water bath is known as the calorimeter. To measure the heat of combustion,the bomb is immersed in the bath. The heat given off during combustion causes an increase in the temperature of the calorimeter, i.e., the water and the vessel, and this temperature rise is used to calculate the heat of combustion. Additional details concerning the apparatus and experimental techniques of bomb calorimetry are described in most Physics and Chemistry texts and laboratory books. Reliable procedures for measuring the explosion temperature (Tex) have not as yet been developed. One difficulty lies in the fact that most temperature recording devices do not follow the temperature with sufficient rapidity, nor have optical methods given satisfactory results. However, the explosion temperature can be calculated with considerable accuracy; the following relationship can be used for this calculation: RT = f ex If the gas constant R is known and the powder force f measured, the temperature Tex may be determined by: T = f/R (1) ex The gas constant R may be determined from Eq. (4) of Section 5.1: pv/T = R The specific gas volume (vo) expressed in units of m /kg is equal to the volume of a unit mass. It may now be introduced to give:

III-33 To For standard conditions: T = 273~K = 0~C po = 760 mm Hg = 1 atm Thus, if vO is known,then R is known. However, vo must be calculated from the reaction equation,which requires a knowledge of the equilibrium conditions for high temperatures, and these values are usuallyknown only approximately. As an alternative method, the value of vO may be measuredbut this procedure includes the same uncertainties. TABLE 5.3 Values of Q, Texi and vo for Three Different Powders Heat of Explosion Specific Gas Powder Combustion, Temperature, Volume, vo Q kcal/kg Tex OK m3/kg Single-Base (Nitrocellulose) 800 — 900 2500 -- 3000 0.900 - 0.970 Double-Base (Nitroglycerine) 1100 — 1200 3000 — 3800 0.800 - 0.860 Diglycol 720 -- 800 2500 -- 2700 1.00 Table 5.3 lists general values for typical powders. However, since the heat of combustion and the explosion temperature vary with the formula of the powder, it is possible for values to lie outside the limits indicated if special powder formulae are used.

III-34 Example 3. Consider a double-base (nitroglycerine) powder for which f = 120,000 kpm/kg, and the specific volume (vo) = 0.850 m3/kg, and calculate: a) the gas constant, R; b) the explosion temperature, T ex 2 3 R = pov/T = 10,332 kp/m (0.850 m /kg) 322 00 273.16 OK kgOK _f 120,000 kpm/kg 3727 oK ex - R 32.2 kpm/kgOK 5.3 THE BURNING PROCESS 5.31 The Linear Burning Velocity The linear burning velocity (ul) is another characteristic constant of a particular powder;this constant is more complex;thus, the process of combustion must now be considered in some detail. If a long cylindrical grain of smokeless powder in the shape of a pencil such as a piece of cordite is ignited at one end in free air, it will burn with a constant velocity. In a certain time period t, a thickness e of the cylindrical body is burnt: e U = - This quantity e/t is the linear burning velocity. If the burning velocity is not constant, then e/t is replaced by the differential rate de/dt: de u rdt

111-35 Propellant powder differs from most other combustible materials because it can burn in an enclosed space without requiring the oxygen present in the air. It is able to do this because its chemical composition contains the oxygen necessary for its complete combustion. If a powder grain is ignited in an enclosed space, it will start burning only at the point where it has been ignited. However, hot combustion gases develop which soon cover the entire surface of the powder grain with flame, and, at this point, all of the powder grain burns in successive layers perpendicular to its surface. The "combustion velocity" concept of linear burning perpendicular to the surface and in the direction normal to the surface is still applicable. However, the differential linear burning velocity de/dt is not a constant. As burning progresses, the powder gases in the enclosed volume increase, causing the pressure to increase. This increase in pressure in turn causes an increase in the rate of reaction and, concomitantly, a further increase in pressure. The over-all process is increased somewhat like an avalanche with an accelerated chain-reaction type of combustion having a velocity that accelerates,and it eventually terminates in what is commonly called an "explosion."' Thus, an "explosion" is a combustion process in which the reaction velocity continuously increases, in contrast to the burning of propellants in a weapon where the reaction velocity increases to a maximum and then decreases. The reaction velocity and its influence on the system of the gun and bullet can only be determined by experimentally burning the powder in the gun. In order to represent the combustion process of smokeless powder by an equation, the French ballistician, Vieille, (who produced the first smokeless powder truly successful as a propellant), introduced the following relationship for the linear burning velocity: u = de/dt = A p (1) where A and v depend on the chemical formulation of the powder, and both are constant for a given powder. For French smokeless powder, Vieille correlated his results using v = 2/3. However, detailed examinations in 1913 by

III-36 Schmitz, a German ballistician at the Krupp laboratories, showed that the value v = 1.0 may be used with many powders. For v = 1.0 Eq. (1) reduces to: u = de/dt = Ap (2) 4 n Eq. (2) A is the combustion velocity at p 1 atm (10 kg/m ). The equation for the linear burning velocity (ul) can be written so that the combustion law now assumes the form: u = U1 p (3) Here, ul is an additional constant which is a characteristic of the powder. The following values have been established for the two most common types of propellant: a. Single-Base (nitrocellulose) -4 m/s powder u1 0.6 — 0.9x 10 b. Double-Base (nitrocellulose plus nitroglycerine) -4 m/s powder u = 0.7 — 1.5x 10 These may be used as guide values for initial calculations. As mentioned previously, the powder grains burn over their entire surface after sufficient pressure and temperature have been reached in the combustion spaceand this burning occurs in layers. The combustion velocity (u) is equal to the thickness burned per unit time, and must be defined at some value of the powder-layer thickness. It is usually taken at u, which is the initial rate and which corresponds to el, the initial thickness. If the combustion velocity were constant, a layer "a" would burn in one unit time period, etc., as indicated in Fig. 7. But, ordinarily, the linear burning velocity is not constant, and the layer thicknesses indicated in Fig. 7 are different. The values of u1 (mentioned above) are theoretical ones since they

III-37 have been calculated for a combustion process at high temperatures using Eq. (3). Actually ul depends on several factors such as the chemical composition of the powder, its moisture content, and its temperature. The value of u1 is generally larger for double-base (nitroglycerine) than for single-base (nitrocellulose) powders. The percentages of solid nitrocellulose and of liquid nitroglycerine both influence the linear burning velocity: the larger the percentage of these two explosives, the larger the value of ul. On the other hand, the addition of camphor or vaseline as plasticizers which are, in addition, "desensitizers," will result in the reduction of ul. Thus, one method of regulating the combustion velocity involves controlling the formula of the powder. Moisture content has a marked influence on ul because water also acts as a desensitizer. Powder with a high moisture content burns more slowly than powder with a normal moisture content. This desensitization can be quite serious. For example if the powder is too damp the burning may be incomplete when the projectile leaves the barrel, causing a sharp reduction in the muzzle velocity of the projectile and erratic grouping at the target. The single-base (nitrocellulose) powders are especially sensitive to moisture because they are quite porous. The average moisture content in powder is about 1 per cent, but can increase to 2 per cent. The temperature of the powder load before firing also plays an important role. Firing tables for artillery weapons are calculated for a selected powder temperature, usually an ambient temperature of 15~C. At higher temperatures, however, the entire combustion process occurs more quickly, the maximum pressure is larger, and, ordinarily, the muzzle velocity of the projectile is slightly increased. Because of these changes, the influence of the temperature must be determined before a set of firing tables can be prepared. For this purpose a series of "cool," "normal" and''warm" loads are fired, and the muzzle velocity and maximum gas pressure are measured.

III-38 Figure 8 shows typical results of such measurements for a range of temperatures. It indicates that the muzzle velocity (vo) varies with the temperature of the load in a nearly linear fashion for the temperature range measured. On the basis of similar measurements for other powders, the influence of temperature and a correction table for its influence may be established for use with artillery tables. Although the peak of the flame temperature during combustion is limited to a very short time interval, its influence on ul should not be overlooked. And,as briefly discussed previously, the thickness of the powder grains has a marked influence on the value of ul. Experimental measurements of the influence of temperature on u! can also be used to determine whether the powder burns uniformly at both low and high temperatures and if gas pressure excursions might occur at high temperatures which would be unsafe. An important phenomenon of internal ballistics, sometimes referred to as the Krupp-Schmitz law, is related to the behavior of gas pressure as a function of time. Schmitz studied this behavior using a bomb with a relatively large volume of 3.35 liters. By this technique he was able to contain a volume of powder gas comparable to the chamber volume used in the smaller artillery weapons. A diagram of a cross section of the Schmitz bomb is shown in Fig. 9. The Schmitz bomb is a high-pressure vessel consisting of a thick-walled, closed steel cylinder. A movable piston (1) is introduced through the front surface and makes contact with a strong spring (2) attached to the outside. This spring carries a mirror (3) which reflects a beam of light onto a rotating drum (4). The drum surface contains a light-sensitive paper. The angle of the light beam is varied with pressure. The time is determined by the angular velocity of the drum. The record of the combustion process obtained from the drum is similar to the curve shown in Fig. 10. The subscript "e" indicates the end of the combustion process; thus, Pe and te are the pressure and the time, respectively, at the end of the

III-39 combustion. The pressure impulse (Ie) is the pressure impulse of the powder gases at the end of the combustion and is given by the integral,which may be calculated from the area under the curve p(t). Thus: t e I = f pdt (4) e 0 The pressure impulse (I) for any time t indicates the time dependent area under curve p(t). It is given by: t I= 1 pdt (5) 0 Schmitz performed numerous experiments with tubular powders using loading densitiesA between 0. 12 and 0.26 kg/dm3, and discovered that for the same powder Ie is constant and independent of the loading density. The term same powder implies that the powder has not only the same chemical composition, but also the same shape, and the same dimensions. Figure 11 gives typical results of three experimental firings using the same powder, but different loading densities. The value shown for Ple is the end pressure for the lowest loading density. With larger loading densities, higher end pressures Pe2 and Pe3 are obtained,while the times,te2 and te3,required for burning become shorter. This occurs because a larger weight of powder produces a more rapid rise in the pressure and thus increases the burning velocity. For the areas under the three curves are proportional to the integrals of the relationship of p(t) and as these areas are equal: Iel Ie2 Ie3 From this Schmitz concluded the validity of the law: u = u1 p

III-40 In one of the following chapters,the reasons supporting the Schmitz law as well as some exceptions will be analyzed more closely. The Schmitz law states that the linear combustion velocity is proportional to the pressure p and to a characteristic constant of the powder ul giving the relationship: u - U1P The pressure impulse Ie of the powder gases for the same powder at the end of the combustion is constant and independent of the loading density. This sentence may be reversed and stated as follows: If for a given powder, the pressure impulse Ie of the powder gases at the end of the combustion is found to be independent of the loading density, the linear burning velocity of the powder may be expressed by the following equation: u = u1 p Another important relationship can be derived from Schmitz's law. This relationship will be illustrated with tubular powder but can be applied to many powders of different shapes. Tubular powder may be assumed to burn uniformly at the same rate from the inside and outside. Thus, the wall thickness may be considered as 2el (see Fig. 12). After a certain time period,a layer of thickness "e" is burnt on the inside and outside. The wall thickness, then, becomes 2el-2e. de dt = ul p gives: de=u p dt

III-41 and thus: te e u1 f pdt = u1 I (6) Rearranging gives: e I (7) e u or: u= e(8) e Equation (8) permits the determination of the powder constant ul because the half-wall thickness e1 is known and Ie can be measured in the bomb. For all powder shapes, the dimension 2e1 controls the burning limit, i.e., when the burnt thickness is 2el all of the powder is burned. 5.3.2 Dependence of Burning Rate on Size and Shape of the Powder Particles The powder charge consists of a large number of powder grains whose burning rate depends not only on the chemical composition of the propellant, but also on individual grain shape. By the proper selection of these two factors,the burning rate characteristic of the powder can be adjusted to give the required rate by the system of a given weapon and projectile. Figure 13 shows six different simple shapes used with powders for small arms. The powder burns perpendicularly to the exposed surface, and for long cylinder (a) it burns both in the direction of its longitudinal axis and in the direction perpendicular to the axis, i.e., radially inward. During combustion, the cylinder becomes both shorter and thinner, so that the ratio of the length to the diameter increases. For the total burning period of long cylinder

III-42 (a), the radius el (for a diameter of 2el) is the factor controlling the burning rather than the height of the cylinder. On the other hand, for the short cylinder (b), the burning period depends on the height of 2el rather than on the diameter. A similar relationship (burning controlled by 2el) is valid for the strip powder (e). Similarly, for tubular powder (d) the burning time also depends on the thickness 2e1. In German ballistic literature the quantity 2el is usually called the "wall thickness" and corresponds to what the American-British literature refers to as "web size." Tubular and leaf shapes are used in most German powders. Figure 13-f shows how the dimension 2e1 may be used in irregularly shaped powder grains. The Soviet Union and the United States use multi-channel powder in addition to simple powder shapes, but even for these more complex shapes the dimension 2e1 is useful. Figure 14 shows the section of a "seven-channel" powder; diagram (a) shows the cross section at the beginning of combustion and diagram (b) at the moment of disintegration. In Fig. 14a, (D) is the outer diameter of the powder "grain" or body, and (d) is the inner diameter of one channel before the start of combustion. The condition at the end of combustion is shown in Fig. 14b. The powder body disintegrates into 12 three-edged columns. In order to make these columns as small as possible,the powder may be given the cross-section shown in Fig. 15. Grains of this shape are called rosette-powder. The combustion space in the gun available to the propellant changes during the combustion process because the volume becomes larger with projectile travel. The question can be raised, of what is the value of regulating powder shape? In answer, note that this increase in volume does not occur uniformly, because the projectile velocity steadily increases. Thus, it is desirable —and here is the basic importance of regulating powder shape —that the powder charge burn slowly at first, with the rate increasing as the projectile travels through the barrel at an increasing rate of speed. The volume of the powder before combustion is indicated by A1. This volume becomes smaller because of combustion during time t; the volume remaining at any time is called A. Thus, the relative fraction of the charge which is burnt at a certain time t is indicated by "y," and can be expressed as follows:

III-43 A1 -A Y= A (1) Consider a spherical powder shape (Fig. 13-c) with a diameter 2el. The initial volume is: 4 3 A1 3 el A layer of thickness e is burnt in time t. The remaining radius is (el-e). Thus, the remaining volume A is: 4 3 A - 4 v(e1-e) Hence: 4/3 e13_ 4/3 (e -e)3 e3 -(e - e)3 y 3 3 4/3 v e1 el y =3 e 3 ( 2 + e 3 (2) - - - (el In Eq. (2) e/el is the relative thickness of the burnt layer. For simplicity it is given the symbol"z," or: Z _ e (3) e Thus: 2 3 y = 3z-3z2 + z (4) Equation (4) is one form of a general relationship that can be shown to hold for all powder shapes, or: y = flz + mz2 + nz (5)

III-44 Equation (5) is valid for any powder as y increases from 0 to 1.0. At the beginning of burning z = 0, and, according to Eq. (5), y = 0. At the end of the combustion z = 1 and Eq. (5) gives: 1 = L + m + n Thus, only two coefficients are independent of each other and Eq. (5) may be written as: y = I (1 +( z + iz ) (6) where "4", "A", and " Ii" are called shape factors. Using this notation for the sphere gives: 1 2 y = 3z (1- z + z ) (7) and 4 = 3, X = -1, M = Table 5.3 gives the relationship between shape and the independent values of a and d of some common powder shapes. The values a and 3 are defined as follows: Strip powder2e 2e a 2b = j 2 (see Fig. 16) 2b 2c Tubular powder2e1 = ratio of wall thickness to length Q, or e Quadratic leaf powder - f = ratio of thickness to edge length

111-45 Solid cylinder powder3 = ratio of diameter to length TABLE 5.3 Table of Powder Shape Factors Powder Shape I A Strip l+ a+ a + + a a _ + Tubular 1 + - 3 0 Quadratic leaf 1 + 2 7 -i 1+22 1+21 Solid cylinder 2 + 1 + 2 d- + 2+1 2+1 Example 4: Determine the expression for the value of y as a function of z for a quadratic leaf powder which has a thickness equal to 0.1 of the edge length. By definition, 1 = ratio of thickness to length, or 1 = 0.1, and: d =- 1+213 = 1+2(0.1) = 1.2 2 +2 _ 2(0.1) + 0.o1 - 0.75 1 +2 1 + 2(0.1) 13 _0.01 = 1+2: 1.2 = 0.0083 = +22 1.2 Hence: y = 1.2 z(1 - 0.175z + 0.008 z2) (Curve 1)

Figure 17, curve 1, shows this function. The curve 2 of the same Fig. also shows the function y for a sphere: y = 3z (1- z + z2 ) (Curve 2) Figure 17 shows that the fraction (y) of load burnt increases more uniformly for the leaf powder than for the spherical powder. Thus, to understand the burning process better, one must know the velocity at which powder is transformed into gas, i.e., the change of y per unit time or, dy/dt. The quantity dy/dt is called the spatial combustion velocity, and it is discussed in the following section. The fraction (y) of load burnt increases a differential amount dy in a differential increment, dt. In this time period, the total powder load of weight "L" has been reduced by "dL." Thus, the following relationship exists: dL = Ldy (8) If "S" is the total surface of the powder charge at time t, the following equation is valid: dL = o Sde (9) where p is the density of the load. Combining both equations yields: dy = L de (10) According to Eq. (3), Section 5.31, de can be written: de = u1 pdt Substituting this into Eq. (10) gives: =t PL Up p(11) dt = LU p

III-47 Using S1 to indicate the initial surface of the total load and expanding the right side of Eq. (11) yields: d p 1 S p(2) dt S(12) This equation gives a relationship between the linear burning velocity and the spatial combustion velocity (dy/dt). The right side can be considered a product of three factors. The first factor consists only of constants and is itself a constant indicated by A: S A = p - u1 (13) The term A gives the "brisance" or the "quickness" constant of the powder. The ratio L/p is the initial volume (A1) of the powder load. Hence: S A - Al u (14) The size of the powder grain influences its characteristics. For example,the ratio Sl/Al becomes larger as the size of individual powder grains is decreased. This is because the surface area S1 varies as the square of the size,whereas the volume A1 varies as the cube of the size. Thus the spatial combustion velocity becomes smaller as the powder burns. The ratios S/S1 and Sl/Al are the same whether they refer to a single powder grain or to all the grains in the load. In the following discussion it is often useful to consider individual powder grains. For a cube with an edge length "a," we have: S1 6a2 6 A 3 a 1 a and, for a cylinder with radius "r" and height "h" we have:

III-48 S1 2 r2 + 2 T rh 2 2 =..2 +- (16) A1 rh h Cylindrical powder is usually in the form of long strings, e.g., cordite. Therefore, since h is much larger than r, in most cases 2/h is so small with respect to 2/r that it can be neglected. For the case under consideration r = el. Hence, the expression for cordite powder may be written as: S1 2 _ _- (17) 1 e For a hollow cylinder (tubular powder) with the outside radius "R," the inner radius "r," and the height "h," the expression becomes: S1 2 fT (R - r )+ 2 fT h (R + r) 1 rT h (R -r) S1 2 2 (R+r) Ai - h 2(18) Ah 2 2 1 R -r S1 2 2 A h R-r 1 where (R-r) = 2el. Ordinarily h is significantly larger than R-r;so, once again, 2/h may be neglected as compared to 2/(R-r). The expression then reduces to: S1 1 ~ 0 ~ (19) A e These considerations apply equally well to other powder shapes,further demonstrating that e1 is a constant of basic importance to the spatial combustion velocity (dy/dt). The larger el, the smaller the spatial combustion velocity.

III-49 A comparison of Eqs. (17) and (19) also shows that the shape,too,plays an important role. In Eq. (17) a 2 appears in the numerator and in Eq. (19) a i is present. However, this consideration is valid only for the beginning of combustion. Besides A, other factors such as S/S1 and p must be considered. For the case of an exponent of unity on the pressure term, the spatial combustion velocity is directly proportional to the pressure. The factor S/S1 is the ratio of the instantaneous (burning) surface to the initial surface. For tubular powder this ratio is equal to 1 if end effects are neglected, i.e., if the length of the grain as compared to its diameter is so large that the burnt-off sections at both ends are not significant. For all other powder forms, the ratio S/S1 is variable. The surface changes of other powder shapes will now be examined more closely, i.e., the behavior of: = (20) 1 where "a" is the relative surface of the powder body. As a simple example we may consider spherically-shaped powder. According to Fig. 13-c we may use el as the radius. After a certain time a layer of thickness e is burnt. Hence: =S =4(e- L4 e e)2 (21) 1 1 where e/e1 is the relative thickness of the burnt layer. We indicate this layer by z whose value changes from 0 to 1. Equation (21) may now be written a = 1 - 2z + z Figure 18 shows a as a function of z for spherically-shaped powder (curve 1). Notice that during the combustion process a decreases rapidly with increasing z. This means that we are dealing with a "degressive" powder shape.

III-50 As a second case, consider a very long solid cylinder such as a "grain" of cordite. In order to simplify the calculation, the burnt-off sections at both ends will be neglected,as will the end surfaces (because of their smallness). Again, the radius is equal to el. Indicating the height of the cylinder by h, we have: S 2 (e -e)r h e1 e a =o... = - = 1 - z (22) so 2 ~r el h e which gives a descending straight line (see Fig. 19, curve 1). An exact calculation, i.e., including the surface reduction and end surface which were neglected above,for a solid cylinder whose length is 20 times its diameter gives (Fig. 19, curve 2) a 1 - 1.073 z + 0.073 z2 (curve 2) (22a) and, for a solid cylinder whose length is 100 times its diameter: a = 1- 1.015z + 0.015 z2 (22b) Curve 1, Fig. 19 corresponds nearly to the straight line, a = 1 - z; curve 2 is also nearly a straight line. This shows that the approximations described previously are valid. This approximation, however, is only valid if the length of the grain is essentially larger than the diameter. A solid cylinder whose length is equal to its diameter gives the following expression: a = 1 - 2z + z2 (curve 3) (22c) The corresponding curve 3 in Fig. 19 has a definite deviation from a straight line. In addition to the curves for spherically shaped powder (curve 1) and for solid cylinder powder where its length is 20 times its diameter (curve 2), Fig. 18 presents curves for quadratic leaf powder with a thickness equal to 1/10 of the edge length (curve 3):

III-51 a = 1 - 0.350 z + 0.025 z2 (23) and for strip powder with the ratio of the edge lengths 1:10:100 (curve 4): a = 1- 0.2 z + 0.003 z2 (24) During the burning of tubular powder,the surface remains practically constant. For long tubular powder where the burnt-off sections at the ends and the end surfaces are negligible, the following expression applies: S _ 2 (r + e)dr h+ 2 (R - e)_7 h a =S1 2 r h + 2R h(25) where "R" is the outer radius, "r" the inner radius, and "h" the height. An exact calculation for tubular powder where the length is 50 times the wall thickness (including the end surfaces and burnt off sections) gives: a = I - 0.039 z (curve 5) (25a) Thus Eq. (25a) demonstrates that during the burning of tubular powder, the surface decreases very slowly and is only 4 per cent smaller at the end of the combustion. The equation for the spatial combustion velocity, Eq. (12), can also be written without applying the previous relationships; thus: dy = Aup (26) dt Aap As pointed out earlier, a continuous increase in the combustion velocity is desirable. At the beginning of combustion in the gun,this is indeed the case,and p increases rapidly. However, the maximum chamber pressure is soon reached and p decreases thereafter. One means of increasing the combustion velocity is

III-52 by increasing the relative surface of the powder body. However, Fig. 18 shows that with nearly all simple powder shapes,the relative surface of the powder body decreases,and that most shapes are, therefore, "degressive." The only exception to this is tubular powder where a = 1 [see Eq. 25 ]. Powder shapes in which a does increase are called "progressive," and these have been the subject of extensive research. One progressivel shape is the seven-channel powder (see Figs. 14 and 15). Here, the central channel serves to compensate for the decrease of the outer surface just as in the case of simple tubular powder. The remaining channels contribute to the enlargement of the burning surface. Figure 20 shows the behavior of a as a function of z for a sevenchannel powder with channel diameters d = el, making a total diameter of lid and a height of 25d. Note that a increases to 1.37 at z = 1. The powder body then disintegrates into columnar slivers; z increases further but a decreases quickly, making the powder degressive after the disintegration of the seven channels. Thus, for seven-channel powder, 85 per cent of the powder body burns progressively, and 15 per cent burns degressively. Test firings, however, show that multichannel powder strongly deviates from theoretical behavior; this has been discussed in Section 5.35. Another type of definitely progressive powder is "coated powder." Such powders are produced by coating the outer surface of tubular powder with a substance called a deterrent. This coating causes the powder to burn more slowly at the outer surface and, as a result, to burn primarily at the inner surface. Deterrents can, however, create new problems. Some perform well except that the coated powder may then develop a large amount of smoke. Dinitrotoluene is used successfully as a deterrent coating in American rifle powders. We have seen that a = S/S1 can be represented as a function of: e

III-53 The same is true for the fraction of the powder burnt (y). Thus. an inter-relationship exists between both equations which permits y to be expressed in terms of z and the powder shape constants. dz = (1 +z X z+ 3z2 (27) By definition we can write: = dy. dt de dz dt de dz By substitution we have: =t SI S dt A S u p de dt- U p de dz = 1 By combination of terms: S 1 S dz A S 1(28) Considering Eq. (27) and substitution of terms yields: -e = l(1 + 2tz + 3 P z) (29) A Sel 1 1 For z = O, S = S1. It follows from Eq. (29): Se SAe (30) A1.=

III-54 Dividing Eq. (29) by Eq. (30) gives: a = 1+2 z + 3 z (31) S1 The function a thus depends on the coefficients X and j and the variable z whereas I is defined by Eq. (30). So far we have considered y and a as functions of the relative burning thickness, z. In some cases, however, it is more convenient to give a as a function of y, or a(y). This approach is found in both the French and German internal ballistic literature. Its origin is the pioneer work of the French ballistic expert Charbonnier, who considered this relationship as a shape function. His method has been used extensively and permits the expression of all calculations in terms of the independent variable y. The curves a(y) can be obtained by calculating the values of y and a for a series of values of z using Eqs. (6) and (31) and cross-plotting with respect to y. Figure 21 shows the curves a(y) for the powder shapes used in Fig. 18. In Fig. 21,curve 1 corresponds to spherical powder, curve 2 to solid cylinder powder, curve 3 to quadratic leaf powder, curve 4 to strip powder, and curve 5 to tubular powder with the same dimension ratios as given for Fig. 18. 5.3.3 The Basic Equation of Pyrostatics Data obtained from test measurements in the manometric bomb can be combined under the heading "Pyrostatics," so named because combustion is studied under static conditions rather than being associated with expanding gases. The relationship between pressure and volume is given by Abel's equation which was derived in Section 5.1. This equation applies to the end conditions when the load is completely burnt.

III-55 In determining the pressure behavior during the combustion process, it should be recognized that the total combustion space of the bomb is not available, since part of this space is still occupied by the volume of unburnt powder. In additionthe covolume of the powder gases already formed must be considered. Figure 22 shows three different stages in the burning process. Before the start of combustion, y = 0. The free space is shown in Fig. 22 to be equal to the combustion space (V.) of the bomb minus the space occupied by the load (L), i.e., L converted to volume as L/6 where "6" is the density of the powder. In the second stage, a fraction y of the load is burnt. The free space (Vy) is reduced by the powder gases, or: V = V --- (-y) - aLy where L(1-y) is the unburnt load and aLy is the covolume of the powder gases already created. The third stage begins at y = 1, the point at which the total load is burnt. At this point: Vy = Vo - aL To use the equation of state Eq. (1) Section 5. 1, the pressure and volume of the gases must be related at the moment when the fraction y of the load is burnt. Thus: p V = RTLy = fly (1) As derived above, Eq. (2) may be written to solve for V y V = V — (1-y)- aLy (2) By substituting for Vy Eq. (3) is obtained: By substituting for VEq. (l-) is obtained: f Y

II1 56 or: LY fLy 1 (3a) Y V L -L(cu )y This gives the dependence of the pressure on y. Introducing the loading density a = L/Vo, Eq. (3a) may be rewritten as: = fay y 1A~ - A(a - 1 )y or Py 1 1 (4a) Equation (4) becomes Abel's equation ) y Equation (4) becomes Abel's equation when y - 1. Figure 23 shows the pressure behavior (p ) for nitrocellulose powder where f = 95,000 kpm/kg, a = 1.Y (10)-3 m3/kg, and, 6 = 1650 kg/m3 at a loading density of 180 kg/m3.

III-57 This figure shows that the pressure as a function of y deviates only slightly from a straight line which goes through the origin (0,0). This is more precisely so if a m 1/6, as is the case for pure nitroglycerine. Values of y as a function of p can be found from Eq. (4) using Eq. (4a). Thus: 1 1 ---- - - P 6 pY Equation (5) permits the calculation of the fraction py of the load burnt y that corresponds to a particular pressure. Since the quantity a - (1/6) can usually be neglected as compared to f, the following simplified equation results: PY 1 (6) f A f) (6) This is sufficiently accurate for most cases. Equation (6) is, then, a simplified version of Abel's equation. Examples: A load of 30 gm of single-base nitrocellulose powder is burnt in a manometric bomb which has a combustion space of Vo = 200 cm3. The powder has the constants f = 101,000 kpm/kg, a = 0.984 (10)-3 m3/kg, and, a - 1620 kg/m3. 1. What pressure exists at the end of the combustion? 2. What fraction of the total charge is burnt when the pressure was equal to 1/2 of the final pressure? With regard to (1) the loading density is: _ L 0.030kd = 130kg/i3 V_ 0.2 dmdj

III-58 Using the simplified version of Abel's equation, the end pressure (Pe) may be calculated from Eq. (6) rewritten as: e f e i - aA = 101,000 kpm/kg (150 kg/m ) e 1 - 0.984 (10)-3 m3/kg (150 kg/m3) 1.515 (10) kp/m Pe 1 - 0.1476 = 1.777 (10)7 kp/m2 = 1.777 (10)3 kp/cm2 With regard to (2), Eq. (5) is used to calculate the fraction y of the total load which is burnt to reach one half of the end pressxu3e is: 1/2 p = 0.8885 (10)7 kp/m - 6.050 (10)-3 m/kg A 5 f =.01 (10)' kpm/kg 1.1368 (10) -2 m3/kg Py 0.8885 (10)7 kp/m2 1 + - 11.735 (10)-3 m 3/kg y 6.050 (10) m /kg = 0.516 110735 (10) m /kg Thus, for the case of a pressure equal to 1/2 final pressure, it turns out that only slightly more than one-half of the charge was burnt.

III-59 5.3.4 The Linear Burning Law of Muraour and Aunis* The representations given so far have been mostly theoretical. In deriving the spatial combustion velocity, geometrical considerations were used as the basis. Also, it was assumed that at any given pressure the powder grains burn uniformly over their total surface with the same linear velocity, and that the powder grains do not influence each other. The linear burning law, u = ulp, was verified experimentally by Schmitz for loading densities of 0.12...0.26 kg/dm3. This showed that the pressure impulse for the same powder has the constant value: t e e f pdt = I -- (1) e u1 Muraour and Aunis have established a different equation for the linear burning velocity based on theoretical considerations. This law is given by: de d- = a + bp (2) where "a" and "b" are constants. In theory, the quantity "a" involves loss in pressure because of the transfer of heat by conduction,and "b" involves the increase in pressure because of the transfer of energy by molecular motion. Calculating the impulse integral for this new law gives: te e= ate +b f pdt (3) e 0 or *Muraour-Aunis, "Verbrennungsgesetz Rolloidaler Pulver," Explosivstoffe 2, 1954, S. 154 ff.

III-60 t e e a I pdt = at (4) This equation does not consider the impulse integral (Ie) a constant, but instead it is considered to be a function of the burning period t as well as of the loading density (A). The result of the subtraction is that Ie becomes smaller; and u1 in Eq. (1) is no longer a constant for the entire burning. Now u1 depends on the length of the burning period, becoming larger as the burning period becomes longer. In experiments conducted in Russia by M. E. Serebrjakow and A. I. Kochanow, changes in Ie were observed only in rather thick types of powder having a considerable burning thickness (el) which had longer burning periods. On the other hand, fast burning powders did not show any influence of Ie on the loading density. This implies that the length of the combustion process plays an important role in slow-burning powders. This might be expected, because in slower combustion processes the heat of the hot combustion gases has more time for transfer and to influence the temperature of the total powder mass. Consequently, the temperature of the interior of the powder grains is increased, which in turn increases ul. The net result is to reduce the dependence of Ie on time. Experiments with loading densities significantly smaller than those used by Schmitz (A < 0. 10 kg/dm3) have verified the statements above regarding the increase of u1. In summary, for rifles with full loads of fast or medium burning powders the following law is valid: u = ulp For longer burning periods such as for cannons and slow burning powder, however, ul is not constant. It becomes larger because of heating during the combustion process, and this increase in turn causes a reduction in the total impulse (Ie). This can be expressed by the following burning relationship: de u = dt = a+bp or by an equation of the form:

III-61 u de = ApV (5) dt with v < 1. The relationship expressed by Eq. (5) may be written: de = Ap dt = Ap p dt = A pdt -V ~ 1-v P P A constant average value, (pl-v)m, obtained through integration may be substituted for pi-". Thus: te el f pdt (p - )m 0 or te e te e1 1-v I = pdt =- (p )m (6) e 0 Increasing the loading density increases (p )m,since increasing the largest value also increases the average value, where u < 1, and Ie is no longer constant. The total impulse becomes larger with increasing loading density and vice versa. 5.3.5 Experimental Examination of Progressivity Degressive and progressive powder shapes were defined in Section 5.3.2. For degressive forms: S a = S 1 The value of a decreases forodegressive forms and increases for progressive forms. At constant pressure p:

III-62 S S dy =1 S dt A U1 S1 The value of dy/dt decreases or increases with degressive or progressive powders, respectively. In order to obtain a measure of progressivity, the relationship dy/dt must be obtained at constant pressure. Of course, dy/dt is the rate of the conversion of powder grains to powder gas and, as indicated previously, this rate is greatly influenced by pressure. If u1 is directly proportional to pressure, however, we can "normalize dy/dt by dividing by p to give: 1 dy p dt For convenience,this relation will be given the symbol Fand called the intensity of the gas formation or: r =! (dy) (1) p dt or: 1 S S 1 1 =p dzt-)=1 u -- = A- u a (2) p dt A i S A 1 The value of I can be determined experimentally on the basis of pressure measurements in the manometric bomb. Figure 24 shows the results of test firings of tubular powder and gives F as a function of y. An experiment with strip powder gave nearly the same results. In Fig. 24 FT is the theoretical curve based upon the following equation: S rT A 1 a The experimental curve of r sometimes deviates considerably from the theoretical curve. As shown in Fig. 24, the curve can be divided into four sections for the purpose of analysis. These sections will now be considered in more detail.

III-63 Section I. The curve does not start with the theoretical value. The reason for this is that the total powder load does not ignite instantaneously as assumed for the theoretical condition. The value of r reaches a maximum at y = 0.05...0.08. (The little excursion above the theoretical value may be related to deviation from the linear relation between pressure and burning rate.) Section II. The value of Fdecreases and slowly approaches the theoretical curve. This section includes the values of y = 0.05...0.08 to y = 0.30, and covers an accelerated combustion process. Section III. This section indicates normal combustion which corresponds to the geometrical law and occurs in the region y = 0.3...0.85 to 0.90. Section IV. In this last sectionthe experimental curve deviates strongly from the theoretical curve. The experimental curve decreases quickly and approaches a value of zero. Figure 25 shows F (y) for a seven-channel powder. Whereas for tubular and strip powders r still lies in the approximate vicinity of the theoretical curve over a comparatively large section, a completely different behavior is now experienced. In the case shown, the predicted progressivity does not occur. The reason for this is that some stoppage occurs in the narrow channels, preventing the powder gases from escaping uniformly. Due to the pressure increase resulting from this confinement, the powder grains are prematurely disintegrated. The location of the peak in the curve of F for tubular and strip powders with respect to the magnitude of r depends largely on the composition of the powder. Powder with solid solvent (Trotyl and Pyroxillin) has a significantly lower peak than powder with a liquid solvent. It is believed that the number and the size of the pores have a great influence on the height of the peak. The behavior of r can be modified significantly if the outer layers of the powder are desensitized. This can be accomplished by coating the powder with a deterrent (see Section 5.3.2). Following this, the more favorable behavior of curve 2 instead of cursve 1 of Fig. 20 occurs.

H H H Fig. 1. Sectional diagram of a pressure bomb for measuring properties of propellants.

III-65 20,000 - Nitroglycerine (double-base) powders f=120,0000m k =0.85(10 kg| / 15,000 ~~~|- -Nitrocellulose (single-base) powders / 15f=95,000 =. kg kg / 10,000 / 7,500 // // 4,000/. 3,000. _, 2,000 1,000 0.1 0.2 0.3 0.4 0.5 0.6 Loading Density, A Fig. 2. Combustion bomb pressures plotted as a function of loading density for single-base nitrocellulose and double-base nitroglycerine powders.

III-66.4 U) P-4 f 0 0 Pressure, p Fig. 3. Pressure as a function of (f+p/L) from pressure bomb firings of one powder using different loading densities.

III-67 150,000 1 - 100,000 3 0 Pr 50,000 30,000 10,000 0 _ 0 Pressure, p Fig. 4. Comparison of data from pressure bomb firings of three types of powder.

PressureLoading Density oqq 0 Id Id4'dl H 00 "tb H0 (D1 n'oq Q H) o oD (D D H U) H O (D (D D (D (d H)~ ~~~~~~~~~~~~~~~~~~ ro~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r H)H 0 C+ — 0

III-69 a /Y go bO'H A rd 0 Pressure, p Fig. 6. Serebrjakow's data for pressure bomb firings with low loading densities. // -,, Fig. 7.-Diagram of power layers astheburnsuccessiveIy Fig. 7. Diagram of powder layers as they burn successively from outside to inside.

III-70 Temperature ~C Fig. 8. The typical influence of powder temperature on muzzle velocity. 1 / 3 4 Fig. 9 T Diagram showing construction of the Schmitz Bomb.

111-71 Pe U Ie 0 0 Time, t te Fig. 10. Curve of pressure versus time as obtained from the Scbmitz Bomb. e 212 o Time, t te te t impulse, I e!

III-72 I l.. _ _ _ " _ A_ I t e % 2e b 1 I Fig. 13. Six different shapes commonly used for rifle powders. 2e 2

III-73 d+2e, I I |-.... —-- D.| | —. D-2el - a) b) Fig. 14. Seven-channel artillery powder: (a) before burning, and (b) at the instant of splintering. 00 b) a) Fig. 15. Rosette artillery powder: (a) before burning, and (b) at the instant of splintering.

III-74 2e1 X 2e/ e'-..,,- - 2b Fig. 16. Significant dimensions for burning of strip powder. 1.0 2 0 0.1 0.3 0.5 1.0 Fraction of thickness burnt, z Fig. 17. Fraction of thickness burnt (z) versus fraction burnt (y) for quadratic leaf powder (Curve 1) ind spherical powder (Curve 2).

III-75 1.0 0.9. 0.8 0 4, 4 0.7 7 0.6 0.5 Frction of thickness burnt, z 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fraction of thickness burnt, z Fig. 18. Influence of the fractional surface factor (a) of the powder grain on the fraction of thickness burnt (z) for different powder shapes.

III-76 1.0 0. 0.8 0.7 0o.6 0 2 ~14 3 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fraction of thickness burnt, z Fig. 19. Powder fractional surface (a) for three cylindrical powder lengths: long (Curve i), intermediate (Curve 2), and short (Curve 3).

III-77 1.37 1.0 I \ powder (a = 1.37 max.). powder (a = 1.37 max.).

III-78 1.0 0.9 0o.8 4 0.7 o 0.6 4 0)5 0.3 0.2 0.1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fraction burnt, y Fig. 21. Powder fractional surface (a) as a function of fraction burnt (y). V =V -(1-Y)-LY 0< O <y<1 | a L V =V - L| y=l y 0o Fig. 22. Three states in the burning of powder at constant volume.

III-79 1,500' 1,000 500 0 0.5 1.0 Fraction burnt, y Fig. 23. Pressure (p) versus fraction burnt (y) for the combustion of nitrocellulose-based powders burnt at constant volume. o.,, o* - I rt 4) 4) Fig. 24. Intensity of gas formation (r) versus fraction burnt (y) for tubular and strip powders.

UNIVERSITY OF MICHIGAN IIIl II111111111111111 1111 11111 111111111l1111111111111 3 9015 03023 2139 o3 3 o 4-4 1-4 0 0 0o.5 1.o Fraction burnt, y Fig. 25. Intensity of gas formation (r) versus fraction burnt (y) for seven-channel powder. 0 o o - 0 0 0.5 1.0 Fraction burnt, y Fig. 26. Intensity of gas formation (r) versus fraction burnt (y) for tubular powders having untreated (Curve l) and deterrent-treated (Curve 2) surfaces.